{"id":114,"date":"2017-02-03T11:58:05","date_gmt":"2017-02-03T11:58:05","guid":{"rendered":"http:\/\/conferences.leeds.ac.uk\/lc2016\/?page_id=114"},"modified":"2017-02-03T14:38:28","modified_gmt":"2017-02-03T14:38:28","slug":"schedule","status":"publish","type":"page","link":"https:\/\/conferences.leeds.ac.uk\/lc2016\/schedule\/","title":{"rendered":"Schedule"},"content":{"rendered":"<style>\r\n.schedule table td, .schedule table th {padding: 0 0.1em 0.1em 0.1em; border-right-width: 0 !important; border-left-width: 0 !important; border-bottom: 1px solid #666 !important}\r\n.schedule table a {border-bottom: 0 !important}\r\n<\/style>\r\n<script type=\"text\/x-mathjax-config\">\r\nMathJax.Hub.Config({\r\n  tex2jax: {inlineMath: [['$','$'], ['\\\\(','\\\\)']]}\r\n});\r\n<\/script>\r\n<script type=\"text\/javascript\" async\r\n  src=\"https:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_CHTML\">\r\n<\/script>\r\n\r\n<script type=\"text\/javascript\" >\r\nvar currently_open_abstract = 0;\r\nfunction openAbstract(id) {\r\n  if (currently_open_abstract)\r\n    currently_open_abstract.style.display='none';\r\n  var this_abs = document.getElementById(id);\r\n  if (this_abs == currently_open_abstract) {\r\n    currently_open_abstract = 0;\r\n  } else {\r\n    this_abs.style.display='block';\r\n    currently_open_abstract = this_abs;\r\n  }\r\n}\r\nfunction closeAbstract(id) {\r\n  document.getElementById(id).style.display='none';\r\n  currently_open_abstract = 0;\r\n}\r\n<\/script>\r\n<style>\r\n    .abstract_box {\r\n        display: none;\r\n        position: absolute;\r\n        left: 30%;\r\n        width: 40%;\r\n        padding: 8px;\r\n        border: 1px solid black;\r\n        background-color: white;\r\n        z-index:1002;\r\n        overflow: auto;\r\n        box-shadow: 1px 2px 4px #000000;\r\n    }\r\n    .abstract_box p {\r\n    line-height: 130%;\r\n    font-size: .9em;\r\n    }\r\n<\/style>\r\n\r\n<div  class=schedule>\r\n<p><table width=\"100%\" style=\"color:black; border: 2px solid black; border-collapse: collapse;\">\r\n<tr style=\"border-top:1px solid black;\">\r\n<td width=\"1px\" bgcolor=\"#AAA\" colspan=\"6\" style=\"text-align:center; padding-left:1ex; padding-right:1ex;\">\r\n<b>Monday 1st August<\/b>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:1ex\">9.00<\/td>\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">9.15<\/td>\r\n<td align=\"left\" bgcolor=\"#FC9\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nOpening\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">9.15<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.15<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary (BLC Lecture): Laurent Bienvenu, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m1')\"><i>Randomized algorithms in computability theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-Bienvenu.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m1\" class=\"abstract_box\">\r\nLaurent Bienvenu, <i>Randomized algorithms in computability theory<\/i>\r\n<br><br>\r\nAre randomized algorithms more powerful than deterministic ones? This is perhaps one of the most important general questions in computational complexity, the problem P ?= BPP perhaps being the best known instance of it. In computability theory, this question is typically less considered because of a theorem of De Leeuw et al., which states that if a given sequence\/language can be probabilistically computed, it can in fact be deterministically computed. However the problem remains interesting if one consider classes of objects: there are some classes $\\mathcal{C}$ containing no computable element but for which there is a probabilistic algorithm which produces an element of $\\mathcal{C}$ with positive probability. We will discuss some some positive and negative examples and will explain how to get a quantitative analysis of such classes using Kolmogorov complexity, with numerous applications to algorithmic randomness. Finally, we will see how the theorem of De Leeuw et al. can be turned around to get the existence of computable objects from probabilistic algorithms.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nKelty Allen, Laurent Bienvenu and Theodore Slaman,\r\nOn zeros of Martin-L\u00f6f random Brownian motion,\r\nJournal of Logic and Analysis, vol.&nbsp;6 (2014).\r\n<br>\r\n[2]\r\nLaurent Bienvenu and Ludovic Patey,\r\nDiagonally Non-Computable Functions and Fireworks,\r\nAvailable at <tt>http:\/\/arxiv.org\/abs\/1411.6846<\/tt>.\r\n<br>\r\n[3]\r\nLaurent Bienvenu and Christopher Porter,\r\nDeep $\\Pi^0_1$ Classes,\r\nBulletin of Symbolic Logic, to appear. Available at <tt>http:\/\/arxiv.org\/abs\/1403.0450<\/tt>.\r\n<br>\r\n[4]\r\nSamuel Epstein and Leonid A. Levin,\r\nSets have simple members,\r\nAvailable at <tt>http:\/\/arxiv.org\/abs\/1107.1458<\/tt>.\r\n<br>\r\n[5]\r\nSteven M. Kautz,\r\nDegrees of Random Sets,\r\nPh.D. dissertation, Cornell University, 1991.\r\n<br>\r\n[6]\r\nAndrei Rumyantsev and Alexander Shen,\r\nProbabilistic Constructions of Computable Objects and a Computable Version of Lov\u00e1sz Local Lemma,\r\nFundamenta Informaticae, vol.&nbsp;132 (2014), no.&nbsp;1, pp.&nbsp;1\u201314.\r\nAvailable at <tt>http:\/\/arxiv.org\/abs\/1305.1535<\/tt>.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m1');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">10.15<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.45<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">10.45<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">11.45<\/td>\r\n<td align=\"left\" bgcolor=\"#C9DDFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nTutorial A: Thierry Coquand, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m3')\"><i>Univalent Type Theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-coquand-1.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m3\" class=\"abstract_box\">\r\nThierry Coquand, <i>Univalent Type Theory<\/i>\r\n<br><br>\r\nThis tutorial will be an introduction to dependent type theory,\r\nand to the univalence axiom (V. Voevodsky, 2010).\r\nSimple type theory, as formulated by A. Church (1940), constitutes\r\nan elegant alternative of set theory for representing formally\r\nmathematics. The stratification of mathematical objects in a type of propositions,\r\na type of individuals and a type of functions between two types is indeed\r\nquite natural. The axiom of extensionality (the first axiom of set theory)\r\ncomes in two forms: the fact that two equivalent propositions are\r\nequal, and the fact that two pointwise equal functions are equal.\r\nSimple type theory as a formal system has however some unnatural limitations,\r\nin that we cannot express the notion of an arbitrary structure, for instance\r\nthe type of an arbitrary group. Dependent type theory solves this issue\r\nby introducing in type theory the notion of universe. What was missing\r\nuntil the work of V. Voevodsky was a formulation of the extensionality\r\naxiom for universe. This is the univalence axiom, which generalizes\r\npropositional extensionality. When introducing universes, we also can use the\r\nprinciple of propositions-as-types and proofs-as-programs, so that the logical\r\noperations themselves, and their proofs, can be represented as type theoretic\r\noperations.\r\nThe lectures will roughly proceed as follows. The first lecture will be an\r\nintroduction to simple type theory and dependent type theory, where we shall\r\ntry to point out the connections and differences with set theory and end\r\nwith a formulation of the univalence axiom. The second lecture will explore\r\nsome consequences of this axiom, such as a proof of a strong form of\r\nthe axiom of \"unique choice\", from which we can derive results that would\r\nrequire the full axiom of choice in set theory. We will end\r\nwith a presentation of a model of the univalence axiom.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m3');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">11.45<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">12.45<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Itay Kaplan, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m5')\"><i>Developments in unstable theories focusing on NIP and NTP$_2$<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-Kaplan.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m5\" class=\"abstract_box\">\r\nItay Kaplan, <i>Developments in unstable theories focusing on NIP and NTP$_2$<\/i>\r\n<br><br>\r\n\r\nFor many years after Morley's celebrated categoricity theorem [2],\r\nand Shelah's discovery [4] of stable theories, abstract\r\nmodel theory <i>was<\/i> stability theory: the study of stable theories\r\nand related subclasses (totally transcendental, strongly minimal,\r\netc.).\r\n<br>\r\nHowever, since the (re)discovery of simple theories [1, 6],\r\nand of $o$-minimal theories [3], there has been much\r\nresearch going into these classes. In recent years there was much\r\nfocus in NIP and NTP$_2$ theories. NIP theories (introduced by Shelah\r\nin [5]) generalize both $o$-minimal and stable theories\r\nand NTP$_2$ theories (defined in [7]) generalize both NIP\r\nand simple theories.\r\n<br>\r\nIn this talk I will review some of the progress done in recent years\r\nin the study of NIP and NTP$_2$. The talk will be aimed at a wide\r\naudience.\r\n\r\n<br><br><b>References<\/b>\r\n\r\n<br>[1]\r\nByunghan Kim.\r\n   Simple first order theories.\r\n  PhD thesis, University of Notre Dame, 1996.\r\n\r\n<br>[2]\r\nMichael Morley.\r\n  Categoricity in power.\r\n   Transaction of the American Mathematical Society, 114:514\u2013538,\r\n  1965.\r\n\r\n<br>[3]\r\nAnand Pillay and Charles Steinhorn.\r\n  Definable sets in ordered structures. I.\r\n   Trans. Amer. Math. Soc., 295(2):565\u2013592, 1986.\r\n\r\n<br>[4]\r\nSaharon Shelah.\r\n  Stable theories.\r\n   Israel J. Math., 7:187\u2013202, 1969.\r\n\r\n<br>[5]\r\n\u2015.\r\n  Stability, the f.c.p., and superstability; model theoretic properties\r\n  of formulas in first order theory.\r\n   Ann. Math. Logic, 3(3):271\u2013362, 1971.\r\n\r\n<br>[6]\r\n\u2015.\r\n  Simple unstable theories.\r\n   Ann. Math. Logic, 19(3):177\u2013203, 1980.\r\n\r\n<br>[7]\r\n\u2015.\r\n   Classification theory and the number of nonisomorphic models,\r\n  volume&nbsp;92 of  Studies in Logic and the Foundations of Mathematics.\r\n  North-Holland Publishing Co., Amsterdam, second edition, 1990.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m5');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:6ex\">12.45<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.15<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nLunch break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td colspan=\"3\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">Special Sessions (titles: see <a href=\"#special1\">below<\/a>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Homogeneous structures <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Set Theory <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Proof theory and reverse mathematics <\/i><\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.15<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">15.00<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m7')\">Ross Willard<\/a>\r\n<div id=\"absbox-m7\" class=\"abstract_box\">\r\nRoss Willard, <i>The decidable discriminator variety problem<\/i>\r\n<br><br>\r\nThis talk is an advertisement for an old, unsolved\r\nproblem in which universal algebra meets homogeneous structures.\r\nAn <i>equational class<\/i> is any class in an algebraic signature\r\n(i.e., constants and function symbols only) which is axiomatized by universally\r\nquantified equations.  Such a class is <i>locally finite<\/i> if every finitely generated\r\nsubstructure of a member is finite.  The problem in question is that of describing\r\nall locally finite equational classes with finite signature whose first-order theory\r\nis  decidable.\r\nThe work of Burris, McKenzie and Valeriote [1, 3] in the 1980s reduced\r\nthis problem to two special kinds of equational classes:\r\n<ul>\r\n<li>\r\nFor a given finite ring $R$, the class ${}_R\\mathcal M$ of all $R$-modules.\r\n<li>\r\nLocally finite \"discriminator varieties\" in a finite signature.\r\n<\/ul>\r\nWhat are discriminator varieties?  They are\r\nequational classes $\\mathcal E$ which resemble the\r\nclass of Boolean algebras in certain ways.  In particular,\r\n(i) the class $\\mathcal S$ of simple algebras in $\\mathcal E$ is\r\n$\\forall_1$-axiomatizable, and\r\n(ii)\r\neach algebra in $\\mathcal E$ has a Stone-like representation via a\r\nsheaf over $\\mathcal S$.\r\nIn practice [4, 2],\r\nthe question of whether a locally finite discriminator variety\r\n$\\mathcal E$ has\r\na decidable first-order theory hinges on how well-structured are the members of\r\n$\\mathcal S$,\r\nand in particular\r\non the degree to which the countable members of $\\mathcal S$\r\nfail to be homogeneous.  In this talk I will make this precise and explain the current\r\nstate of the problem.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nStanley Burris and Ralph McKenzie,\r\nDecidability and Boolean representations,\r\nMemoirs of the American Mathematical Society,\r\nvol. 32 (1981), no. 246.\r\n<br>\r\n[2]\r\nDejan Deli\u0107,\r\nDecidable discriminator varieties arising from dihedral varieties of groups,\r\nJournal of Pure and Applied Algebra,\r\nvol. 198 (2005), no. 1\u20133, pp. 75\u201392.\r\n<br>\r\n[3]\r\nRalph McKenzie and Matthew Valeriote,\r\nThe Structure of Decidable Locally Finite Varieties,\r\nProgress in Mathematics, Birkh\u00e4user, Boston, 1989.\r\n<br>\r\n[4]\r\nRoss Willard,\r\nDecidable discriminator varieties from unary classes,\r\nTransactions of the American Mathematical Society,\r\nvol. 336 (1993), no. 1, 311-333.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m7');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m9')\">David Schrittesser<\/a>\r\n<div id=\"absbox-m9\" class=\"abstract_box\">\r\nDavid Schrittesser, <i>Definable discrete sets, forcing, and Ramsey theory<\/i>\r\n<br><br>\r\nLet $\\mathcal R$ be a family of finitary relations on a set $X$. A set $A\\subseteq X$ is called $\\mathcal R$-discrete if no relation $R \\in \\mathcal R$ relates any elements of $A$;\r\n$A$ is called maximal discrete if it is maximal with respect to subset-inclusion among $\\mathcal R$-discrete subsets of $X$.\r\nFor any family of relations $\\mathcal R$, maximal $\\mathcal R$-discrete sets exist by the axiom of choice; whether such sets can be <i>definable<\/i> is contentious.\r\nMaximal discrete sets have been widely studied: instances are  maximal co-finitary groups, maximal almost disjoint families, and maximal orthogonal families of measures. In many (but not all) cases one can show such objects cannot be analytic (i.e. projections of closed sets).\r\nOn the contrary, certain definable (in fact, co-analytic) maximal discrete sets have been shown to exist under the assumption that every set is constructible.\r\nWe present some new results, exhibiting definable maximal discrete sets in forcing extensions, e.g. extensions where the continuum hypothesis fails.\r\nIn most cases, these results rely on Ramsey theoretic considerations.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m9');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m11')\">Sam Sanders<\/a>\r\n<div id=\"absbox-m11\" class=\"abstract_box\">\r\nSam Sanders, <i>The unreasonable effectiveness of Nonstandard Analysis<\/i>\r\n<br><br>\r\nAs suggested by the title, we will uncover the vast <i>computational content<\/i> of <i>classical<\/i> Nonstandard Analysis.\r\nTo this end, we formulate a template $\\mathfrak{CI}$ which converts a theorem of `pure' Nonstandard Analysis, i.e. formulated solely with the <i>nonstandard<\/i> definitions (of continuity, integration, differentiability, convergence, compactness, et cetera), into the associated <i>effective<\/i> theorem.  The latter constitutes a theorem of computable mathematics <i>no longer involving Nonstandard Analysis<\/i>.  The template often produces theorems of Bishop's <i>Constructive Analysis<\/i> ([1]).\r\n\\medskip\r\nTo establish the vast scope of $\\mathfrak{CI}$, we apply this template to representative theorems from the <i>Big Five<\/i> categories from <i>Reverse Mathematics<\/i> ([3, 5]).   The latter foundational program provides a classification of the majority of theorems from `ordinary', that is non-set theoretical, mathematics into the aforementioned five categories.  The <i>Reverse Mathematics zoo<\/i> ([2]) gathers exceptions to this classification, and is studied in [4] using $\\mathfrak{CI}$.  Hence, the template $\\mathfrak{CI}$ is seen to apply to essentially <i>all of ordinary mathematics<\/i>, thanks to the Big Five classification (and associated zoo) from Reverse Mathematics.\r\nFinally, we establish that certain `highly constructive' theorems, called Herbrandisations, imply the original theorem of Nonstandard Analysis from which they were obtained via $\\mathfrak{CI}$.\r\n<br>\r\n<b>Acknowledgement.<\/b> This research is generously sponsored by the John Templeton Foundation and the Alexander Von Humboldt Foundation.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nE. Bishop,\r\nD. Bridgess\r\nConstructive analysis,\r\nGrundlehren der Mathematischen Wissenschaften, vol. 279, Springer, 1985, xii+477\r\n<br>\r\n[2]\r\nD. Dzhafarov, The Reverse Mathematics zoo, {http:\/\/rmzoo.uconn.edu\/}\r\n<br>\r\n[3]\r\nU. Kohlenbach,\r\nHigher-order Reverse Mathematics,\r\nLect. Notes Log., Reverse Mathematics 2001, vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 281-295\r\n<br>\r\n[4]\r\nS. Sanders,\r\nThe refining of the taming of the Reverse Mathematics zoo, to appear in\r\nNotre Dame Journal for formal logic, 2016\r\n<br>\r\n[5]\r\nS. Simpson,\r\nSubsystems of Second-order Arithmetic,\r\n2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m11');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">15.00<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">15.45<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m13')\">Josh Wiscons<\/a>\r\n<div id=\"absbox-m13\" class=\"abstract_box\">\r\nJosh Wiscons, <i>The status of Cherlin's conjecture for primitive structures of relational complexity 2<\/i>\r\n<br><br>\r\nThe relational complexity of a structure $\\mathbf{X}$ is the least $k&lt;\\omega$ for which the orbits of $\\operatorname{Aut}(\\mathbf{X})$ on $X^k$ \"determine\" the orbits of $\\operatorname{Aut}(\\mathbf{X})$ on $X^n$ for all $n&lt;\\omega$. This invariant originated in Lachlan's classification theory for homogeneous finite \u2013and more generally, countable stable\u2013 relational structures, but not much was known about the complexities of specific structures until de work of Cherlin, Martin and Saracino in the 1990's. In this talk, I will present some background on relational complexity and discuss recent work on Cherlin's conjecture regarding the classification of the finite primitive structures of complexity 2.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m13');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m15')\">Anush Tserunyan<\/a>\r\n<div id=\"absbox-m15\" class=\"abstract_box\">\r\nAnush Tserunyan, <i>Integer cost and ergodic actions<\/i>\r\n<br><br>\r\nA countable Borel equivalence relation $E$ on a probability space can always be generated in two ways: as the orbit equivalence relation of a Borel action of a countable group and as the connectedness relation of a locally countable Borel graph, called a <i>graphing<\/i> of $E$. Assuming that $E$ is measure-preserving, graphings provide a numerical invariant called <i>cost<\/i>, whose theory has been largely developed and used by Gaboriau and others in establishing rigidity results. A well-known theorem of Hjorth states that when $E$ is ergodic, treeable (admits an acyclic graphing), and has integer cost $n \\ge 1$, then it is generated by an a.e. free measure-preserving action of the free group $\\mathbf{F}_n$ on $n$ generators. We give a simpler proof of this theorem and the technique of our proof, combined with a recent theorem of Tucker-Drob, yields a strengthening of Hjorth's theorem: the action of $\\mathbf{F}_n$ can be arranged so that each of the $n$ generators acts ergodically.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m15');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m17')\">David R. Belanger<\/a>\r\n<div id=\"absbox-m17\" class=\"abstract_box\">\r\nDavid R. Belanger, <i>A computable perfect-set theorem<\/i>\r\n<br><br>\r\nWe gauge the difficulty of finding a perfect subtree in a tree of a given Cantor-Bendixson rank.  To simplify the analysis we introduce <i>half-derivative<\/i>, and extend the definition of rank to include values of the form $n$-and-a-half; each increase of one-half in the rank corresponds to one added jump in the perfect-subtree problem.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m17');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">15.45<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">16.15<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:5ex\">16.15<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">17.30<\/td>\r\n<td align=\"left\" bgcolor=\"#AADDDD\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nContributed talks (see <a href=\"#contrib1\">below<\/a>)\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td colspan=\"6\" style=\"height:2ex\"><\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:8ex\">18.00<\/td>\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">20.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FC9\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nReception (Parkinson Court)\r\n<\/td>\r\n<\/tr>\r\n<\/table><\/p>\r\n<p><table width=\"100%\" style=\"color:black; border: 2px solid black; border-collapse: collapse;\">\r\n<tr style=\"border-top:1px solid black;\">\r\n<td width=\"1px\" bgcolor=\"#AAA\" colspan=\"6\" style=\"text-align:center; padding-left:1ex; padding-right:1ex;\">\r\n<b>Tuesday 2nd August<\/b>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">9.00<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.00<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Toniann Pitassi, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m19')\"><i>Connections between proof complexity, circuit complexity and polynomial identity testing<\/i><\/a>\r\n<div id=\"absbox-m19\" class=\"abstract_box\">\r\nToniann Pitassi, <i>Connections between proof complexity, circuit complexity and polynomial identity testing<\/i>\r\n<br><br>\r\nIn this talk we discuss new variants on algebraic proof systems, and establish\r\ntight connections to central questions in (algebraic) circuit complexity.\r\nIn particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system\r\nimplies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP).\r\nAs a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in\r\nPolynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Prior to our work, there\r\nwas no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound.\r\nOur proof system helps clarify the relationships\r\nbetween previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various\r\nproof systems have been so much harder than lower bounds on the corresponding circuit classes.\r\nIn doing so, we  highlight the importance of the polynomial identity testing problem (PIT) for understanding proof complexity.\r\nThis is joint work with Joshua A. Grochow.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m19');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">10.00<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.30<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">10.30<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">11.30<\/td>\r\n<td align=\"left\" bgcolor=\"#C9DDFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nTutorial A: Thierry Coquand, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m21')\"><i>Univalent Type Theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-coquand-2.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m21\" class=\"abstract_box\">\r\nThierry Coquand, <i>Univalent Type Theory<\/i>\r\n<br><br>\r\nThis tutorial will be an introduction to dependent type theory,\r\nand to the univalence axiom (V. Voevodsky, 2010).\r\nSimple type theory, as formulated by A. Church (1940), constitutes\r\nan elegant alternative of set theory for representing formally\r\nmathematics. The stratification of mathematical objects in a type of propositions,\r\na type of individuals and a type of functions between two types is indeed\r\nquite natural. The axiom of extensionality (the first axiom of set theory)\r\ncomes in two forms: the fact that two equivalent propositions are\r\nequal, and the fact that two pointwise equal functions are equal.\r\nSimple type theory as a formal system has however some unnatural limitations,\r\nin that we cannot express the notion of an arbitrary structure, for instance\r\nthe type of an arbitrary group. Dependent type theory solves this issue\r\nby introducing in type theory the notion of universe. What was missing\r\nuntil the work of V. Voevodsky was a formulation of the extensionality\r\naxiom for universe. This is the univalence axiom, which generalizes\r\npropositional extensionality. When introducing universes, we also can use the\r\nprinciple of propositions-as-types and proofs-as-programs, so that the logical\r\noperations themselves, and their proofs, can be represented as type theoretic\r\noperations.\r\nThe lectures will roughly proceed as follows. The first lecture will be an\r\nintroduction to simple type theory and dependent type theory, where we shall\r\ntry to point out the connections and differences with set theory and end\r\nwith a formulation of the univalence axiom. The second lecture will explore\r\nsome consequences of this axiom, such as a proof of a strong form of\r\nthe axiom of \"unique choice\", from which we can derive results that would\r\nrequire the full axiom of choice in set theory. We will end\r\nwith a presentation of a model of the univalence axiom.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m21');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">11.30<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">12.30<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Farmer Schlutzenberg, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m23')\"><i>Ordinal definability in extender models<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-schlutzenberg.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m23\" class=\"abstract_box\">\r\nFarmer Schlutzenberg, <i>Ordinal definability in extender models<\/i>\r\n<br><br>\r\nG\u00f6del's universe $L$ of constructible sets admits\r\na detailed analysis, and $\\mathrm{ZFC}$ decides much of its first-order theory.\r\nThis makes the study of $L$ tractable and interesting.\r\nHowever, many natural,\r\ndesirable set theoretic principles \u2013 in particular, moderate strength large cardinal and\r\ndeterminacy principles \u2013 must fail in $L$.\r\nExtender models $L[\\mathbb{E}]$ are generalizations\r\nof $L$, which still admit a detailed analysis,\r\nbut can also satisfy large cardinal principles.\r\nThe predicate $\\mathbb{E}$ is a sequence of <i>extenders<\/i>,\r\nwhich witness large cardinals.\r\nThe more canonical $L[\\mathbb{E}]$ are called <i>iterable<\/i>; iterability is witnessed in $V$ by an\r\n<i>iteration strategy<\/i>. It is known that if $L[\\mathbb{E}]$ has Woodin cardinals then it cannot know\r\ntoo much of this iteration strategy.\r\nOne can ask whether $\\mathbb{E}$ can be defined over $L[\\mathbb{E}]$, possibly from some parameter. Related to\r\nthis, one can ask about the structure of $\\mathrm{HOD}^{L[\\mathbb{E}]}$ (that is, the universe $\\mathrm{HOD}$\r\nof hereditarily ordinal definable sets, as computed in $L[\\mathbb{E}]$). I will survey what is known to\r\nthe author regarding these questions.\r\nIn simple cases, $L[\\mathbb{E}]$ satisfies \"$V=\\mathrm{HOD}$\". This holds in $L$,\r\nand, for example, in the\r\nminimal proper class $L[\\mathbb{E}]$\r\nwith a measurable cardinal. But in the presence of Woodin cardinals,\r\nthe question becomes harder to understand.\r\nI will cover the following recent results.\r\nLet $L[\\mathbb{E}]$ be iterable and satisfy $\\mathrm{ZFC}$.\r\nThen (i) $\\mathbb{E}$ is definable from the parameter\r\n$\\mathbb{E}\\upharpoonright\\omega_1^{L[\\mathbb{E}]}$ in $L[\\mathbb{E}]$; and\r\n(ii) in many circumstances, $L[\\mathbb{E}]$ is a small forcing extension of $\\mathrm{HOD}^{L[\\mathbb{E}]}$ and\r\n$\\mathrm{HOD}^{L[\\mathbb{E}]}$\r\nadmits a detailed analysis above\r\n$\\omega_2^{L[\\mathbb{E}]}$. However, the full structure of $\\mathrm{HOD}^{L[\\mathbb{E}]}$ is an\r\nopen question, even in rather basic cases.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m23');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:6ex\">12.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nLunch break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td colspan=\"3\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">Special Sessions (titles: see <a href=\"#special2\">below<\/a>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Homogeneous structures <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Set Theory <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Proof theory and reverse mathematics <\/i><\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.00<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.45<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m25')\">Jan Hubicka<\/a>\r\n<div id=\"absbox-m25\" class=\"abstract_box\">\r\nJan Hubicka, <i>All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)<\/i>\r\n<br><br>\r\nClass $\\mathcal K$ of finite structures is Ramsey class if for every choice of\r\n$\\mathbf A,\\mathbf B\\in K$ there exists $\\mathbf C\\in \\mathcal K$ such that for every\r\ncoloring of its substructures isomorphic to $\\mathbf A$ with 2 colors there\r\nexists an isomorphic copy of $\\mathbf B$ in $\\mathbf C$ where all copies of\r\n$\\mathbf A$ are monochromatic.\r\nIt is a classical result that for every purely relational language $L$ the class of all\r\nfinite ordered $L$-structures  is Ramsey&nbsp;[5, 1].\r\nWe extend this theorem for languages containing both relations and functions.\r\nWe also give a new sufficient condition for subclass of a Ramsey class to be Ramsey.\r\nBy verifying this condition we prove Ramsey property of many classes such as\r\nconvexly ordered $S$-metric spaces (solving an open problem&nbsp;[6]), totally ordered structures (structures with linear order on both vertices and relations), and\r\nordered single constraint Cherlin Shelah Shi classes&nbsp;[2].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nF.&nbsp;G.&nbsp;Abramson, L.&nbsp;A.&nbsp;Harrington,\r\nModels without indiscernibles,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;43 (1978), no.&nbsp;3, pp.&nbsp;572\u2013600.\r\n<br>\r\n[2]\r\nG.&nbsp;Cherlin, S.&nbsp;Shelah, N.&nbsp;Shi,\r\nUniversal graphs with forbidden subgraphs and algebraic closure,\r\nAdvances in Applied Mathematics,\r\nvol.&nbsp;22 (1999), no.&nbsp;4, pp.&nbsp;454\u2013491.\r\n<br>\r\n[3]\r\nJ.&nbsp;Hubi\u010dka, J.&nbsp;Ne\u0161et\u0159il,\r\nBowtie-free graphs have a {R}amsey lift,\r\narXiv preprint, arXiv:1402.2700.\r\n<br>\r\n[4]\r\nJ.&nbsp;Hubi\u010dka, J.&nbsp;Ne\u0161et\u0159il,\r\nAll those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms),\r\nin preparation.\r\n<br>\r\n[5]\r\nJ.&nbsp;Ne\u0161et\u0159il, V.&nbsp;R\u00f6dl,\r\nPartitions of Finite Relational and Set Systems,\r\nJournal Combinatorial Theory, Series A,\r\nvol.&nbsp;22 (1977), no.&nbsp;3, pp.&nbsp;289\u2013312.\r\n<br>\r\n[6]\r\nL.&nbsp;Nguyen Van Th\u00e9,\r\nStructural {R}amsey Theory of Metric Spaces and Topological Dynamics of Isometry Groups,\r\nMemoirs of the American Mathematical Society,\r\nAmerican Mathematical Society,\r\n2010.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m25');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m27')\">Nam Trang<\/a>\r\n<div id=\"absbox-m27\" class=\"abstract_box\">\r\nNam Trang, <i>Large cardinals, determinacy, and forcing axioms<\/i>\r\n<br><br>\r\nWe discuss some recent progress in descriptive inner model theory. In particular, we discuss some current results concerning connections of the three hierarchies of models: canonical models of large cardinals (pure extender models), canonical models of determinacy, and strategic hybrid models (e.g. HOD of determinacy models). These structural results can be used to improve (lower-bound) consistency strength of strong combinatorial principles such as The Proper Forcing Axiom ($\\sf{PFA}$), strong forms of the tree property etc. In particular, I proved that $\\sf{PFA}$ implies the existence of a transitive model of ``$\\sf{AD}_\\mathbb{R} + \\Theta$ is regular\". Building on this and structural results above, G. Sargsyan and I have constructed models of theory $\\sf{LSA} =_{\\rm{def}} ``\\sf{AD}^+ + $there is an $\\alpha$ such that $\\Theta=\\theta_{\\alpha+1} + \\theta_\\alpha$ is the largest Suslin cardinal\" from $\\sf{PFA}$. This result is the strongest of its kind and reflects our current understanding of HOD of models of determinacy.\r\n\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m27');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m29')\">Lev Gordeev<\/a>\r\n<div id=\"absbox-m29\" class=\"abstract_box\">\r\nLev Gordeev, <i>On Harvey Friedman's finite phase transitions<\/i>\r\n<br><br>\r\n<b>Definition<\/b> (H. Friedman).\r\nThe<i> proof theoretic integer<\/i> of formal system $\\mathbf{T}$\r\n(abbreviation: $\\mathrm{PTI}\\left( \\mathbf{T}\\right) $ ) is the least\r\ninteger $n$ such that every $\\Sigma _{1}^{0}$ sentence\r\n\\begin{equation*}\r\n\\exists x_{1}\\cdots \\exists x_{m}A\\left( x_{1},\\cdots ,x_{m}\\right)\r\n\\end{equation*}\r\nthat has a proof in $\\mathbf{T}$ with at most $10,000$ symbols, has\r\nwitnesses $x_{1},\\cdots ,x_{m}&lt;n$. (Actually $m=1$ would suffice.)\r\n<br><br>\r\nA good source of examples is in the area surrounding Kruskal's theorem. This\r\ntalk is devoted to $\\mathrm{PTI}\\left( \\mathbf{T}\\right) $'s basic\r\nproperties, examples, comparisons and related phase transitions.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m29');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.45<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">15.30<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m31')\">Libor Barto<\/a>\r\n<div id=\"absbox-m31\" class=\"abstract_box\">\r\nLibor Barto, <i>The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems<\/i>\r\n<br><br>\r\nWe prove that an $\\omega$-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations $\\alpha$, $\\beta$, $s$ satisfying the identity $\\alpha s(x,y,x,z,y,z) \\approx \\beta s(y,x,z,x,z,y)$.\r\nThis establishes an algebraic criterion equivalent to the conjectured borderline between P and NP-complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case.\r\nOur theorem is also of independent mathematical interest, characterizing a topological property of any $\\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).\r\nThis is a joint work with Michael Pinsker.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m31');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m33')\">Andrew Marks<\/a>\r\n<div id=\"absbox-m33\" class=\"abstract_box\">\r\nAndrew Marks, <i>Borel and measurable matchings<\/i>\r\n<br><br>\r\nWe discuss several results related to the question of when a\r\nBorel graph has a Borel matching. Here, the analogue of Hall's\r\nmatching theorem fails, but there are positive results giving Borel\r\nmatchings in several contexts if we are willing to discard null or\r\nmeager sets, or restrict the types of graphs we consider. We also\r\ndiscuss some applications to geometrical paradoxes.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m33');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m35')\">Florian Pelupessy<\/a>\r\n<div id=\"absbox-m35\" class=\"abstract_box\">\r\nFlorian Pelupessy, <i>Ramsey like principles and well-foundedness of $d$-height $\\omega$-towers<\/i>\r\n<br><br>\r\nRecently it has been highlighted by Kreuzer and Yokoyama [1] that, over $\\mathrm{RCA}_0$, there are many principles equivalent to the well foundedness of the ordinal $\\omega^\\omega$. We will observe that there are Ramsey-like principles which are equivalent to the well-foundedness of $\\omega_d$, where $\\omega_0=1$ and $\\omega_{n+1}=\\omega^{\\omega_n}$. One of these examples is based on the relativised Paris\u2013Harrington principle as mentioned in [1], but for dimension $d$. The more interesting example is the restricton of Friedman's adjacent Ramsey theorem to fixed dimension $d$, which is equivalent to the well-foundedness of $\\omega_{d+1}$.\r\n<br><br>\r\n<b>Definition<\/b> (adjacent Ramsey in dimension $d$).\r\nFor every $C\\colon \\mathbb{N}^d \\rightarrow \\mathbb{N}^r$ there exist $x_0 &lt; \\dots &lt; x_{d+1}$ such that $C(x_1, \\dots , x_d) \\leq C(x_2, \\dots , x_{d+1})$, where $\\leq$ is the coordinatewise ordering.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAlexander P. Kreuzer and Keita Yokoyama,\r\nOn principles between $\\Sigma_1$ and $\\Sigma_2$ induction and monotone enumerations,\r\narXiv:1306.1936v5.\r\n<br>\r\n[2]\r\nStephen G. Simpson,\r\nSubsystems of second order arithmetic,\r\nPerspectives in logic (2nd edition),\r\nCambridge University Press,\r\n2009.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m35');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">15.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">16.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:8.33333333333333ex\">16.00<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">18.05<\/td>\r\n<td align=\"left\" bgcolor=\"#AADDDD\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nContributed talks (see <a href=\"#contrib2\">below<\/a>)\r\n<\/td>\r\n<\/tr>\r\n<\/table><\/p>\r\n<p><table width=\"100%\" style=\"color:black; border: 2px solid black; border-collapse: collapse;\">\r\n<tr style=\"border-top:1px solid black;\">\r\n<td width=\"1px\" bgcolor=\"#AAA\" colspan=\"6\" style=\"text-align:center; padding-left:1ex; padding-right:1ex;\">\r\n<b>Wednesday 3rd August<\/b>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">9.00<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.00<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary (G\u00f6del Lecture): Stevo Todorcevic, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m37')\"><i>Basis Problems in Set Theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-todorcevic.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m37\" class=\"abstract_box\">\r\nStevo Todorcevic, <i>Basis Problems in Set Theory<\/i>\r\n<br><br>\r\nGiven a class $\\mathcal{C}$ of mathematical structures we are interested in finding a\r\nlist $\\mathcal{C}_0 \\subseteq \\mathcal{C}$ of <i>critical structures<\/i> in $\\mathcal{C},$ a list with the property that\r\nevery structure in $\\mathcal{C}$ is in some way related to or built from some elements of $\\mathcal{C}_0.$  Such results are of course useful if $\\mathcal{C}_0$ is small (typically finite) and\r\nwhen elements of $\\mathcal{C}$ have strong relationship to the elements of $\\mathcal{C}_0.$ We give an overview of the results of this area concentrating on the more recent ones. We also list some open problems.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m37');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">10.00<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.30<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">10.30<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">11.30<\/td>\r\n<td align=\"left\" bgcolor=\"#C9DDFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nTutorial A: Thierry Coquand, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m39')\"><i>Univalent Type Theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-coquand-3.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m39\" class=\"abstract_box\">\r\nThierry Coquand, <i>Univalent Type Theory<\/i>\r\n<br><br>\r\nThis tutorial will be an introduction to dependent type theory,\r\nand to the univalence axiom (V. Voevodsky, 2010).\r\nSimple type theory, as formulated by A. Church (1940), constitutes\r\nan elegant alternative of set theory for representing formally\r\nmathematics. The stratification of mathematical objects in a type of propositions,\r\na type of individuals and a type of functions between two types is indeed\r\nquite natural. The axiom of extensionality (the first axiom of set theory)\r\ncomes in two forms: the fact that two equivalent propositions are\r\nequal, and the fact that two pointwise equal functions are equal.\r\nSimple type theory as a formal system has however some unnatural limitations,\r\nin that we cannot express the notion of an arbitrary structure, for instance\r\nthe type of an arbitrary group. Dependent type theory solves this issue\r\nby introducing in type theory the notion of universe. What was missing\r\nuntil the work of V. Voevodsky was a formulation of the extensionality\r\naxiom for universe. This is the univalence axiom, which generalizes\r\npropositional extensionality. When introducing universes, we also can use the\r\nprinciple of propositions-as-types and proofs-as-programs, so that the logical\r\noperations themselves, and their proofs, can be represented as type theoretic\r\noperations.\r\nThe lectures will roughly proceed as follows. The first lecture will be an\r\nintroduction to simple type theory and dependent type theory, where we shall\r\ntry to point out the connections and differences with set theory and end\r\nwith a formulation of the univalence axiom. The second lecture will explore\r\nsome consequences of this axiom, such as a proof of a strong form of\r\nthe axiom of \"unique choice\", from which we can derive results that would\r\nrequire the full axiom of choice in set theory. We will end\r\nwith a presentation of a model of the univalence axiom.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m39');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">11.30<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">12.30<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Henry Towsner, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m41')\"><i>A Concrete View of Ultraproducts<\/i><\/a>\r\n<div id=\"absbox-m41\" class=\"abstract_box\">\r\nHenry Towsner, <i>A Concrete View of Ultraproducts<\/i>\r\n<br><br>\r\nUltraproducts are one of the tools from logic most widely used in mathematics, playing a role in functional analysis, differential algebra, algebraic geometry, and recently combinatorics.  We describe how to reinterpret proofs which use ultraproducts to reveal the underlying constructive calculations.  In particular, this makes it possible to replace proofs which use ultraproducts with constructive, explicit proofs which avoid the use of the axiom of choice.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m41');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:5ex\">12.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">13.45<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nLunch break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FC9\" width=\"1px\" colspan=\"3\" style=\"height:17.3333333333333ex;\"><\/d>\r\n<td align=\"left\" bgcolor=\"#FC9\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; \">\r\nExcursion (departure from Parkinson Steps at 13.45)\r\n<\/td>\r\n<\/tr>\r\n<\/table><\/p>\r\n<p><table width=\"100%\" style=\"color:black; border: 2px solid black; border-collapse: collapse;\">\r\n<tr style=\"border-top:1px solid black;\">\r\n<td width=\"1px\" bgcolor=\"#AAA\" colspan=\"6\" style=\"text-align:center; padding-left:1ex; padding-right:1ex;\">\r\n<b>Thursday 4th August<\/b>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">9.00<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.00<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Dima Sinapova, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m43')\"><i>Compactness-type combinatorial principles<\/i><\/a>\r\n<div id=\"absbox-m43\" class=\"abstract_box\">\r\nDima Sinapova, <i>Compactness-type combinatorial principles<\/i>\r\n<br><br>\r\nCompactness-type combinatorial principles like the tree property and failure of square are remnants of large cardinals but can hold at successor cardinals. They test how much can be obtained from forcing and large cardinals versus how L-like the universe is. It is especially difficult to force these properties at small cardinals. We will introduce some background and then discuss some new results on obtaining the tree property and related combinatorial principles at smaller cardinals.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m43');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">10.00<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.30<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">10.30<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">11.30<\/td>\r\n<td align=\"left\" bgcolor=\"#C9DDFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nTutorial B: Uri Andrews, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m45')\"><i>Computable model theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-Andrews-1.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m45\" class=\"abstract_box\">\r\nUri Andrews, <i>Computable model theory<\/i>\r\n<br><br>\r\nIn this tutorial series, I will talk about many instances where ideas from computability theory and ideas from model theory intertwine. The focus will be mostly on questions related to computation of models or theories, but I will also try to highlight how computability can help refine our understanding of model theoretic ideas.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m45');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">11.30<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">12.30<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Benno van den Berg, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m47')\"><i>Homotopy type theory via path categories<\/i><\/a>\r\n<div id=\"absbox-m47\" class=\"abstract_box\">\r\nBenno van den Berg, <i>Homotopy type theory via path categories<\/i>\r\n<br><br>\r\nHomotopy type theory is based on the fact that similar categorical\r\nstructures appear in both type theory and homotopy theory. Paths and higher\r\nhomotopies give every topological space the structure of an $\\infty$-groupoid,\r\nand so does the identity type in type theory. Quillen model categories are\r\nthe most common abstract framework for homotopy theory, but also give rise\r\nto models of the identity type (modulo coherence problems relating to\r\nsubstitution).\r\nThis talk starts from another link: in homotopy theory a well-known\r\nweakening of the notion of a Quillen model structure is that of a category\r\nof fibrant objects, due to Kenneth Brown. A slight variation of this\r\nnotion, which I will call a path category, corresponds to a natural\r\nweaking of the rule for the identity type, where we ask for the computation\r\nrule for J to hold only in a propositional form. Indeed, this weakening has\r\nbeen considered by Coquand and collaborators in their attempt to build\r\nconstructive models of homotopy type theory. Ignoring the coherence\r\nproblems again, we can say that path categories provide a sound and\r\ncompleteness semantics for these weak identity types.\r\nIn constructive mathematics one often avoids taking quotients; instead, one\r\nconsiders sets together with an arbitrary equivalence relation. In type\r\ntheory such an object is called a setoid. In this talk I will show that the\r\ncategory of setoids can be seen as a two-step construction, where one first\r\nbuilds a new path category out of an old one and then takes the\r\nhomotopy category. It turns out that the intermediate path category\r\nhas interesting properties: for example, it satisfies functional\r\nextensionality even when the original one does not.\r\nIf time permits, I also plan to talk about algebraic set theory and models\r\nof Aczel's constructive set theory {\\bf CZF} from weak universes, also in the\r\ncontext of path categories. (This is joint work with Ieke Moerdijk and based on the preprints\r\n[1, 2].)\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nB. van den Berg,\r\nPath categories and propositional identity types,\r\narXiv:1605.02534, April 2016.\r\n<br>\r\n[2]\r\nB. van den Berg and I. Moerdijk,\r\nExact completion of path categories and algebraic set theory,\r\narXiv:1603.02456, March 2016.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m47');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:6ex\">12.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nLunch break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td colspan=\"3\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">Special Sessions (titles: see <a href=\"#special4\">below<\/a>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Model theory and limit structures <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Computability theory <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Formal theories of truth <\/i><\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.00<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.45<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m49')\">Patrice Ossona de Mendez<\/a>\r\n<div id=\"absbox-m49\" class=\"abstract_box\">\r\nPatrice Ossona de Mendez, <i>Structural Limits and Clustering near Infinity<\/i>\r\n<br><br>\r\nStructural limits (including FO-limits and X-limits for various fragments X) arise as a natural generalization of limits of bothe dense and bounded degree graphs, yet having a distinctive model theoretic flavour. It is in a way dual approach which allows to prove distributional limits in a full generality. More recently it leads to clustering which seems to be interesting from both analytic and model theoretic perspective. This can be also outlined as follows:\r\nThe cluster analysis of very large objects is an important problem, which spans several\r\ntheoretical as well as applied branches of mathematics and computer science.\r\nHere we suggest a novel approach: under assumption of local convergence of a sequence of\r\nfinite structures we derive an asymptotic clustering. This is achieved by a blend of\r\nanalytic and geometric techniques, and particularly by a new interpretation of the authors?\r\nrepre-sentation theorem for limits of local convergent sequences, which serves as a\r\nguidance for the whole process. Our study may be seen as an effort to describe connectivity\r\nstructure at the limit (without having a defined explicit limit structure) and to pull this\r\nconnectivity structure back to the finite structures in the sequence in a continuous way.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m49');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m51')\">Rupert H\u00f6lzl<\/a>\r\n<div id=\"absbox-m51\" class=\"abstract_box\">\r\nRupert H\u00f6lzl, <i>Randomness for computable measures and initial segment complexity<\/i>\r\n<br><br>\r\nThe Levin-Schnorr theorem establishes the equivalence of a certain measure-theoretic notion of typicality for infinite sequences (known as Martin-L\u00f6f randomness) with a notion of incompressibility given in terms of Kolmogorov complexity.  Although the Levin-Schnorr theorem is usually formulated for sequences that are random with respect to the Lebesgue measure on $2^\\omega$, it is well known that the theorem can be generalized to hold for any computable probability measure on $2^\\omega$.  More specifically, a sequence $X\\in2^\\omega$ is Martin-L\u00f6f random with respect to a computable measure&nbsp;$\\mu$ if and only if the initial segment complexity of $X\\upharpoonright n$ is bounded from below by $-\\log\\mu(X\\upharpoonright n)$.  Thus we see that certain values of the measure $\\mu$ constrain the possible values of the initial segment complexities of the $\\mu$-random sequences.\r\nIn this study, we further explore the interaction between computable measures and the initial segment complexity of the sequences that are random with respect to these measures (hereafter, we will refer to those sequences that are random with respect to a computable measure as <i>proper<\/i> sequences, following the terminology of Zvonkin and Levin&nbsp;[6]).  We focus in particular on the growth rates of functions of the form $f(X,n)=-\\log\\mu(X\\upharpoonright n)$ for various computable measures $\\mu$ and $\\mu$-random sequences $X$.  As we demonstrate, these growth rates can vary widely, depending on the choice of the underlying measure $\\mu$.\r\nIn the first half of the paper, we focus on the relationship between a class of sequences known as <i>complex sequences<\/i> and those sequences that are random with respect to a computable, continuous measure.\r\nFirst studied systematically by Kjos-Hanssen et al.&nbsp;[4] (but also studied earlier by Kanovi\u010d&nbsp;[2]), complex sequences are those sequences whose initial segment complexities are bounded below by some computable function.   We characterize the complex proper sequences as the sequences that are random with respect to some computable <i>continuous<\/i> measure. This is done by studying the \"removability\" of $\\mu$-atoms, that is, sequences&nbsp;$X$ such that $\\mu(\\{X\\})>0$. We show that if a sequence $X$ is complex and random with respect to some computable measure $\\mu$, we can define a computable, continuous measure $\\nu$ such that $X$ is random with respect to $\\nu$ by removing the $\\mu$-atoms that are in some sense near $X$.  It is natural to ask whether this removal of atoms can always be carried out while preserving all non-atomic random sequences simultaneously, again assuming that all of these random sequences are complex. We show that this is not the case.\r\nUsing this characterization of complex sequences through computable continuous measures, we establish new results on the relationship between the notions of avoidability, hyperavoidability, semigenericity, and not being random for any computable, continuous measure.  More specifically, when restricted to the collection of proper sequences, we show that these four notions are equivalent to being complex.  We also study the granularity of a computable, continuous measure&nbsp;$\\mu$ and show that the inverse of the granularity function provides a uniform lower bound for the initial segment complexity of $\\mu$-random sequences.\r\nIn the second half of the paper, we turn our attention to atomic computable measures, i.e., computable measures $\\mu$ that have $\\mu$-atoms.  First, we study atomic measures $\\mu$ with the property that every $\\mu$-random sequence is either a $\\mu$-atom or is complex.  We show that for such measures&nbsp;$\\mu$, even though the initial segment complexity of each non-atom $\\mu$-random sequence is bounded from below by some computable function, there is in general no uniform computable lower bound for every non-atom $\\mu$-random sequence.  Next, we construct a computable atomic measure $\\mu$ with the property that the initial segment complexity of each $\\mu$-random sequence dominates no computable function, and a computable atomic measure $\\nu$ with the property that the initial segment complexity of each $\\nu$-random sequence\r\nis dominated by all computable functions.  The former sequences are called <i>infinitely often anti-complex<\/i>, while the latter are known simply as <i>anti-complex<\/i>.\r\nLastly, we study two specific kinds of atomic measures:  diminutive measures and trivial measures. Here, a measure $\\mu$ is <i>trivial<\/i> if $\\mu(\\mathrm{Atoms}_\\mu)=1$, and diminutive measures are defined as follows.\r\n<br><br>\r\n<b>Definition<\/b>.   Let $\\mathcal{C}\\subseteq2^\\omega$.\r\n<br>\r\n(i) $\\mathcal{C}$ is <i>diminutive<\/i> if it does not contain a computably perfect subclass.\r\n<br>\r\n(ii) Let $\\mu$ be a computable measure, and let $(\\mathcal{U}_i)_{i\\in\\omega}$ be the universal $\\mu$-Martin-L\u00f6f test.  Then we say that $\\mu$ is <i>diminutive<\/i> if $\\mathcal{U}_i^c$ is a diminutive $\\Pi^0_1$ class for every $i$.\r\n<br><br>\r\nWe show that while every computable trivial measure is diminutive, the converse does not hold. The proof of this last statement gives an alternative, priority-free proof of the following known result.\r\n<br><br>\r\n<b>Corollary<\/b> (Kautz&nbsp;[3]).\r\nThere is a computable, non-trivial measure $\\mu$ such that there is no $\\Delta^0_2$, non-computable $X\\in\\mathrm{MLR}_\\mu$.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nR.&nbsp;H\u00f6lzl and C.&nbsp;P. Porter,\r\nRandomness for Computable Measures and Initial Segment Complexity,\r\nArXiv e-prints, October 2015, 1510.07202.\r\n<br>\r\n[2]\r\nMax&nbsp;I. Kanovi\u010d,\r\nThe complexity of the enumeration and solvability of predicates,\r\nDoklady Akademii nauk SSSR,\r\nvol.&nbsp;190 (1970), pp.&nbsp;23\u201326.\r\n<br>\r\n[3]\r\nSteven&nbsp;M. Kautz,\r\nDegrees of random sets,\r\nPhD thesis, Cornell University, 1991.\r\n<br>\r\n[4]\r\nBj\u00f8rn Kjos-Hanssen, Wolfgang Merkle, and Frank Stephan,\r\nKolmogorov complexity and the recursion theorem,\r\nTransactions of the American Mathematical Society,\r\nvol.&nbsp;363 (2011), no.&nbsp;10, pp.&nbsp;5465\u20135480.\r\n<br>\r\n[5]\r\nChristopher Porter,\r\nMathematical and Philosophical Perspectives on Algorithmic\r\nRandomness,\r\nPhD thesis, University of Notre Dame, 2012.\r\n<br>\r\n[6]\r\nAlexander&nbsp;K. Zvonkin and Leonid&nbsp;A. Levin,\r\nThe complexity of finite objects and the basing of the concepts of\r\ninformation and randomness on the theory of algorithms,\r\nUspekhi Matematicheskikh Nauk,\r\nvol.&nbsp;25 (1970), no.&nbsp;6(156), pp.&nbsp;85\u2013127.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m51');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m53')\">Volker Halbach<\/a>\r\n<div id=\"absbox-m53\" class=\"abstract_box\">\r\nVolker Halbach, <i>Axiomatizing Kripke's theory of truth in classical and nonclassical logic<\/i>\r\n<br><br>\r\nWe consider axiomatizations of Kripke's [3] theory of truth, more precisely, of all fixed points of Kripke's operator based on Strong Kleene logic over the standard model over arithmetic. We also consider analogous models with gluts as considered by [4]. Feferman [1] axiomatized Kripke's theory in classical logic. Feferman's axiomatization is sound in the sense that whenever $T\\ulcorner\\phi \\urcorner$ is provable then $\\phi $ holds in all of Kripke's fixed-point models.  Halbach and Horsten [2] provided an axiomatization that is directly sound: $\\phi $ holds in all of Kripke's fixed-point models, if $\\phi $ is provable in their system. Halbach and Horsten [2] showed that their system is proof-theoretically much weaker than Feferman's system and lacks many arithmetical theorems provable in Feferman's system. There are many sentences such that $T\\ulcorner\\phi \\urcorner$ is provable in Feferman's system, while $\\phi $ isn't provable in the Halbach\u2013Horsten system.\r\nWe pinpoint the source of the deductive weakness of the Halbach\u2013Horsten system in the induction rule. Both systems contain the same rule of induction. If it is removed from both systems, then $T\\ulcorner\\phi \\urcorner$ is provable in Feferman's system iff $\\phi $ is provable in the Halbach\u2013Horsten system. Thus the effect of switching from classical logic to the nonclassical logic of the Halbach\u2013Horsten system limits the usability of the mathematical principle of induction. We take this as evidence that the often advocated strategy of restricting classical logic for semantic vocabulary doesn't necessarily affect our semantic reasoning, but it can cripple our mathematical reasoning.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSolomon Feferman,\r\nReflecting on incompleteness,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;56 (1991), no.&nbsp;1, pp.&nbsp;1\u201349.\r\n<br>\r\n[2]\r\nVolker Halbach and Leon Horsten,\r\nAxiomatizing Kripke's theory of truth,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;71 (2006), no.&nbsp;2, pp.&nbsp;677\u2013712.\r\n<br>\r\n[3]\r\nSaul A. Kripke,\r\nOutline of a Theory of Truth,\r\nJournal of Philosophy,\r\nvol.&nbsp;72 (1975), no.&nbsp;19, pp.&nbsp;690\u2013716.\r\n<br>\r\n[4]\r\nAlbert Visser,\r\nFour-valued semantics and the liar,\r\nJournal of Philosophical Logic,\r\nvol.&nbsp;13 (1984), no.&nbsp;2, pp.&nbsp;181\u2013212.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m53');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.45<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">15.30<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m55')\">Ove Ahlman<\/a>\r\n<div id=\"absbox-m55\" class=\"abstract_box\">\r\nOve Ahlman, <i>Simple structures axiomatized by almost sure theories<\/i>\r\n<br><br>\r\nStudying simple structures, one of the nicest examples of a simple yet not stable structure is the Rado graph.\r\nThe Rado graph is the unique countable graph $\\mathcal{G}$ which satisfies that for any finite disjoint subsets $A$ and $B$ of $\\mathcal{G}$ there is a vertex $c$ such that $c$ is adjacent to all vertices in $A$ and no vertices in $B$. Some of the properties of the Rado graph which makes it especially nice include $\\omega-$categoricity (even homogeneity), SU-rank $1$ and trivial algebraic closure.\r\n<br>\r\nFor each $n\\in\\mathbb{N}$ let $\\mathbf{K}_n$ be a set of finite structures and let $\\mu_n$ be the uniform probability measure on $\\mathbf{K}_n$, assigning the same probability to each structure. This meas-sure induce a probability measure on first order sentences $\\varphi$ by putting $\\mu_n(\\varphi)=\\mu(\\{\\mathcal{M}\\in\\mathbf{K}_n: \\mathcal{M}\\models \\varphi\\})$. Put $T$ to be the theory, called the almost sure theory, of all first order sentences $\\varphi$ such that $\\lim_{n\\rightarrow\\infty} \\mu_n(\\varphi)= 1$. We directly see that $T$ is complete if and only if each sentence has asymptotic probability $0$ or $1$, in which case we say that the pair $(\\mathbf{K}_n,\\mu_n)_{n\\in\\mathbb{N}}$ has a $0-1$ law. Sets which have a $0-1$ law under the uniform probability measure include all finite structures, all finite partial orders, all finite $l-$colorable graphs and many more.\r\n<br>\r\nOne can easily show that the Rado graph is not only axiomatized by the vertex-extension property described above, but can also be seen as the unique countable structure satisfying the almost sure theory coming from $\\mathbf{K}_n$ consisting of all graphs with universe $\\{1,\\ldots,n\\}$. Further more many other almost sure theories, including all of the above mentioned examples, are $\\omega-$categorical and simple with $SU-$rank 1. In this talk we will discuss why this connection occurs and see that the binary simple $\\omega-$categorical structures with SU-rank $1$ is in direct connection with almost sure theories.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m55');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m57')\">Chris Conidis<\/a>\r\n<div id=\"absbox-m57\" class=\"abstract_box\">\r\nChris Conidis, <i>New directions in reverse algebra<\/i>\r\n<br><br>\r\n\r\nWe will survey some old results in the reverse mathematics of Artinian and Noetherian rings, showing how computability-theoretic insights can give rise to new algebraic techniques. Finally, we will conclude with some new results concerning the reverse mathematics of the (very general) class of Noetherian rings.\r\n<br><br><b>References<\/b>\r\n<br>[1]\r\n Conidis, C.J.,\r\n Chain conditions in computable rings,\r\n Transactions of the American Mathematical Society,\r\nvol.&nbsp;362 (2010), no.&nbsp;12, pp.&nbsp;6523\u20136550.\r\n<br>[2]\r\n\u2015\r\n The computability, definability, and proof theory of Artinian rings,\r\n To appear.\r\n<br>[3]\r\n\u2015\r\n The meta-metamathematics of Neotherian rings,\r\n In preparation.\r\n<br>\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m57');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m59')\">Lavinia Picollo<\/a>\r\n<div id=\"absbox-m59\" class=\"abstract_box\">\r\nLavinia Picollo, <i>Truth, Reference and Disquotation<\/i>\r\n<br><br>\r\nI first provide intuitively appealing notions of reference, self-reference, and well-foundedness of sentences of the language of first-order Peano arithmetic extended with a truth predicate. They are intended as a tool for studying reference patterns that underlie expressions leading to semantic paradox, and thus to shed light on the debate on whether every paradox formulated in a first-order language involves self-reference or some other vicious reference pattern.\r\nI use the new notions to formulate sensible restrictions on the acceptable instances of the T-schema, to carry out the disquotationalist project. Since the concept of reference I put forward is proof-theoretic\\textemdash i.e., it turns to the provability predicate rather than the truth predicate\\textemdash and, therefore, arithmetically definable,  it can be used to provide recursive axiomatizations of truth. I show the resulting systems are $\\omega$-consistent and as strong as Tarski's theory of ramified truth iterated up to $\\epsilon_0$.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m59');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">15.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">16.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:8.33333333333333ex\">16.00<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">18.05<\/td>\r\n<td align=\"left\" bgcolor=\"#AADDDD\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nContributed talks (see <a href=\"#contrib4\">below<\/a>)\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td colspan=\"6\" style=\"height:3.66666666666667ex\"><\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">19.00<\/td>\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FC9\" width=\"1px\" style=\"text-align:left; padding-right:1ex;\">...<\/td>\r\n<td align=\"left\" bgcolor=\"#FC9\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nConference dinner\r\n<\/td>\r\n<\/tr>\r\n<\/table><\/p>\r\n<p><table width=\"100%\" style=\"color:black; border: 2px solid black; border-collapse: collapse;\">\r\n<tr style=\"border-top:1px solid black;\">\r\n<td width=\"1px\" bgcolor=\"#AAA\" colspan=\"6\" style=\"text-align:center; padding-left:1ex; padding-right:1ex;\">\r\n<b>Friday 5th August<\/b>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">9.00<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.00<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Timothy Williamson, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m61')\"><i>Alternative Logics and Abductive Methodology<\/i><\/a>\r\n<div id=\"absbox-m61\" class=\"abstract_box\">\r\nTimothy Williamson, <i>Alternative Logics and Abductive Methodology<\/i>\r\n<br><br>\r\nSometimes we have to make genuine choices between classical logic and various non-classical alternatives, e.g. as to which we rely on for establishing meta-logical results or for reasoning about extra-logical matters. Discussion is needed, of a not purely formal kind, of the methodology we should use to make such a choice between logics. It can be understood as a special case of theory choice in science, governed by similar broadly abductive criteria, such as elegance, simplicity, strength, explanatory power, and fit with evidence. There is no need to appeal to a pre-given relation of logical consequence, since a candidate logic's consistency with evidence can be understood as the closure of the set of evidence under that logic's consequence relation. There is also no requirement for the logic to be weak enough to be `neutral' in some sense, because there is no useful standard of neutrality. The idea that sub-classical logics can recapture the strength of classical logic `when they need it', e.g. by adding instances of excluded middle as auxiliary assumptions, will be criticized on the grounds that it degrades a wide range of scientific explanations by introducing extraneous ad hoc assumptions. The most promising rivals to classical logic are those preserving simple and strong principles that classical logic renders inconsistent, e.g. disquotational principles in the case of semantic paradoxes and tolerance principles in the case of sorites paradoxes. However, even these examples are arguably bad bargains in which local benefits are gained at the expense of global costs.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m61');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">10.00<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.30<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">10.30<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">11.30<\/td>\r\n<td align=\"left\" bgcolor=\"#C9DDFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nTutorial B: Uri Andrews, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m63')\"><i>Computable model theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-Andrews-2.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m63\" class=\"abstract_box\">\r\nUri Andrews, <i>Computable model theory<\/i>\r\n<br><br>\r\nIn this tutorial series, I will talk about many instances where ideas from computability theory and ideas from model theory intertwine. The focus will be mostly on questions related to computation of models or theories, but I will also try to highlight how computability can help refine our understanding of model theoretic ideas.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m63');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">11.30<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">12.30<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Richard Garner, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m65')\"><i>Non-standard arities<\/i><\/a>\r\n<div id=\"absbox-m65\" class=\"abstract_box\">\r\nRichard Garner, <i>Non-standard arities<\/i>\r\n<br><br>\r\nFinitary equational theories admit a presentation-independent\r\nrealisation through the notion of <i>finitary monad<\/i> on the\r\ncategory of sets. It is well-understood that the monad-theoretic\r\npresentation of algebraic theories is apt for generalisation to other\r\nsettings (= other base categories); it is less well-appreciated that,\r\neven without leaving the world of sets and functions, the monadic\r\napproach allows one to detect richer and more complex structure\r\nassociated to a theory than just that of its derived $n$-ary\r\noperations and the equations they satisfy. This talk will investigate\r\nsome of the <i>non-standard arities<\/i> detectable in this way, and\r\nexamine how these impinge on other parts of logic.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nA.&nbsp;Blass,\r\nExact functors and measurable cardinals,\r\nPacific Journal of Mathematics,\r\nvol.&nbsp;63 (1976), no.&nbsp;2, pp.&nbsp;335\u2013346.\r\n<br>\r\n[2]\r\nM.&nbsp;Fiore, G.&nbsp;Plotkin, and D.&nbsp;Turi,\r\nAbstract syntax and variable binding,\r\nLogic in Computer Science 14\r\n(Trento),\r\nIEEE Computer Society Press,\r\n1999,\r\npp.&nbsp;193\u2013202.\r\n<br>\r\n[3]\r\nM.&nbsp;Hyland, M.&nbsp;Nagayama, J.&nbsp;Power, and G.&nbsp;Rosolini,\r\nA category theoretic formulation for Engeler-style models of the\r\nuntyped $\\lambda$-calculus,\r\nElectronic Notes in Theoretical Computer Science,\r\nvol.&nbsp;161 (2006), pp.&nbsp;43\u201357.\r\n<br>\r\n[4]\r\nA.&nbsp;Joyal,\r\nFoncteurs analytiques et esp\u00e8ces de structures,\r\nSpringer Lecture Notes in Mathematics,\r\nvol.&nbsp;1234 (1986), pp.&nbsp;126-159.\r\n<br>\r\n[5]\r\nM.&nbsp;Makkai,\r\nThe topos of types,\r\nSpringer Lecture Notes in Mathematics,\r\nvol.&nbsp;859 (1981), pp.&nbsp;157\u2013201.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m65');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:6ex\">12.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nLunch break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td colspan=\"3\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">Special Sessions (titles: see <a href=\"#special5\">below<\/a>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" colspan=\"3\" style=\"height:0ex;\"><\/d>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Model theory and limit structures <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Computability theory <\/i><\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><i>Formal theories of truth <\/i><\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.00<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">14.45<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m67')\">Caroline Terry<\/a>\r\n<div id=\"absbox-m67\" class=\"abstract_box\">\r\nCaroline Terry, <i>An application of model theoretic Ramsey theory<\/i>\r\n<br><br>\r\nChudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs.  In this talk we present joint work with Malliaris, in which we reprove their theorem using many of the same ideas, but with the key model theoretic ingredient of first determining the so-called amount of stability of the graph.  This approach changes the applicable Ramsey theorem, improves the bounds, and offers a different structural perspective on the graphs in question.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m67');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m69')\">Andre Nies<\/a>\r\n<div id=\"absbox-m69\" class=\"abstract_box\">\r\nAndre Nies, <i>Describing finite groups by first-order sentences of polylogarithmic length<\/i>\r\n<br><br>\r\nI discuss recent research with Katrin Tent [1] that connects group theory, logic, and the idea of Kolmogorov complexity.\r\nWe call a class of finite structures for a finite first-order signature R-compressible, for an unbounded function $R$ on the natural numbers, if each structure $G$ in the class has a first-order description of length  at most $O(R(|G|))$. We show that the class of finite simple groups is $\\log$-compressible, and the class of all finite groups  is $\\log^3$  compressible.\r\nThe   results rely on the classification of finite simple groups,   the existence of   profinite presentations with few relators for finite groups, and group cohomology.   We also indicate why the bounds  are  close to optimal.\r\nA much easier result that still conveys the flavour of our  research is the following: for each $n$ there is a first-order sentence of length $O(\\log n)$  expressing that a group has size $n$  (see [2]).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nA.&nbsp;Nies and K.&nbsp;Tent.\r\nDescribing finite groups by short first-order sentences.\r\nIsrael J. of Matematics, to appear.\r\nAvailable at arXiv:1409.8390.\r\n<br>\r\n[2]\r\nA.&nbsp;Nies (editor).\r\nLogic Blog 2016.\r\nAvailable at <tt>cs.auckland.ac.nz\/~nies<\/tt>.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m69');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m71')\">Theodora Achourioti<\/a>\r\n<div id=\"absbox-m71\" class=\"abstract_box\">\r\nTheodora Achourioti, <i>Truth, Intensionality and Paradox<\/i>\r\n<br><br>\r\nWe know since Tarski that the truth of a sufficiently strong first-order theory cannot be captured by means of a truth predicate which satisfies the full $T$-schema. A number of formal theories of type-free truth have been developed that restrict the instances of the $T$-schema in some interesting way so that a consistent theory of truth can be obtained. We present an untyped satisfaction predicate restricted to the geometric fragment of the language and motivated by semantic structures that embody a model of falsification on which the truth of some theory resides. This is a predicate with intensional meaning that it inherits from its semantic environment where the existence of objects and interpretation of predicates depend on what can be seen as some decision process. We show how the consistency of this predicate follows from a certain form of groundedness conditions and we explain what these mean in our context. Finally, we turn to complexity issues and show that even though syntactically severely restricted, this truth predicate does not amount to a simple notion of truth.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m71');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px dotted black;\">\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:3ex\">14.45<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#99CCCC\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">15.30<\/td>\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m73')\">Nathanael Ackerman<\/a>\r\n<div id=\"absbox-m73\" class=\"abstract_box\">\r\nNathanael Ackerman, <i>Hypergraphons, invariant measures, and model theory<\/i>\r\n<br><br>\r\nOver the last decade a rich theory of the limits of sequences of dense finite structures has developed. The limiting objects, known as hypergraphons, lie at a crossroads between probability theory, combinatorics and model theory. In particular, hypergraphons are closely tied to $S_\\infty$-invariant measures on the space of countable structures, and can be seen as a variant of the Aldous-Hoover-Kallenberg representation of such a measure.\r\nIn this talk we will review the basic theory of hypergraphons and will discuss the effects the model theoretic assumption of stability can have on their structure.\r\nThis is joint work with Cameron Freer and Rehana Patel.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m73');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m75')\">Yang Yue<\/a>\r\n<div id=\"absbox-m75\" class=\"abstract_box\">\r\nYang Yue, <i>A Computation Model on Real Numbers<\/i>\r\n<br><br>\r\nThe study of computation models on real numbers has a history almost as long as the one of recursion theory.  Various models have been proposed, for instance, the type-two theory of effectivity (TTE) based on oracle Turing machines and the Blum-Shub-Smale model of computation (BSS).\r\nIn this talk I will identify a class of functions over real numbers, which can be characterized by three equivalent ways: by functional schemes similar to the class of partial recursive functions; by master-slave machines which are generalizations of Turing machines; and by $\\lambda$-calculus with extra $\\delta$-reductions.\r\nThese equivalent characterizations seem to suggest something intrinsic behind this class of functions. The talk is based on joint work with Keng Meng Ng\r\nfrom Nanyang Technological University, Singapore and Nazanin Tavana from Amirkabir University of Technology, Iran; Duccio Pianigiani and Andrea Sorbi from University of Siena, Italy and Jiangjie Qiu from Renming University, China.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m75');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<td align=\"left\" bgcolor=\"#99CCCC\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\"><a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m77')\">Roy Cook<\/a>\r\n<div id=\"absbox-m77\" class=\"abstract_box\">\r\nRoy Cook, <i>Embracing Inference: Three Revenge-Free Deductive Systems for Truth<\/i>\r\n<br><br>\r\nThe Embracing Revenge account, which provides a logical and semantic treatment of the Liar paradox and the revenge phenomena, has been developed in a series of papers by Roy T Cook (2008, 2009), Nicholas Tourville (under review), and Philippe Schlenker (2010). In this paper, after surveying the semantics developed in the most recent such work, I present three deductive systems for the Embracing Revenge view. Each of these three systems reflect distinct ways to \"read\" the semantics, modelled roughly on many-valued (or \"gappy\"), dialethic (or \"glutty\"), and degree-theoretic approaches to semantics.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nRoy T Cook,\r\nEmbracing Revenge: On the Indefinite Extensibility of Language,\r\nRevenge of the Liar\r\n(JC Beall, editors),\r\nOxford University Press\r\n2008,\r\npp.&nbsp;31\u201352.\r\n<br>\r\n[2]\r\n\u2015\r\nWhat is a Truth Value, and How Many Are There?,\r\nStudia Logica,\r\nvol.&nbsp;92 (2009), pp.&nbsp;183\u2013201.\r\n<br>\r\n[3]\r\n\u2015\r\nP. Schlenker,\r\nSuper-Liars,\r\nReview of Symbolic Logic,\r\nvol.&nbsp;3 (2010), no.&nbsp;3, pp.&nbsp;374\u2013414.\r\n<br>\r\n[4]\r\n\u2015\r\nN. Tourville and R. Cook,\r\nEmbracing the Technicalities: Expressive Completeness and Revenge,\r\nunder review,\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m77');\">[Close]<\/a><\/div>\r\n<\/td>\r\n\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">15.30<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">16.00<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:8.33333333333333ex\">16.00<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AADDDD\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">18.05<\/td>\r\n<td align=\"left\" bgcolor=\"#AADDDD\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nContributed talks (see <a href=\"#contrib5\">below<\/a>)\r\n<\/td>\r\n<\/tr>\r\n<\/table><\/p>\r\n<p><table width=\"100%\" style=\"color:black; border: 2px solid black; border-collapse: collapse;\">\r\n<tr style=\"border-top:1px solid black;\">\r\n<td width=\"1px\" bgcolor=\"#AAA\" colspan=\"6\" style=\"text-align:center; padding-left:1ex; padding-right:1ex;\">\r\n<b>Saturday 6th August<\/b>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">9.00<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.00<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Rob Goldblatt, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m79')\"><i>Spatial logic of tangled closure and derivative operators<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-Goldblatt.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m79\" class=\"abstract_box\">\r\nRob Goldblatt, <i>Spatial logic of tangled closure and derivative operators<\/i>\r\n<br><br>\r\nThe tangled closure of a collection of sets is the largest set in which each member of the collection is dense. This operation models a generalised modality that was introduced by Dawar and Otto [1], who showed that its addition to basic propositional modal  logic produces a language that is expressively equivalent over certain classes of finite transitive structures to the bisimulation-invariant fragments of both first-order logic  and monadic second-order logic. (By contrast, over arbitrary structures the bisimulation-invariant fragment of monadic second-order logic is  equivalent to the more powerful modal mu-calculus.) The name `tangle' and its spatial meaning are due to Fern\u00e1ndez-Duque [2].\r\nThis talk surveys joint work [3, 4] with Ian Hodkinson on interpretations of the tangle modality, including a variant in which topological closure is replaced by the derivative (= set of limit points) operation. We prove the finite model property for, and provide complete axiomatisations of, the logics of a range of topological spaces in a number of languages, some with the universal modality. This includes results for all Euclidean spaces $\\mathbb{R}^n$, and all zero-dimensional dense-in-themselves metric spaces. The methods used involve new kinds of `dissections' of metric spaces in the sense of McKinsey and Tarski [5].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAnuj Dawar and Martin Otto,\r\nModal characterisation theorems over special classes of frames,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;161 (2009),  pp.&nbsp;1\u201342.\r\n<br>\r\n[2]\r\nDavid Fern\u00e1ndez-Duque,\r\nTangled modal logic for spatial reasoning,\r\nProceedings of the Twenty-Second International Joint Conference\r\non Artificial Intelligence (IJCAI),\r\n(Toby Walsh, editor),\r\nAAAI Press\/IJCAI,\r\n2011,\r\npp.&nbsp;857\u2013862.\r\n<br>\r\n[3]\r\nRobert Goldblatt and Ian Hodkinson,\r\nSpatial Logic of Modal Mu-Calculus and Tangled Closure Operators (2014),\r\n<tt>arXiv.org\/abs\/1603.01766<\/tt>\r\n<br>\r\n[4]\r\n\u2015\r\nThe Tangled Derivative Logic of the Real Line and Zero-Dimensional Spaces,\r\nAdvances in Modal Logic\r\n(Budapest),\r\n(Lev Beklemishev, St\u00e9phane Demri and Andr\u00e1s M\u00e1t\u00e9, editors),\r\nvol.&nbsp;11,\r\nCollege Publications,\r\n2016.\r\n<br>\r\n[5]\r\nJ. C. C. McKinsey and Alfred Tarski,\r\nThe algebra of topology,\r\nAnnals of Mathematics,\r\nvol.&nbsp;45 (1944), no.&nbsp;1, pp.&nbsp;141\u2013191.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m79');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:2ex\">10.00<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#FED\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">10.30<\/td>\r\n<td align=\"left\" bgcolor=\"#FED\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nCoffee break\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">10.30<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#C9DDFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">11.30<\/td>\r\n<td align=\"left\" bgcolor=\"#C9DDFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nTutorial B: Uri Andrews, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m81')\"><i>Computable model theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-Andrews-3.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m81\" class=\"abstract_box\">\r\nUri Andrews, <i>Computable model theory<\/i>\r\n<br><br>\r\nIn this tutorial series, I will talk about many instances where ideas from computability theory and ideas from model theory intertwine. The focus will be mostly on questions related to computation of models or theories, but I will also try to highlight how computability can help refine our understanding of model theoretic ideas.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m81');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"border-top:1px solid black;\">\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-left:1ex; height:4ex\">11.30<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:center;\">-<\/td>\r\n<td bgcolor=\"#AACCFF\" width=\"1px\" style=\"text-align:right; padding-right:1ex;\">12.30<\/td>\r\n<td align=\"left\" bgcolor=\"#AACCFF\" colspan=\"3\" style=\"padding-left:1ex; padding-right:1ex; border-left: 1px dotted black;\">\r\nPlenary: Boris Zilber, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-m83')\"><i>On the semantics of algebraic quantum mechanics and the role of model theory<\/i><\/a> <a href=\"http:\/\/conferences.leeds.ac.uk\/lc2016\/wp-content\/uploads\/sites\/4\/2017\/01\/s-zilber.pdf\">[Slides]<\/a>\r\n<div id=\"absbox-m83\" class=\"abstract_box\">\r\nBoris Zilber, <i>On the semantics of algebraic quantum mechanics and the role of model theory<\/i>\r\n<br><br>\r\nI will talk about the methods and results of my recent paper `The semantics of the canonical commutation relation' (arxiv.org\/abs\/1604.07745).\r\nThe particular emphasis in this talk will be on how the model-theoretic approach is leading to a novel interpretation of quantum mechanics.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-m83');\">[Close]<\/a><\/div>\r\n\r\n\r\n<\/td>\r\n<\/tr>\r\n<\/table><\/p>\r\n<\/div>\r\n            <h2>Schedule of Special Sessions<\/h2>\r\n\r\n\r\n<table border=\"0\" style=\"border-spacing: 1em .5ex;\" class=noborders>\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"special1\"><\/a>Monday 1st August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Homogeneous Structures <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.15<\/td>\r\n<td style=\"vertical-align: top;\">Ross Willard, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s1')\"><i>The decidable discriminator variety problem<\/i><\/a>\r\n<div id=\"absbox-s1\" class=\"abstract_box\">\r\nRoss Willard, <i>The decidable discriminator variety problem<\/i>\r\n<br><br>\r\nThis talk is an advertisement for an old, unsolved\r\nproblem in which universal algebra meets homogeneous structures.\r\nAn <i>equational class<\/i> is any class in an algebraic signature\r\n(i.e., constants and function symbols only) which is axiomatized by universally\r\nquantified equations.  Such a class is <i>locally finite<\/i> if every finitely generated\r\nsubstructure of a member is finite.  The problem in question is that of describing\r\nall locally finite equational classes with finite signature whose first-order theory\r\nis  decidable.\r\nThe work of Burris, McKenzie and Valeriote [1, 3] in the 1980s reduced\r\nthis problem to two special kinds of equational classes:\r\n<ul>\r\n<li>\r\nFor a given finite ring $R$, the class ${}_R\\mathcal M$ of all $R$-modules.\r\n<li>\r\nLocally finite \"discriminator varieties\" in a finite signature.\r\n<\/ul>\r\nWhat are discriminator varieties?  They are\r\nequational classes $\\mathcal E$ which resemble the\r\nclass of Boolean algebras in certain ways.  In particular,\r\n(i) the class $\\mathcal S$ of simple algebras in $\\mathcal E$ is\r\n$\\forall_1$-axiomatizable, and\r\n(ii)\r\neach algebra in $\\mathcal E$ has a Stone-like representation via a\r\nsheaf over $\\mathcal S$.\r\nIn practice [4, 2],\r\nthe question of whether a locally finite discriminator variety\r\n$\\mathcal E$ has\r\na decidable first-order theory hinges on how well-structured are the members of\r\n$\\mathcal S$,\r\nand in particular\r\non the degree to which the countable members of $\\mathcal S$\r\nfail to be homogeneous.  In this talk I will make this precise and explain the current\r\nstate of the problem.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nStanley Burris and Ralph McKenzie,\r\nDecidability and Boolean representations,\r\nMemoirs of the American Mathematical Society,\r\nvol. 32 (1981), no. 246.\r\n<br>\r\n[2]\r\nDejan Deli\u0107,\r\nDecidable discriminator varieties arising from dihedral varieties of groups,\r\nJournal of Pure and Applied Algebra,\r\nvol. 198 (2005), no. 1\u20133, pp. 75\u201392.\r\n<br>\r\n[3]\r\nRalph McKenzie and Matthew Valeriote,\r\nThe Structure of Decidable Locally Finite Varieties,\r\nProgress in Mathematics, Birkh\u00e4user, Boston, 1989.\r\n<br>\r\n[4]\r\nRoss Willard,\r\nDecidable discriminator varieties from unary classes,\r\nTransactions of the American Mathematical Society,\r\nvol. 336 (1993), no. 1, 311-333.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s1');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">15.00<\/td>\r\n<td style=\"vertical-align: top;\">Josh Wiscons, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s3')\"><i>The status of Cherlin's conjecture for primitive structures of relational complexity 2<\/i><\/a>\r\n<div id=\"absbox-s3\" class=\"abstract_box\">\r\nJosh Wiscons, <i>The status of Cherlin's conjecture for primitive structures of relational complexity 2<\/i>\r\n<br><br>\r\nThe relational complexity of a structure $\\mathbf{X}$ is the least $k&lt;\\omega$ for which the orbits of $\\operatorname{Aut}(\\mathbf{X})$ on $X^k$ \"determine\" the orbits of $\\operatorname{Aut}(\\mathbf{X})$ on $X^n$ for all $n&lt;\\omega$. This invariant originated in Lachlan's classification theory for homogeneous finite \u2013and more generally, countable stable\u2013 relational structures, but not much was known about the complexities of specific structures until de work of Cherlin, Martin and Saracino in the 1990's. In this talk, I will present some background on relational complexity and discuss recent work on Cherlin's conjecture regarding the classification of the finite primitive structures of complexity 2.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s3');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Set Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.15<\/td>\r\n<td style=\"vertical-align: top;\">David Schrittesser, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s5')\"><i>Definable discrete sets, forcing, and Ramsey theory<\/i><\/a>\r\n<div id=\"absbox-s5\" class=\"abstract_box\">\r\nDavid Schrittesser, <i>Definable discrete sets, forcing, and Ramsey theory<\/i>\r\n<br><br>\r\nLet $\\mathcal R$ be a family of finitary relations on a set $X$. A set $A\\subseteq X$ is called $\\mathcal R$-discrete if no relation $R \\in \\mathcal R$ relates any elements of $A$;\r\n$A$ is called maximal discrete if it is maximal with respect to subset-inclusion among $\\mathcal R$-discrete subsets of $X$.\r\nFor any family of relations $\\mathcal R$, maximal $\\mathcal R$-discrete sets exist by the axiom of choice; whether such sets can be <i>definable<\/i> is contentious.\r\nMaximal discrete sets have been widely studied: instances are  maximal co-finitary groups, maximal almost disjoint families, and maximal orthogonal families of measures. In many (but not all) cases one can show such objects cannot be analytic (i.e. projections of closed sets).\r\nOn the contrary, certain definable (in fact, co-analytic) maximal discrete sets have been shown to exist under the assumption that every set is constructible.\r\nWe present some new results, exhibiting definable maximal discrete sets in forcing extensions, e.g. extensions where the continuum hypothesis fails.\r\nIn most cases, these results rely on Ramsey theoretic considerations.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s5');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">15.00<\/td>\r\n<td style=\"vertical-align: top;\">Anush Tserunyan, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s7')\"><i>Integer cost and ergodic actions<\/i><\/a>\r\n<div id=\"absbox-s7\" class=\"abstract_box\">\r\nAnush Tserunyan, <i>Integer cost and ergodic actions<\/i>\r\n<br><br>\r\nA countable Borel equivalence relation $E$ on a probability space can always be generated in two ways: as the orbit equivalence relation of a Borel action of a countable group and as the connectedness relation of a locally countable Borel graph, called a <i>graphing<\/i> of $E$. Assuming that $E$ is measure-preserving, graphings provide a numerical invariant called <i>cost<\/i>, whose theory has been largely developed and used by Gaboriau and others in establishing rigidity results. A well-known theorem of Hjorth states that when $E$ is ergodic, treeable (admits an acyclic graphing), and has integer cost $n \\ge 1$, then it is generated by an a.e. free measure-preserving action of the free group $\\mathbf{F}_n$ on $n$ generators. We give a simpler proof of this theorem and the technique of our proof, combined with a recent theorem of Tucker-Drob, yields a strengthening of Hjorth's theorem: the action of $\\mathbf{F}_n$ can be arranged so that each of the $n$ generators acts ergodically.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s7');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Proof Theory and Reverse Mathematics <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.15<\/td>\r\n<td style=\"vertical-align: top;\">Sam Sanders, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s9')\"><i>The unreasonable effectiveness of Nonstandard Analysis<\/i><\/a>\r\n<div id=\"absbox-s9\" class=\"abstract_box\">\r\nSam Sanders, <i>The unreasonable effectiveness of Nonstandard Analysis<\/i>\r\n<br><br>\r\nAs suggested by the title, we will uncover the vast <i>computational content<\/i> of <i>classical<\/i> Nonstandard Analysis.\r\nTo this end, we formulate a template $\\mathfrak{CI}$ which converts a theorem of `pure' Nonstandard Analysis, i.e. formulated solely with the <i>nonstandard<\/i> definitions (of continuity, integration, differentiability, convergence, compactness, et cetera), into the associated <i>effective<\/i> theorem.  The latter constitutes a theorem of computable mathematics <i>no longer involving Nonstandard Analysis<\/i>.  The template often produces theorems of Bishop's <i>Constructive Analysis<\/i> ([1]).\r\n\\medskip\r\nTo establish the vast scope of $\\mathfrak{CI}$, we apply this template to representative theorems from the <i>Big Five<\/i> categories from <i>Reverse Mathematics<\/i> ([3, 5]).   The latter foundational program provides a classification of the majority of theorems from `ordinary', that is non-set theoretical, mathematics into the aforementioned five categories.  The <i>Reverse Mathematics zoo<\/i> ([2]) gathers exceptions to this classification, and is studied in [4] using $\\mathfrak{CI}$.  Hence, the template $\\mathfrak{CI}$ is seen to apply to essentially <i>all of ordinary mathematics<\/i>, thanks to the Big Five classification (and associated zoo) from Reverse Mathematics.\r\nFinally, we establish that certain `highly constructive' theorems, called Herbrandisations, imply the original theorem of Nonstandard Analysis from which they were obtained via $\\mathfrak{CI}$.\r\n<br>\r\n<b>Acknowledgement.<\/b> This research is generously sponsored by the John Templeton Foundation and the Alexander Von Humboldt Foundation.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nE. Bishop,\r\nD. Bridgess\r\nConstructive analysis,\r\nGrundlehren der Mathematischen Wissenschaften, vol. 279, Springer, 1985, xii+477\r\n<br>\r\n[2]\r\nD. Dzhafarov, The Reverse Mathematics zoo, {http:\/\/rmzoo.uconn.edu\/}\r\n<br>\r\n[3]\r\nU. Kohlenbach,\r\nHigher-order Reverse Mathematics,\r\nLect. Notes Log., Reverse Mathematics 2001, vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 281-295\r\n<br>\r\n[4]\r\nS. Sanders,\r\nThe refining of the taming of the Reverse Mathematics zoo, to appear in\r\nNotre Dame Journal for formal logic, 2016\r\n<br>\r\n[5]\r\nS. Simpson,\r\nSubsystems of Second-order Arithmetic,\r\n2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s9');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">15.00<\/td>\r\n<td style=\"vertical-align: top;\">David R. Belanger, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s11')\"><i>A computable perfect-set theorem<\/i><\/a>\r\n<div id=\"absbox-s11\" class=\"abstract_box\">\r\nDavid R. Belanger, <i>A computable perfect-set theorem<\/i>\r\n<br><br>\r\nWe gauge the difficulty of finding a perfect subtree in a tree of a given Cantor-Bendixson rank.  To simplify the analysis we introduce <i>half-derivative<\/i>, and extend the definition of rank to include values of the form $n$-and-a-half; each increase of one-half in the rank corresponds to one added jump in the perfect-subtree problem.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s11');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\"><br><\/td><td style=\"vertical-align: top;\"><br><\/td><\/tr>\r\n\r\n\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"special2\"><\/a>Tuesday 2nd August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Homogeneous Structures <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Jan Hubicka, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s13')\"><i>All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)<\/i><\/a>\r\n<div id=\"absbox-s13\" class=\"abstract_box\">\r\nJan Hubicka, <i>All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)<\/i>\r\n<br><br>\r\nClass $\\mathcal K$ of finite structures is Ramsey class if for every choice of\r\n$\\mathbf A,\\mathbf B\\in K$ there exists $\\mathbf C\\in \\mathcal K$ such that for every\r\ncoloring of its substructures isomorphic to $\\mathbf A$ with 2 colors there\r\nexists an isomorphic copy of $\\mathbf B$ in $\\mathbf C$ where all copies of\r\n$\\mathbf A$ are monochromatic.\r\nIt is a classical result that for every purely relational language $L$ the class of all\r\nfinite ordered $L$-structures  is Ramsey&nbsp;[5, 1].\r\nWe extend this theorem for languages containing both relations and functions.\r\nWe also give a new sufficient condition for subclass of a Ramsey class to be Ramsey.\r\nBy verifying this condition we prove Ramsey property of many classes such as\r\nconvexly ordered $S$-metric spaces (solving an open problem&nbsp;[6]), totally ordered structures (structures with linear order on both vertices and relations), and\r\nordered single constraint Cherlin Shelah Shi classes&nbsp;[2].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nF.&nbsp;G.&nbsp;Abramson, L.&nbsp;A.&nbsp;Harrington,\r\nModels without indiscernibles,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;43 (1978), no.&nbsp;3, pp.&nbsp;572\u2013600.\r\n<br>\r\n[2]\r\nG.&nbsp;Cherlin, S.&nbsp;Shelah, N.&nbsp;Shi,\r\nUniversal graphs with forbidden subgraphs and algebraic closure,\r\nAdvances in Applied Mathematics,\r\nvol.&nbsp;22 (1999), no.&nbsp;4, pp.&nbsp;454\u2013491.\r\n<br>\r\n[3]\r\nJ.&nbsp;Hubi\u010dka, J.&nbsp;Ne\u0161et\u0159il,\r\nBowtie-free graphs have a {R}amsey lift,\r\narXiv preprint, arXiv:1402.2700.\r\n<br>\r\n[4]\r\nJ.&nbsp;Hubi\u010dka, J.&nbsp;Ne\u0161et\u0159il,\r\nAll those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms),\r\nin preparation.\r\n<br>\r\n[5]\r\nJ.&nbsp;Ne\u0161et\u0159il, V.&nbsp;R\u00f6dl,\r\nPartitions of Finite Relational and Set Systems,\r\nJournal Combinatorial Theory, Series A,\r\nvol.&nbsp;22 (1977), no.&nbsp;3, pp.&nbsp;289\u2013312.\r\n<br>\r\n[6]\r\nL.&nbsp;Nguyen Van Th\u00e9,\r\nStructural {R}amsey Theory of Metric Spaces and Topological Dynamics of Isometry Groups,\r\nMemoirs of the American Mathematical Society,\r\nAmerican Mathematical Society,\r\n2010.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s13');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Libor Barto, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s15')\"><i>The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems<\/i><\/a>\r\n<div id=\"absbox-s15\" class=\"abstract_box\">\r\nLibor Barto, <i>The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems<\/i>\r\n<br><br>\r\nWe prove that an $\\omega$-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations $\\alpha$, $\\beta$, $s$ satisfying the identity $\\alpha s(x,y,x,z,y,z) \\approx \\beta s(y,x,z,x,z,y)$.\r\nThis establishes an algebraic criterion equivalent to the conjectured borderline between P and NP-complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case.\r\nOur theorem is also of independent mathematical interest, characterizing a topological property of any $\\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).\r\nThis is a joint work with Michael Pinsker.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s15');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Set Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Nam Trang, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s17')\"><i>Large cardinals, determinacy, and forcing axioms<\/i><\/a>\r\n<div id=\"absbox-s17\" class=\"abstract_box\">\r\nNam Trang, <i>Large cardinals, determinacy, and forcing axioms<\/i>\r\n<br><br>\r\nWe discuss some recent progress in descriptive inner model theory. In particular, we discuss some current results concerning connections of the three hierarchies of models: canonical models of large cardinals (pure extender models), canonical models of determinacy, and strategic hybrid models (e.g. HOD of determinacy models). These structural results can be used to improve (lower-bound) consistency strength of strong combinatorial principles such as The Proper Forcing Axiom ($\\sf{PFA}$), strong forms of the tree property etc. In particular, I proved that $\\sf{PFA}$ implies the existence of a transitive model of ``$\\sf{AD}_\\mathbb{R} + \\Theta$ is regular\". Building on this and structural results above, G. Sargsyan and I have constructed models of theory $\\sf{LSA} =_{\\rm{def}} ``\\sf{AD}^+ + $there is an $\\alpha$ such that $\\Theta=\\theta_{\\alpha+1} + \\theta_\\alpha$ is the largest Suslin cardinal\" from $\\sf{PFA}$. This result is the strongest of its kind and reflects our current understanding of HOD of models of determinacy.\r\n\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s17');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Andrew Marks, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s19')\"><i>Borel and measurable matchings<\/i><\/a>\r\n<div id=\"absbox-s19\" class=\"abstract_box\">\r\nAndrew Marks, <i>Borel and measurable matchings<\/i>\r\n<br><br>\r\nWe discuss several results related to the question of when a\r\nBorel graph has a Borel matching. Here, the analogue of Hall's\r\nmatching theorem fails, but there are positive results giving Borel\r\nmatchings in several contexts if we are willing to discard null or\r\nmeager sets, or restrict the types of graphs we consider. We also\r\ndiscuss some applications to geometrical paradoxes.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s19');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Proof Theory and Reverse Mathematics <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Lev Gordeev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s21')\"><i>On Harvey Friedman's finite phase transitions<\/i><\/a>\r\n<div id=\"absbox-s21\" class=\"abstract_box\">\r\nLev Gordeev, <i>On Harvey Friedman's finite phase transitions<\/i>\r\n<br><br>\r\n<b>Definition<\/b> (H. Friedman).\r\nThe<i> proof theoretic integer<\/i> of formal system $\\mathbf{T}$\r\n(abbreviation: $\\mathrm{PTI}\\left( \\mathbf{T}\\right) $ ) is the least\r\ninteger $n$ such that every $\\Sigma _{1}^{0}$ sentence\r\n\\begin{equation*}\r\n\\exists x_{1}\\cdots \\exists x_{m}A\\left( x_{1},\\cdots ,x_{m}\\right)\r\n\\end{equation*}\r\nthat has a proof in $\\mathbf{T}$ with at most $10,000$ symbols, has\r\nwitnesses $x_{1},\\cdots ,x_{m}&lt;n$. (Actually $m=1$ would suffice.)\r\n<br><br>\r\nA good source of examples is in the area surrounding Kruskal's theorem. This\r\ntalk is devoted to $\\mathrm{PTI}\\left( \\mathbf{T}\\right) $'s basic\r\nproperties, examples, comparisons and related phase transitions.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s21');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Florian Pelupessy, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s23')\"><i>Ramsey like principles and well-foundedness of $d$-height $\\omega$-towers<\/i><\/a>\r\n<div id=\"absbox-s23\" class=\"abstract_box\">\r\nFlorian Pelupessy, <i>Ramsey like principles and well-foundedness of $d$-height $\\omega$-towers<\/i>\r\n<br><br>\r\nRecently it has been highlighted by Kreuzer and Yokoyama [1] that, over $\\mathrm{RCA}_0$, there are many principles equivalent to the well foundedness of the ordinal $\\omega^\\omega$. We will observe that there are Ramsey-like principles which are equivalent to the well-foundedness of $\\omega_d$, where $\\omega_0=1$ and $\\omega_{n+1}=\\omega^{\\omega_n}$. One of these examples is based on the relativised Paris\u2013Harrington principle as mentioned in [1], but for dimension $d$. The more interesting example is the restricton of Friedman's adjacent Ramsey theorem to fixed dimension $d$, which is equivalent to the well-foundedness of $\\omega_{d+1}$.\r\n<br><br>\r\n<b>Definition<\/b> (adjacent Ramsey in dimension $d$).\r\nFor every $C\\colon \\mathbb{N}^d \\rightarrow \\mathbb{N}^r$ there exist $x_0 &lt; \\dots &lt; x_{d+1}$ such that $C(x_1, \\dots , x_d) \\leq C(x_2, \\dots , x_{d+1})$, where $\\leq$ is the coordinatewise ordering.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAlexander P. Kreuzer and Keita Yokoyama,\r\nOn principles between $\\Sigma_1$ and $\\Sigma_2$ induction and monotone enumerations,\r\n\r\narXiv:1306.1936v5.\r\n<br>\r\n[2]\r\nStephen G. Simpson,\r\nSubsystems of second order arithmetic,\r\nPerspectives in logic (2nd edition),\r\nCambridge University Press,\r\n2009.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s23');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\"><br><\/td><td style=\"vertical-align: top;\"><br><\/td><\/tr>\r\n\r\n\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"special4\"><\/a>Thursday 4th August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory and Limit Structures <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Patrice Ossona de Mendez, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s25')\"><i>Structural Limits and Clustering near Infinity<\/i><\/a>\r\n<div id=\"absbox-s25\" class=\"abstract_box\">\r\nPatrice Ossona de Mendez, <i>Structural Limits and Clustering near Infinity<\/i>\r\n<br><br>\r\nStructural limits (including FO-limits and X-limits for various fragments X) arise as a natural generalization of limits of bothe dense and bounded degree graphs, yet having a distinctive model theoretic flavour. It is in a way dual approach which allows to prove distributional limits in a full generality. More recently it leads to clustering which seems to be interesting from both analytic and model theoretic perspective. This can be also outlined as follows:\r\nThe cluster analysis of very large objects is an important problem, which spans several\r\ntheoretical as well as applied branches of mathematics and computer science.\r\nHere we suggest a novel approach: under assumption of local convergence of a sequence of\r\nfinite structures we derive an asymptotic clustering. This is achieved by a blend of\r\nanalytic and geometric techniques, and particularly by a new interpretation of the authors?\r\nrepre-sentation theorem for limits of local convergent sequences, which serves as a\r\nguidance for the whole process. Our study may be seen as an effort to describe connectivity\r\nstructure at the limit (without having a defined explicit limit structure) and to pull this\r\nconnectivity structure back to the finite structures in the sequence in a continuous way.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s25');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Ove Ahlman, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s27')\"><i>Simple structures axiomatized by almost sure theories<\/i><\/a>\r\n<div id=\"absbox-s27\" class=\"abstract_box\">\r\nOve Ahlman, <i>Simple structures axiomatized by almost sure theories<\/i>\r\n<br><br>\r\nStudying simple structures, one of the nicest examples of a simple yet not stable structure is the Rado graph.\r\nThe Rado graph is the unique countable graph $\\mathcal{G}$ which satisfies that for any finite disjoint subsets $A$ and $B$ of $\\mathcal{G}$ there is a vertex $c$ such that $c$ is adjacent to all vertices in $A$ and no vertices in $B$. Some of the properties of the Rado graph which makes it especially nice include $\\omega-$categoricity (even homogeneity), SU-rank $1$ and trivial algebraic closure.\r\n<br>\r\nFor each $n\\in\\mathbb{N}$ let $\\mathbf{K}_n$ be a set of finite structures and let $\\mu_n$ be the uniform probability measure on $\\mathbf{K}_n$, assigning the same probability to each structure. This meas-sure induce a probability measure on first order sentences $\\varphi$ by putting $\\mu_n(\\varphi)=\\mu(\\{\\mathcal{M}\\in\\mathbf{K}_n: \\mathcal{M}\\models \\varphi\\})$. Put $T$ to be the theory, called the almost sure theory, of all first order sentences $\\varphi$ such that $\\lim_{n\\rightarrow\\infty} \\mu_n(\\varphi)= 1$. We directly see that $T$ is complete if and only if each sentence has asymptotic probability $0$ or $1$, in which case we say that the pair $(\\mathbf{K}_n,\\mu_n)_{n\\in\\mathbb{N}}$ has a $0-1$ law. Sets which have a $0-1$ law under the uniform probability measure include all finite structures, all finite partial orders, all finite $l-$colorable graphs and many more.\r\n<br>\r\nOne can easily show that the Rado graph is not only axiomatized by the vertex-extension property described above, but can also be seen as the unique countable structure satisfying the almost sure theory coming from $\\mathbf{K}_n$ consisting of all graphs with universe $\\{1,\\ldots,n\\}$. Further more many other almost sure theories, including all of the above mentioned examples, are $\\omega-$categorical and simple with $SU-$rank 1. In this talk we will discuss why this connection occurs and see that the binary simple $\\omega-$categorical structures with SU-rank $1$ is in direct connection with almost sure theories.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s27');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Computability Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Rupert H\u00f6lzl, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s29')\"><i>Randomness for computable measures and initial segment complexity<\/i><\/a>\r\n<div id=\"absbox-s29\" class=\"abstract_box\">\r\nRupert H\u00f6lzl, <i>Randomness for computable measures and initial segment complexity<\/i>\r\n<br><br>\r\nThe Levin-Schnorr theorem establishes the equivalence of a certain measure-theoretic notion of typicality for infinite sequences (known as Martin-L\u00f6f randomness) with a notion of incompressibility given in terms of Kolmogorov complexity.  Although the Levin-Schnorr theorem is usually formulated for sequences that are random with respect to the Lebesgue measure on $2^\\omega$, it is well known that the theorem can be generalized to hold for any computable probability measure on $2^\\omega$.  More specifically, a sequence $X\\in2^\\omega$ is Martin-L\u00f6f random with respect to a computable measure&nbsp;$\\mu$ if and only if the initial segment complexity of $X\\upharpoonright n$ is bounded from below by $-\\log\\mu(X\\upharpoonright n)$.  Thus we see that certain values of the measure $\\mu$ constrain the possible values of the initial segment complexities of the $\\mu$-random sequences.\r\nIn this study, we further explore the interaction between computable measures and the initial segment complexity of the sequences that are random with respect to these measures (hereafter, we will refer to those sequences that are random with respect to a computable measure as <i>proper<\/i> sequences, following the terminology of Zvonkin and Levin&nbsp;[6]).  We focus in particular on the growth rates of functions of the form $f(X,n)=-\\log\\mu(X\\upharpoonright n)$ for various computable measures $\\mu$ and $\\mu$-random sequences $X$.  As we demonstrate, these growth rates can vary widely, depending on the choice of the underlying measure $\\mu$.\r\nIn the first half of the paper, we focus on the relationship between a class of sequences known as <i>complex sequences<\/i> and those sequences that are random with respect to a computable, continuous measure.\r\nFirst studied systematically by Kjos-Hanssen et al.&nbsp;[4] (but also studied earlier by Kanovi\u010d&nbsp;[2]), complex sequences are those sequences whose initial segment complexities are bounded below by some computable function.   We characterize the complex proper sequences as the sequences that are random with respect to some computable <i>continuous<\/i> measure. This is done by studying the \"removability\" of $\\mu$-atoms, that is, sequences&nbsp;$X$ such that $\\mu(\\{X\\})>0$. We show that if a sequence $X$ is complex and random with respect to some computable measure $\\mu$, we can define a computable, continuous measure $\\nu$ such that $X$ is random with respect to $\\nu$ by removing the $\\mu$-atoms that are in some sense near $X$.  It is natural to ask whether this removal of atoms can always be carried out while preserving all non-atomic random sequences simultaneously, again assuming that all of these random sequences are complex. We show that this is not the case.\r\nUsing this characterization of complex sequences through computable continuous measures, we establish new results on the relationship between the notions of avoidability, hyperavoidability, semigenericity, and not being random for any computable, continuous measure.  More specifically, when restricted to the collection of proper sequences, we show that these four notions are equivalent to being complex.  We also study the granularity of a computable, continuous measure&nbsp;$\\mu$ and show that the inverse of the granularity function provides a uniform lower bound for the initial segment complexity of $\\mu$-random sequences.\r\nIn the second half of the paper, we turn our attention to atomic computable measures, i.e., computable measures $\\mu$ that have $\\mu$-atoms.  First, we study atomic measures $\\mu$ with the property that every $\\mu$-random sequence is either a $\\mu$-atom or is complex.  We show that for such measures&nbsp;$\\mu$, even though the initial segment complexity of each non-atom $\\mu$-random sequence is bounded from below by some computable function, there is in general no uniform computable lower bound for every non-atom $\\mu$-random sequence.  Next, we construct a computable atomic measure $\\mu$ with the property that the initial segment complexity of each $\\mu$-random sequence dominates no computable function, and a computable atomic measure $\\nu$ with the property that the initial segment complexity of each $\\nu$-random sequence\r\nis dominated by all computable functions.  The former sequences are called <i>infinitely often anti-complex<\/i>, while the latter are known simply as <i>anti-complex<\/i>.\r\nLastly, we study two specific kinds of atomic measures:  diminutive measures and trivial measures. Here, a measure $\\mu$ is <i>trivial<\/i> if $\\mu(\\mathrm{Atoms}_\\mu)=1$, and diminutive measures are defined as follows.\r\n<br><br>\r\n<b>Definition<\/b>.   Let $\\mathcal{C}\\subseteq2^\\omega$.\r\n<br>\r\n(i) $\\mathcal{C}$ is <i>diminutive<\/i> if it does not contain a computably perfect subclass.\r\n<br>\r\n(ii) Let $\\mu$ be a computable measure, and let $(\\mathcal{U}_i)_{i\\in\\omega}$ be the universal $\\mu$-Martin-L\u00f6f test.  Then we say that $\\mu$ is <i>diminutive<\/i> if $\\mathcal{U}_i^c$ is a diminutive $\\Pi^0_1$ class for every $i$.\r\n<br><br>\r\nWe show that while every computable trivial measure is diminutive, the converse does not hold. The proof of this last statement gives an alternative, priority-free proof of the following known result.\r\n<br><br>\r\n<b>Corollary<\/b> (Kautz&nbsp;[3]).\r\nThere is a computable, non-trivial measure $\\mu$ such that there is no $\\Delta^0_2$, non-computable $X\\in\\mathrm{MLR}_\\mu$.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nR.&nbsp;H\u00f6lzl and C.&nbsp;P. Porter,\r\nRandomness for Computable Measures and Initial Segment Complexity,\r\nArXiv e-prints, October 2015, 1510.07202.\r\n<br>\r\n[2]\r\nMax&nbsp;I. Kanovi\u010d,\r\nThe complexity of the enumeration and solvability of predicates,\r\nDoklady Akademii nauk SSSR,\r\nvol.&nbsp;190 (1970), pp.&nbsp;23\u201326.\r\n<br>\r\n[3]\r\nSteven&nbsp;M. Kautz,\r\nDegrees of random sets,\r\nPhD thesis, Cornell University, 1991.\r\n<br>\r\n[4]\r\nBj\u00f8rn Kjos-Hanssen, Wolfgang Merkle, and Frank Stephan,\r\nKolmogorov complexity and the recursion theorem,\r\nTransactions of the American Mathematical Society,\r\nvol.&nbsp;363 (2011), no.&nbsp;10, pp.&nbsp;5465\u20135480.\r\n<br>\r\n[5]\r\nChristopher Porter,\r\nMathematical and Philosophical Perspectives on Algorithmic\r\nRandomness,\r\nPhD thesis, University of Notre Dame, 2012.\r\n<br>\r\n[6]\r\nAlexander&nbsp;K. Zvonkin and Leonid&nbsp;A. Levin,\r\nThe complexity of finite objects and the basing of the concepts of\r\ninformation and randomness on the theory of algorithms,\r\nUspekhi Matematicheskikh Nauk,\r\nvol.&nbsp;25 (1970), no.&nbsp;6(156), pp.&nbsp;85\u2013127.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s29');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Chris Conidis, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s31')\"><i>New directions in reverse algebra<\/i><\/a>\r\n<div id=\"absbox-s31\" class=\"abstract_box\">\r\nChris Conidis, <i>New directions in reverse algebra<\/i>\r\n<br><br>\r\n\r\nWe will survey some old results in the reverse mathematics of Artinian and Noetherian rings, showing how computability-theoretic insights can give rise to new algebraic techniques. Finally, we will conclude with some new results concerning the reverse mathematics of the (very general) class of Noetherian rings.\r\n<br><br><b>References<\/b>\r\n<br>[1]\r\n Conidis, C.J.,\r\n Chain conditions in computable rings,\r\n Transactions of the American Mathematical Society,\r\nvol.&nbsp;362 (2010), no.&nbsp;12, pp.&nbsp;6523\u20136550.\r\n<br>[2]\r\n\u2015\r\n The computability, definability, and proof theory of Artinian rings,\r\n To appear.\r\n<br>[3]\r\n\u2015\r\n The meta-metamathematics of Neotherian rings,\r\n In preparation.\r\n<br>\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s31');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Formal Theories of Truth <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Volker Halbach, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s33')\"><i>Axiomatizing Kripke's theory of truth in classical and nonclassical logic<\/i><\/a>\r\n<div id=\"absbox-s33\" class=\"abstract_box\">\r\nVolker Halbach, <i>Axiomatizing Kripke's theory of truth in classical and nonclassical logic<\/i>\r\n<br><br>\r\nWe consider axiomatizations of Kripke's [3] theory of truth, more precisely, of all fixed points of Kripke's operator based on Strong Kleene logic over the standard model over arithmetic. We also consider analogous models with gluts as considered by [4]. Feferman [1] axiomatized Kripke's theory in classical logic. Feferman's axiomatization is sound in the sense that whenever $T\\ulcorner\\phi \\urcorner$ is provable then $\\phi $ holds in all of Kripke's fixed-point models.  Halbach and Horsten [2] provided an axiomatization that is directly sound: $\\phi $ holds in all of Kripke's fixed-point models, if $\\phi $ is provable in their system. Halbach and Horsten [2] showed that their system is proof-theoretically much weaker than Feferman's system and lacks many arithmetical theorems provable in Feferman's system. There are many sentences such that $T\\ulcorner\\phi \\urcorner$ is provable in Feferman's system, while $\\phi $ isn't provable in the Halbach\u2013Horsten system.\r\nWe pinpoint the source of the deductive weakness of the Halbach\u2013Horsten system in the induction rule. Both systems contain the same rule of induction. If it is removed from both systems, then $T\\ulcorner\\phi \\urcorner$ is provable in Feferman's system iff $\\phi $ is provable in the Halbach\u2013Horsten system. Thus the effect of switching from classical logic to the nonclassical logic of the Halbach\u2013Horsten system limits the usability of the mathematical principle of induction. We take this as evidence that the often advocated strategy of restricting classical logic for semantic vocabulary doesn't necessarily affect our semantic reasoning, but it can cripple our mathematical reasoning.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSolomon Feferman,\r\nReflecting on incompleteness,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;56 (1991), no.&nbsp;1, pp.&nbsp;1\u201349.\r\n<br>\r\n[2]\r\nVolker Halbach and Leon Horsten,\r\nAxiomatizing Kripke's theory of truth,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;71 (2006), no.&nbsp;2, pp.&nbsp;677\u2013712.\r\n<br>\r\n[3]\r\nSaul A. Kripke,\r\nOutline of a Theory of Truth,\r\nJournal of Philosophy,\r\nvol.&nbsp;72 (1975), no.&nbsp;19, pp.&nbsp;690\u2013716.\r\n<br>\r\n[4]\r\nAlbert Visser,\r\nFour-valued semantics and the liar,\r\nJournal of Philosophical Logic,\r\nvol.&nbsp;13 (1984), no.&nbsp;2, pp.&nbsp;181\u2013212.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s33');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Lavinia Picollo, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s35')\"><i>Truth, Reference and Disquotation<\/i><\/a>\r\n<div id=\"absbox-s35\" class=\"abstract_box\">\r\nLavinia Picollo, <i>Truth, Reference and Disquotation<\/i>\r\n<br><br>\r\nI first provide intuitively appealing notions of reference, self-reference, and well-foundedness of sentences of the language of first-order Peano arithmetic extended with a truth predicate. They are intended as a tool for studying reference patterns that underlie expressions leading to semantic paradox, and thus to shed light on the debate on whether every paradox formulated in a first-order language involves self-reference or some other vicious reference pattern.\r\nI use the new notions to formulate sensible restrictions on the acceptable instances of the T-schema, to carry out the disquotationalist project. Since the concept of reference I put forward is proof-theoretic\\textemdash i.e., it turns to the provability predicate rather than the truth predicate\\textemdash and, therefore, arithmetically definable,  it can be used to provide recursive axiomatizations of truth. I show the resulting systems are $\\omega$-consistent and as strong as Tarski's theory of ramified truth iterated up to $\\epsilon_0$.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s35');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\"><br><\/td><td style=\"vertical-align: top;\"><br><\/td><\/tr>\r\n\r\n\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"special5\"><\/a>Friday 5th August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory and Limit Structures <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Caroline Terry, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s37')\"><i>An application of model theoretic Ramsey theory<\/i><\/a>\r\n<div id=\"absbox-s37\" class=\"abstract_box\">\r\nCaroline Terry, <i>An application of model theoretic Ramsey theory<\/i>\r\n<br><br>\r\nChudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs.  In this talk we present joint work with Malliaris, in which we reprove their theorem using many of the same ideas, but with the key model theoretic ingredient of first determining the so-called amount of stability of the graph.  This approach changes the applicable Ramsey theorem, improves the bounds, and offers a different structural perspective on the graphs in question.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s37');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Nathanael Ackerman, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s39')\"><i>Hypergraphons, invariant measures, and model theory<\/i><\/a>\r\n<div id=\"absbox-s39\" class=\"abstract_box\">\r\nNathanael Ackerman, <i>Hypergraphons, invariant measures, and model theory<\/i>\r\n<br><br>\r\nOver the last decade a rich theory of the limits of sequences of dense finite structures has developed. The limiting objects, known as hypergraphons, lie at a crossroads between probability theory, combinatorics and model theory. In particular, hypergraphons are closely tied to $S_\\infty$-invariant measures on the space of countable structures, and can be seen as a variant of the Aldous-Hoover-Kallenberg representation of such a measure.\r\nIn this talk we will review the basic theory of hypergraphons and will discuss the effects the model theoretic assumption of stability can have on their structure.\r\nThis is joint work with Cameron Freer and Rehana Patel.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s39');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Computability Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Andre Nies, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s41')\"><i>Describing finite groups by first-order sentences of polylogarithmic length<\/i><\/a>\r\n<div id=\"absbox-s41\" class=\"abstract_box\">\r\nAndre Nies, <i>Describing finite groups by first-order sentences of polylogarithmic length<\/i>\r\n<br><br>\r\nI discuss recent research with Katrin Tent [1] that connects group theory, logic, and the idea of Kolmogorov complexity.\r\nWe call a class of finite structures for a finite first-order signature R-compressible, for an unbounded function $R$ on the natural numbers, if each structure $G$ in the class has a first-order description of length  at most $O(R(|G|))$. We show that the class of finite simple groups is $\\log$-compressible, and the class of all finite groups  is $\\log^3$  compressible.\r\nThe   results rely on the classification of finite simple groups,   the existence of   profinite presentations with few relators for finite groups, and group cohomology.   We also indicate why the bounds  are  close to optimal.\r\nA much easier result that still conveys the flavour of our  research is the following: for each $n$ there is a first-order sentence of length $O(\\log n)$  expressing that a group has size $n$  (see [2]).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nA.&nbsp;Nies and K.&nbsp;Tent.\r\nDescribing finite groups by short first-order sentences.\r\nIsrael J. of Matematics, to appear.\r\nAvailable at arXiv:1409.8390.\r\n<br>\r\n[2]\r\nA.&nbsp;Nies (editor).\r\nLogic Blog 2016.\r\nAvailable at <tt>cs.auckland.ac.nz\/~nies<\/tt>.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s41');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Yang Yue, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s43')\"><i>A Computation Model on Real Numbers<\/i><\/a>\r\n<div id=\"absbox-s43\" class=\"abstract_box\">\r\nYang Yue, <i>A Computation Model on Real Numbers<\/i>\r\n<br><br>\r\nThe study of computation models on real numbers has a history almost as long as the one of recursion theory.  Various models have been proposed, for instance, the type-two theory of effectivity (TTE) based on oracle Turing machines and the Blum-Shub-Smale model of computation (BSS).\r\nIn this talk I will identify a class of functions over real numbers, which can be characterized by three equivalent ways: by functional schemes similar to the class of partial recursive functions; by master-slave machines which are generalizations of Turing machines; and by $\\lambda$-calculus with extra $\\delta$-reductions.\r\nThese equivalent characterizations seem to suggest something intrinsic behind this class of functions. The talk is based on joint work with Keng Meng Ng\r\nfrom Nanyang Technological University, Singapore and Nazanin Tavana from Amirkabir University of Technology, Iran; Duccio Pianigiani and Andrea Sorbi from University of Siena, Italy and Jiangjie Qiu from Renming University, China.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s43');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Formal Theories of Truth <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">14.00<\/td>\r\n<td style=\"vertical-align: top;\">Theodora Achourioti, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s45')\"><i>Truth, Intensionality and Paradox<\/i><\/a>\r\n<div id=\"absbox-s45\" class=\"abstract_box\">\r\nTheodora Achourioti, <i>Truth, Intensionality and Paradox<\/i>\r\n<br><br>\r\nWe know since Tarski that the truth of a sufficiently strong first-order theory cannot be captured by means of a truth predicate which satisfies the full $T$-schema. A number of formal theories of type-free truth have been developed that restrict the instances of the $T$-schema in some interesting way so that a consistent theory of truth can be obtained. We present an untyped satisfaction predicate restricted to the geometric fragment of the language and motivated by semantic structures that embody a model of falsification on which the truth of some theory resides. This is a predicate with intensional meaning that it inherits from its semantic environment where the existence of objects and interpretation of predicates depend on what can be seen as some decision process. We show how the consistency of this predicate follows from a certain form of groundedness conditions and we explain what these mean in our context. Finally, we turn to complexity issues and show that even though syntactically severely restricted, this truth predicate does not amount to a simple notion of truth.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s45');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">14.45<\/td>\r\n<td style=\"vertical-align: top;\">Roy Cook, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-s47')\"><i>Embracing Inference: Three Revenge-Free Deductive Systems for Truth<\/i><\/a>\r\n<div id=\"absbox-s47\" class=\"abstract_box\">\r\nRoy Cook, <i>Embracing Inference: Three Revenge-Free Deductive Systems for Truth<\/i>\r\n<br><br>\r\nThe Embracing Revenge account, which provides a logical and semantic treatment of the Liar paradox and the revenge phenomena, has been developed in a series of papers by Roy T Cook (2008, 2009), Nicholas Tourville (under review), and Philippe Schlenker (2010). In this paper, after surveying the semantics developed in the most recent such work, I present three deductive systems for the Embracing Revenge view. Each of these three systems reflect distinct ways to \"read\" the semantics, modelled roughly on many-valued (or \"gappy\"), dialethic (or \"glutty\"), and degree-theoretic approaches to semantics.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nRoy T Cook,\r\nEmbracing Revenge: On the Indefinite Extensibility of Language,\r\nRevenge of the Liar\r\n(JC Beall, editors),\r\nOxford University Press\r\n2008,\r\npp.&nbsp;31\u201352.\r\n<br>\r\n[2]\r\n\u2015\r\nWhat is a Truth Value, and How Many Are There?,\r\nStudia Logica,\r\nvol.&nbsp;92 (2009), pp.&nbsp;183\u2013201.\r\n<br>\r\n[3]\r\n\u2015\r\nP. Schlenker,\r\nSuper-Liars,\r\nReview of Symbolic Logic,\r\nvol.&nbsp;3 (2010), no.&nbsp;3, pp.&nbsp;374\u2013414.\r\n<br>\r\n[4]\r\n\u2015\r\nN. Tourville and R. Cook,\r\nEmbracing the Technicalities: Expressive Completeness and Revenge,\r\nunder review,\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-s47');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n<\/table>\r\n\r\n\t\r\n<h2>Schedule of Contributed Talks<\/h2>\r\n\r\n\r\n\t\t<table border=\"0\" style=\"border-spacing: 1em .5ex;\" class=noborders>\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"contrib1\"><\/a>Monday 1st August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Set Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Hazel Brickhill, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c1')\"><i>A Generalisation of Closed Unbounded Sets and Square Sequences<\/i><\/a>\r\n<div id=\"absbox-c1\" class=\"abstract_box\">\r\nHazel Brickhill, <i>A generalisation of closed unbounded sets and square sequences<\/i>\r\n<br><br>\r\nI will introduce a generalisation of the notion closed unbounded (club) set which ties in\r\nwith the generalisation of stationary set given in [1]. We can use these $n$-clubs\r\nto define a generalisation of Jensen\u2019s $\\Box$ sequence below a non-weakly\r\ncompact (constructed in [2]). The main result is that in $L$ we have such a $\\Box^n$ sequence below every cardinal\r\nthat is $\\Pi^1_n$-indescribable but not $\\Pi^1_{n+1}$-indescribable. It is interesting to note that construction of the $\\Box^n$ sequence does not require fine-structure.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nBagaria, J.,  Magidor, M., and Sakai, H.,\r\nReflection and indescribability in the constructible universe.,\r\nIsrael Journal of Mathematics,\r\nVol. 208 (2015), no. 1,  pp. 1-11.\r\n<br>\r\n[2]\r\nJensen, R. B,\r\nThe fine structure of the constructible hierarchy.,\r\nAnnals of Mathematical Logic ,\r\nVol. 4 (1972), no. 3, pp. 229-308.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c1');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">Sam Roberts, <i>A strong reflection principle<\/i><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">17.05<\/td>\r\n<td style=\"vertical-align: top;\">Sy-David Friedman, Radek Honzik and Sarka Stejskalova, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c4')\"><i>The tree property at the double successor of a singular cardinal with a larger gap<\/i><\/a>\r\n<div id=\"absbox-c4\" class=\"abstract_box\">\r\nSy-David Friedman, Radek Honzik and Sarka Stejskalova, <i>The tree property at the double successor of a singular cardinal with a larger gap<\/i>\r\n<br><br>\r\nAll three authors were supported by FWF\/GA{\\v C}R grant I&nbsp;1921-N25.\r\n<br>\r\nStarting from a Laver-indestructible supercompact $\\kappa$ and a weakly compact $\\lambda$ above $\\kappa$, we show there is a forcing extension where $\\kappa$ is a strong limit singular cardinal with cofinality $\\omega$, $2^\\kappa = \\kappa^{+3} = \\lambda^+$, and the tree property holds at $\\kappa^{++} = \\lambda$. Next we generalize this result to an arbitrary cardinal $\\mu$ such that $\\kappa &lt;\\mathrm{cf}(\\mu)$ and $\\lambda^+ \\le \\mu$. This result provides more information about possible relationships between the tree property and the continuum function.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c4');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Proof Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Guido Gherardi, Paolo Maffezioli and Eugenio Orlandelli, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c6')\"><i>Interpolation theorem for first-order theories<\/i><\/a>\r\n<div id=\"absbox-c6\" class=\"abstract_box\">\r\nGuido Gherardi, Paolo Maffezioli and Eugenio Orlandelli, <i>Interpolation theorem for first-order theories<\/i>\r\n<br><br>\r\nIn this work we prove the interpolation theorem for some first-order theory. Previous results by Negri and von Plato [2] and Dyckhoff and Negri [1] have shown how to prove cut elimination in the presence of non-logical rules. Often cut elimination allows a constructive proof of the interpolation theorem. Thus, the question as to whether and to what extent interpolation holds in first-order theories naturally arises. Our aim is to give an answer to this question. First, we show that the standard Takeuti-Maehara techinique fails when first-order axioms are formulated as rules. Then, using an alternative strategy based on negative normal forms, we identify a class of theories for which interpolation holds. The proof we  give is entirely constructive. As a case study, we consider the theory of strict partial orders and we show how to extend the result to linear orders as well.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nR. Dyckhoff, S. Negri,\r\nGeometrization of first-order logic,\r\nThe Bulletin of Symbolic Logic,\r\nvol.&nbsp;21 (2015), no.&nbsp;2, pp.&nbsp;123\u2013163.\r\n<br>\r\n[2]\r\nS. Negri, J. von Plato,\r\nProof Analysis,\r\nCambridge University Press,\r\n2011.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c6');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">Andreas Weiermann, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c8')\"><i>On generalized Goodstein sequences<\/i><\/a>\r\n<div id=\"absbox-c8\" class=\"abstract_box\">\r\nAndreas Weiermann, <i>On generalized Goodstein sequences<\/i>\r\n<br><br>\r\nThe termination property of the classical Goodstein sequences provides\r\na simple number-theoretic assertion which is true but independent of\r\nfirst order Peano arithmetic $\\mathrm{P}\\mathrm{A}$.\r\nIn this talk we consider Goodstein sequences which are defined relative to\r\nAckermannian functions. We discuss the following two results.\r\n{\\bf Theorem A.} When the zero-th branch of the Ackermann function\r\nis the successor function then the induced Goodstein principle will\r\nbe equivalent to the one consistency of PA.\r\n{\\bf Theorem B.} When the zero-th branch of the Ackermann function\r\nis an exponential function $k\\mapsto k^b$ then the induced Goodstein principle will\r\nbe equivalent to the one consistency of $\\mathrm{A}\\mathrm{T}\\mathrm{R}_0$. (Theorem B is joint work with Tosiyasu Arai and Stan Wainer).\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c8');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory: Homogeneous Structures <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Roger Villemaire, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c10')\"><i>$\\aleph_0$-categorical structures for which Forth suffices<\/i><\/a>\r\n<div id=\"absbox-c10\" class=\"abstract_box\">\r\nRoger Villemaire, <i>$\\aleph_0$-categorical structures for which Forth suffices<\/i>\r\n<br><br>\r\nCantor's original proof that countable dense linear orders are isomorphic maps elements in a single direction. This method has been named <i>Forth<\/i> by P. J. Cameron, who furthermore showed, answering a question of A. Mathias, that Forth fails to yield an onto mapping for some $\\aleph_0$-categorical structures. In Cameron's terminology, <i>Forth does not suffice<\/i> for those structures.\r\nIn [1] Cameron gave a sufficient condition for Forth to suffice that was later generalized by McLeish [2]. However, none of these conditions are necessary.\r\nA necessary and sufficient condition for Forth to suffice has been given in [3] in terms of an countable ordinal rank on types. While [3] gives, for any countable ordinal $\\alpha$, a homogeneous structure with the property that the ranks of its types form $\\alpha$, the considered languages are infinite for $\\alpha\\geq\\omega$. These structures are unfortunately not $\\aleph_0$-categorical.\r\nThis talk will present results on the ranks that occur in $\\aleph_0$-categorical structures, with an emphasis on the combinatorial constraints that the existence of a rank imposes on types.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nP. J. Cameron,\r\nOligomorphic Permutation Groups,\r\nLondon Mathematical Society Lecture Note Series,\r\nCambridge University Press,\r\n1990.\r\n<br>\r\n[2]\r\nS. J. McLeish,\r\nThe forth part of the back and forth map in countable homogeneous structures,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;62 (1997), no.&nbsp;3, pp.&nbsp;873\u2013890.\r\n<br>\r\n[3]\r\nR. Villemaire,\r\nHomogeneity and Fix-Points: Going Forth!,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;80 (2015), no.&nbsp;2, pp.&nbsp;636\u2013660, DOI {10.1017\/jsl.2014.72}.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c10');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">David Hartman and Jan Hubicka, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c12')\"><i>Towards relational complexity of graphs<\/i><\/a>\r\n<div id=\"absbox-c12\" class=\"abstract_box\">\r\nDavid Hartman and Jan Hubicka, <i>Towards relational complexity of graphs<\/i>\r\n<br><br>\r\nHomogeneity of relational structures represents property that every local isomorphism can be extended to an automorphism. This strong symmetry condition usually results in relatively small number of families having this property, see example of undirected graphs&nbsp;[4]. For this reason it is valuable to study a process of extending a language of studied structure via introducing new relations in order to make the resulting extension homogeneous. If we additionally require this extension to preserve the automorphism group of the original structure we get notion of relational complexity. This notion evaluate maximal arity of used relation in described homogenization. Relational complexity has been introduced in slightly modified way in&nbsp;[1] and later reviewed in&nbsp;[2]. Motivated by these results we are interested in relational complexity of finite structures acquiring small values like $1$ or $2$ as well as structure maximizing its value. Another interesting area is that of countable universal structures defined by families of forbidden structures, see some details in&nbsp;[3].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nCherlin, G., Martin, G. and Saracino, D.,\r\nArities of permutation groups: Wreath products and k-sets,\r\nJournal of Combinatorial Theory, Series A,\r\nvol.&nbsp;74 (1996), pp.&nbsp;249\u2013286.\r\n<br>\r\n[2]\r\nCherlin, G.,\r\nSporadic Homogeneous Structures,\r\nThe Gelfand Mathematical Seminars, 1996\u20131999\r\n(I. M. Gelfand and V. S. Retakh, editors),\r\nBirkh\u00e4user Boston,\r\nBirkh\u00e4user Publishing Ltd., 675 Massachusetts Avenue, Cambridge,\r\n2000,\r\npp.&nbsp;15\u201348.\r\n<br>\r\n[3]\r\nHartman, D., Hubi\u010dka, J. and Ne\u0161et\u0159il, J.,\r\nComplexities of relational structure,\r\nMathematica Slovaca,\r\nvol.&nbsp;65 (2015), no.&nbsp;2, pp.&nbsp;229\u2013246.\r\n<br>\r\n[4]\r\nLachlan, A. H and Woodrow, R. E.,\r\nCountable ultrahomogeneous undirected graphs,\r\nTransactions of the American Mathematical Society,\r\nvol.&nbsp;262 (1980), pp.&nbsp;51\u201394.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c12');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.05<\/td>\r\n<td style=\"vertical-align: top;\">Samuel Braunfeld, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c14')\"><i>The Lattice of Definable Equivalence Relations in Homogeneous n-Dimensional Permutation Structures<\/i><\/a>\r\n<div id=\"absbox-c14\" class=\"abstract_box\">\r\nSamuel Braunfeld, <i>The Lattice of Definable Equivalence Relations in Homogeneous n-Dimensional Permutation Structures<\/i>\r\n<br><br>\r\nAbstract: In [1], Cameron classified the homogeneous permutations (structures in a language of 2 linear orders). In working toward the classification of the homogeneous <i>n<\/i>-dimensional permutation structures (structures in a language of <i>n<\/i> linear orders), consideration of the lattice of definable equivalence relations leads to a generalization of Cameron's structures, giving a large class of new imprimitive examples, and places some constraints on lattices that can appear.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nPeter Cameron,\r\nHomogeneous Permutations,\r\nThe Electronic Journal of\r\nCombinatorics,\r\nvol.&nbsp;9 (2002), no.&nbsp;2, Research Paper 2.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c14');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Roman Wencel, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c16')\"><i>Open covers of open definable sets in o-minimal structures<\/i><\/a>\r\n<div id=\"absbox-c16\" class=\"abstract_box\">\r\nRoman Wencel, <i>Open covers of definable sets in o-minimal structures<\/i>\r\n<br><br>\r\nIt is well known that a set definable in an o-minimal structure is a finite union\r\nof pairwise disjoint cells. During my talk I am going to discuss\r\nthe main ideas of the proof of a\r\ntheorem saying that an open bounded set definable in an arbitrary\r\no-minimal structure is a union of finitely many open definable cells,\r\nnot necessarily pairwise disjoint.\r\nThis generalizes earlier analogous results of A. Wilkie\r\n(for o-minimal expansions of real closed fields)\r\nand of M. Edmundo, P. Eleftheriou and L. Prelli\r\n(for o-minimal expansions of ordered groups).\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c16');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">Philip Ehrlich and Elliot Kaplan, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c18')\"><i>Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II<\/i><\/a>\r\n<div id=\"absbox-c18\" class=\"abstract_box\">\r\nPhilip Ehrlich and Elliot Kaplan, <i>Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II<\/i>\r\n<br><br>\r\nIn [1], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\\bf No$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of $\\bf No$, i.e. a subfield of $\\bf No$ that is an initial subtree of $\\bf No$. In this sequel to [1], which is joint work with Elliot Kaplan, analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\\bf No}$ that are themselves initial. It is further shown that an initial subdomain of ${\\bf No}$ is discrete if and only if it is an initial subdomain of ${\\bf No}$'s canonical integer part ${\\bf Oz}$ of omnific integers. Finally, extending results of [1], the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are shown to be the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\\bf No}$.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nPhilip Ehrlich,\r\nNumber Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers,\r\nThe Journal of Symbolic Logic,\r\nvol. 66 (2001), no. 3, pp. 1231-1258.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c18');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Complexity Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Oliver Kullmann, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c20')\"><i>Understanding (minimal) unsatisfiability<\/i><\/a>\r\n<div id=\"absbox-c20\" class=\"abstract_box\">\r\nOliver Kullmann, <i>Understanding (minimal) unsatisfiability<\/i>\r\n<br><br>\r\nIn my talk I want to give an overview on <i>understanding<\/i> unsatisfiability of propositional CNFs. One application area is where some kind of artefact (for example a car or a Linux system) is requested, using many options, which is translated into CNF, solved by a SAT solver, and where it then turns out the request was inconsistent \u2013 now the user wants to \"understand\" the inconsistency.\r\nIn general a CNF $F$ can be unsatisfiable for \"many reasons\", where we treat a \\underline{m}inimally \\underline{u}nsatisfiable $F' \\subseteq F$ as \"one reason\". In my talk I concentrate on such MUs $F'$ (alone). Another motivation is that such MUs can be considered as the hardest instances for SAT solvers and propositional proof systems (not having additional helper clauses). See [2] for a general overview.\r\nThe classical paper, which started the investigations, is [1], showing that, using modern terminology, the \"deficiency\" of an MU is at least $1$, i.e., there is at least one more clause than there are variables. The central conjecture of the field is that for every fixed deficiency all MUs can be described by finitely many patterns. See the extensive introduction of [3] for a recent overview.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nRon Aharoni and Nathan Linial,\r\nMinimal Non-Two-Colorable Hypergraphs and Minimal Unsatisfiable Formulas,\r\nJournal of Combinatorial Theory, Series A,\r\nvol.&nbsp;43 (1986), no.&nbsp;2, pp.&nbsp;196\u2013204.\r\n<br>\r\n[2]\r\n{Hans {Kleine B\u00fcning} and Oliver Kullmann},\r\nMinimal Unsatisfiability and Autarkies,\r\nHandbook of Satisfiability\r\n(Armin Biere and Marijn J.H. Heule and Hans van Maaren and Toby Walsh, editors),\r\nIOS Press,\r\n2009,\r\npp.&nbsp;339\u2013401.\r\n<br>\r\n[3]\r\nOliver Kullmann and Xishun Zhao,\r\nBounds for variables with few occurrences in conjunctive normal forms,\r\narXiv:1408.0629,\r\n2016.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c20');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">Olaf Beyersdorff and J\u00e1n Pich, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c22')\"><i>Understanding Gentzen and Frege systems for QBF<\/i><\/a>\r\n<div id=\"absbox-c22\" class=\"abstract_box\">\r\nOlaf Beyersdorff and J\u00e1n Pich, <i>Understanding Gentzen and Frege systems for QBF<\/i>\r\n<br><br>\r\nRecently Beyersdorff, Bonacina, and Chew [1] introduced a natural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of their new extended Frege system, denoted EF+$\\forall$red, which is a natural extension of classical extended Frege EF.\r\nOur main results are the following:\r\nFirstly, we prove that the standard Gentzen-style system $G_1^*$ p-simulates EF+$\\forall$red and that $G_1^*$ is strictly stronger under standard complexity-theoretic hardness assumptions.\r\nSecondly, we show a correspondence of EF+$\\forall$red to bounded arithmetic: EF+$\\forall$red can be seen as the non-uniform propositional version of intuitionistic $S^1_2$. Specifically, intuitionistic $S^1_2$ proofs of arbitrary statements in prenex form translate to polynomial-size EF+$\\forall$red proofs, and EF+$\\forall$red is in a sense the weakest system with this property.\r\nFinally, we show that unconditional lower bounds for EF+$\\forall$red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the system EF+$\\forall$red naturally unites the central problems from circuit and proof complexity.\r\nTechnically, our results rest on a formalised strategy extraction theorem for EF+$\\forall$red akin to witnessing in intuitionistic $S^1_2$ and a normal form for EF+$\\forall$red proofs.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nO.&nbsp;Beyersdorff, I.&nbsp;Bonacina, L.&nbsp;Chew,\r\nLower bounds: From circuits to QBF proof systems,\r\nProc. ACM Conference on Innovations in Theoretical Computer Science (ITCS),\r\nACM,\r\n2016,\r\npp.&nbsp;249\u2013260.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c22');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Categorical Logic and Type Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Murdoch Gabbay, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c24')\"><i>What are variables of first-order logic and the lambda-calculus?<\/i><\/a>\r\n<div id=\"absbox-c24\" class=\"abstract_box\">\r\nMurdoch Gabbay, <i>What are variables of first-order logic and the lambda-calculus?<\/i>\r\n<br><br>\r\nIntuitively, conjunction corresponds to sets intersection; disjunction corresponds to sets union; and negation corresponds to sets complement.\r\nIf we axiomatise these connectives in universal algebra then we obtain Boolean algebras.\r\nEvery Boolean algebra can be presented as a field of sets, and in this sense conjunction <i>is<\/i> sets intersection; disjunction <i>is<\/i> sets union; and negation <i>is<\/i> sets complement.\r\nFirst-order logic has variables.\r\nThe usual approach to variables is to assign them values with a <i>valuation<\/i>, which is just a lookup table.\r\nAn alternative is offered by <i>nominal algebra<\/i>, an extension of universal algebra with nominal-style names and binding.\r\nThis admits a simple finite axiomatisation of variables and substitution as just another `connective'.\r\nThat is, substitution is a single logical connective, with an associated standard finite algebraic theory (no axiom-schemes).\r\nFurthermore, we can finitely axiomatise quantifiers in nominal algebra, including:\r\n<ul>\r\n<li>\r\nthe $\\forall$ universal quantifier of first-order logic,\r\n<li>\r\nthe $\\lambda$ of the untyped lambda-calculus, and\r\n<li>\r\nthe sets comprehension binder in set theory.\r\n<\/ul>\r\nRemarkably, we can extend the presentation as a field of sets to these algebraic theories, and even extend Stone duality to give full topological dualities for first-order logic and the untyped lambda-calculus (the treatment of set theory is the topic of a separate abstract).\r\nBy this account, substitution, quantification, and $\\lambda$-abstraction become\r\n<ul>\r\n<li>\r\naxiomatisations in nominal algebra alongside those for conjunction, disjunction, and negation; and dually, they become\r\n<li>\r\noperations on sets of points of topological spaces\u2013which turn out to be fairly elementary\u2013existing alongside sets intersection, union, and complement.\r\n<\/ul>\r\nThis approach to semantics differs from what is usually found in the textbooks, and it offers some technical advantages: the axioms for $\\forall$ and $\\lambda$ do not reference the axioms for substitution, or vice versa,\r\nand this <i>decoupling<\/i> of the algebraic theories is reflected in the concrete proofs and modular constructions.\r\nA general method seems to be emerging here.\r\nThe end results are clean, elegent, non-evident, and suggestive of future work.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c24');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">Noam Zeilberger, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c26')\"><i>Linear lambda terms as invariants of rooted trivalent maps<\/i><\/a>\r\n<div id=\"absbox-c26\" class=\"abstract_box\">\r\nNoam Zeilberger, <i>Linear lambda terms as invariants of rooted trivalent maps<\/i>\r\n<br><br>\r\nRecent work on the combinatorics of linear lambda calculus (also known\r\nas BCI combinatory logic) has uncovered a variety of surprising\r\nconnections to the theory of graphs on surfaces (also known as\r\n\"maps\").  The main purpose of the talk will be to convey a simple and\r\nconceptual account for one of these connections, namely the\r\ncorrespondence (originally described by Bodini, Gardy, and Jacquot)\r\nbetween $\\alpha$-equivalence classes of closed linear lambda terms and\r\nisomorphism classes of rooted trivalent maps on compact oriented\r\nsurfaces without boundary, as an instance of a more general\r\ncorrespondence between linear lambda terms with a context of free\r\nvariables and rooted trivalent maps with a boundary of free edges.\r\nAfter recalling some basic definitions as well as a familiar\r\ndiagrammatic representation for linear lambda terms, I'll explain how\r\nthe \"easy\" direction of the correspondence is a simple forgetful\r\noperation which erases annotations on the diagram of a linear lambda\r\nterm to produce a rooted trivalent map.  The other more surprising\r\ndirection views linear lambda terms as topological invariants of their\r\nunderlying rooted trivalent maps, reconstructing the missing\r\ninformation through a Tutte-style recurrence on maps with free edges.\r\nAs an application in combinatorics, I'll show how to use this analysis\r\nto enumerate bridgeless rooted trivalent maps as linear lambda terms\r\ncontaining no closed subterms, and conclude by giving a natural\r\nreformulation of the Four Color Theorem as a statement about typing in\r\nlambda calculus.\r\n\\bibliographystyle{abbrvnat}\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nH. P. Barendregt. The Lambda Calculus: Its Syntax and Semantics, Studies in Logic 103, second, revised edition, North-Holland, Amsterdam, 1984.\r\n<br>\r\n[2]\r\nO. Bodini, D. Gardy, and A. Jacquot. Asymptotics and random sampling for BCI and BCK lambda terms. Theoretical Computer Science, 502:227\u2013238, 2013.\r\n<br>\r\n[3]\r\nSergei K. Lando and Alexander K. Zvonkin. Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences 141, Springer-Verlag, 2004.\r\n<br>\r\n[4]\r\nNoam Zeilberger and Alain Giorgetti. A correspondence between rooted planar maps and normal planar lambda terms. Logical Methods in Computer Science, 11(3:22):1\u201339, 2015.\r\n<br>\r\n[5]\r\nNoam Zeilberger. Counting isomorphism classes of $\\beta$-normal linear lambda terms.\r\nSeptember 25, 2015. arXiv:1509.07596\r\n<br>\r\n[6]\r\nNoam Zeilberger. Linear lambda terms as invariants of rooted trivalent maps.\r\nDecember 21, 2015. arXiv:1512.06751\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c26');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.05<\/td>\r\n<td style=\"vertical-align: top;\">Paolo Torrini, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c28')\"><i>Dependent types and linear scoping<\/i><\/a>\r\n<div id=\"absbox-c28\" class=\"abstract_box\">\r\nPaolo Torrini, <i>Dependent Types and Linear Scoping<\/i>\r\n<br><br>\r\nIn languages based on nominal types, a type is characterised by its\r\nabstract structure together with its name. In pure languages, an\r\nabstract characterisation of nominal types can rely on usual nominal\r\ntechniques. Things can get more complicated, for instance, when we\r\nconsider records with mutable fields, where a name refers to a\r\nresource and may represent a physical address. This is also the case\r\nof class types in object-oriented languages.\r\nAn abstract characterisation of effectful nominality can be useful in\r\ndealing with encapsulation and inheritance, to specify shape\r\ntransformation, and to express locality of resources with respect to\r\ncode blocks. In [1] we extended a system of linear dependent\r\ntypes by allowing linear types to depend on locally non-linear terms\r\n(L-terms). L-variables and L-terms can be interpreted as addresses of\r\nabstract locations. L-variables can also be understood as names, as\r\nequivariant renaming is admissible, with the implicit guarantee that\r\nthe same name cannot be associated with different locations. The\r\nevaluation of L-terms is orthogonal to that of value terms.\r\nLinear scoping is defined as L-variable binding, based on a linear\r\nform of $\\Sigma$-types that combines effectfulness and scoping. This\r\nnotion of binding associates system components, represented as linear\r\ntypes, to the abstract locations they can access, thus enforcing\r\nlocality of free L-terms. Linear scoping can be used to specify\r\nmutable record types compatibly with their nominal\r\ncharacterisation. More generally, this approach could give us a simple\r\nway to model abstractly aspects of object orientation and resource\r\nvirtualisation, by using dependent types to type statically programs\r\nthat involve dynamic, effectful operations such as reallocation and\r\noptimisation of memory use.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nP. Torrini,\r\nLinear types and locality,\r\nJournal of Logic and Computation,\r\nvol.&nbsp;24 (2014), no.&nbsp;3, pp.&nbsp;655\u2013685.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c28');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>History of Logic <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.15<\/td>\r\n<td style=\"vertical-align: top;\">Ryszard Mirek, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c30')\"><i>Renaissance Geometry<\/i><\/a>\r\n<div id=\"absbox-c30\" class=\"abstract_box\">\r\nRyszard Mirek, <i>Renaissance Geometry<\/i>\r\n<br><br>\r\nRenaissance mathematicians and geometers like Piero della Francesca and Luca Pacioli refers directly or indirectly to Euclidean geometry. For instance, Piero della Francesca in his proofs refers to the similarity of the triangles.  In <i>Elements<\/i> discussion of these issues is in the Book VI, Proposition 4 to 8.  There is also no doubt that Piero is familiar with Book XIII of <i>Elements<\/i>. Piero constructs each of the regular solids from its circumsphere,  but unlike Euclid, he gives a numerical value for the diameter of the sphere.  In turn,  in Proposition I.8 he shows that the perspective images of orthogonals converge to a <i>centric point<\/i> what follows from Euclid's theorem from  the Book VI, Proposition 21. On the other hand, Luca Pacioli  provides a direct reference to Elements and the precise information are given to where the result is proved by Euclid.\r\n<br>\r\nMy goal here is to provide a detailed analysis of the methods of inference that are employed in the Renaissance treatises. For this purpose one can use a formal system <i>EF<\/i> that seems to present in a precise and  visually readable way the  geometrical systems.\r\n<br>\r\nThis research is supported by the NCN research grant 2012\/07\/B\/HS1\/01986.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c30');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.40<\/td>\r\n<td style=\"vertical-align: top;\">Jeff Paris and Alena Vencovska, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c32')\"><i>The Indian Syllogism as Analogical Reasoning<\/i><\/a>\r\n<div id=\"absbox-c32\" class=\"abstract_box\">\r\nJeff Paris and Alena Vencovska, <i>The Indian Syllogism as analogical reasoning<\/i>\r\n<br><br>\r\nWhen H.T.Colebrooke first introduced to the West the so called Indian or Hindu Syllogism  from Gotama's  Nyaya-Sutra (c.100CE) at his lecture to the Royal Asiatic Society in 1824 it caused a flurry of excitement, not least amongst the main logicians at the time, Babbage, De Morgan, and Boole. For it seemed that here was some independent\r\ndevelopment of logic within the subcontinent, breaking the Aristotelian  monopoly and providing the space for new ideas to develop.\r\nSubsequently however a section of Britain's Victorian Society, perhaps reluctant to acknowledge such advanced thinking in what was after all one of its colonies, contemptuously  `downgraded'\r\nthe syllogism to the status of analogy, an example of reasoning from particular to particular without any genuine soundness or even justification. Whilst most commentators today hold this reading to be incorrect I will argue that notwithstanding even when viewed as a schema of analogical reasoning the syllogism has a demonstrably rational justification within the context of Pure Inductive Logic.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c32');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.05<\/td>\r\n<td style=\"vertical-align: top;\">Sara L. Uckelman, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c34')\"><i>The Logic of Where and While in the 13th and 14th Centuries<\/i><\/a>\r\n<div id=\"absbox-c34\" class=\"abstract_box\">\r\nSara L. Uckelman, <i>The Logic of Where and While in the 13th and 14th Centuries<\/i>\r\n<br><br>\r\nMedieval analyses of molecular propositions include many non-truth-func\\-tional connectives in addition to the standard modern binary connectives (conjunction, disjunction, and conditional). Two types of non-truthfunctional molecular propositions considered by a number of 13th- and 14th-century authors are temporal and local propositions, which combine atomic propositions with `while' and `where'.  Despite modern interest in the historical roots of temporal and tense logic, medieval analyses of `while' propositions are rarely discussed in modern literature, and analyses of `where' propositions are almost completely overlooked.  In this paper we introduce 13th- and 14th-century views on temporal and local propositions, and connect the medieval theories with modern temporal and spatial counterparts.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c34');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\"><br><\/td><td style=\"vertical-align: top;\"><br><\/td><\/tr>\r\n\r\n\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"contrib2\"><\/a>Tuesday 2nd August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Set Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Radek Honzik and Sarka Stejskalova, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c36')\"><i>The tree property and the continuum function below $\\aleph_\\omega$<\/i><\/a>\r\n<div id=\"absbox-c36\" class=\"abstract_box\">\r\nRadek Honzik and Sarka Stejskalova, <i>The tree property and the continuum function below $\\aleph_\\omega$<\/i>\r\n<br><br>\r\nBoth authors were supported by FWF\/GA\u010cR grant I&nbsp;1921-N25.\r\n<br>\r\nWe say that a regular cardinal $\\kappa$, $\\kappa> \\aleph_0$, has the tree property if there are no $\\kappa$-Aronszajn trees; we say that $\\kappa$ has the weak tree property if there are no special $\\kappa$-Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal $\\aleph_{2n}$, $0&lt;n&lt;\\omega$, is consistent with an arbitrary continuum function below $\\aleph_\\omega$ which satisfies $2^{\\aleph_{2n}}>\\aleph_{2n+1}$, $n&lt;\\omega$. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal $\\aleph_n$, $1 &lt; n &lt;\\omega$, is consistent with an arbitrary continuum function which satisfies $2^{\\aleph_n} > \\aleph_{n+1}$, $n&lt;\\omega$. Thus the tree property has no provable effect on the continuum function below $\\aleph_\\omega$ except for the trivial requirement that the tree property at $\\kappa^{++}$ implies $2^\\kappa>\\kappa^+$ for every infinite $\\kappa$.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c36');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Tanmay Inamdar, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c38')\"><i>A fragment of PFA consistent with large continuum<\/i><\/a>\r\n<div id=\"absbox-c38\" class=\"abstract_box\">\r\nTanmay Inamdar, <i>A fragment of PFA consistent with large continuum<\/i>\r\n<br><br>\r\nThe side condition method of Todor\u010devi\u0107 is an important technique to build proper partial orders. It has been used to establish several consequences of PFA, the most important of which are the Open Graph Axiom as well as the P-Ideal Dichotomy. Many of these applications can be reformulated to assert that given a graph on an uncountable set, if there is a proper $\\sigma$-ideal which is well behaved in a certain way with respect to this graph, then a certain partial order to add an uncountable clique to this graph is proper. On the other hand, in recent years Asper\u00f3 and Mota have developed new techniques to iterate proper partial orders which have allowed them to establish the consistency of several consequences of PFA with the continuum large. In my talk I shall talk about how using the methods of Asper\u00f3 and Mota, one can get models where the continuum is arbitrarily large, Martin's Axiom holds, and a certain `side condition forcing axiom' holds for graphs on $\\omega_1$ and $\\omega_1$-generated $\\sigma$-ideals. For example, such models have no S-spaces, $\\omega_1 \\rightarrow (\\omega_1, \\alpha)^2$ holds for any countable ordinal $\\alpha$, and certain restricted forms of OGA and PID hold.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c38');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Sherwood Hachtman, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c40')\"><i>Determinacy and admissible reflection<\/i><\/a>\r\n<div id=\"absbox-c40\" class=\"abstract_box\">\r\nSherwood Hachtman, <i>Determinacy and admissible reflection<\/i>\r\n<br><br>\r\nFor transitive sets $X$, let $\\mathfrak{M}(X)$ denote the least admissible set containing $X$ as an element.  Say $\\kappa$ <i>reflects admissibly<\/i> if for any $\\Pi_1$ formula $\\varphi$ and $A \\subseteq V_\\kappa$, if $(V_{\\kappa+1}, \\in, A) \\models \\varphi$, then for some $\\alpha &lt; \\kappa$, $(V_{\\alpha+1} \\cap \\mathfrak{M}(V_\\alpha), \\in, A \\cap V_\\alpha) \\models \\varphi$.\r\nNote that if we replaced $\\mathfrak{M}(V_\\alpha)$ with $V_{\\alpha+1}$, this would simply be weak compactness of $\\kappa$.  But admissible reflection is much weaker.\r\nOur interest in this principle is due to its use in calibrating the strength of determinacy hypotheses.  We show that the minimal model of NBG+\"ON reflects admissibly\" satisfies clopen, but not open, determinacy for proper class games; this answers a question of Gitman &amp; Hamkins.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c40');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Philipp Schlicht and Fabiana Castiblanco, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c42')\"><i>Tree forcings and sharps<\/i><\/a>\r\n<div id=\"absbox-c42\" class=\"abstract_box\">\r\nPhilipp Schlicht and Fabiana Castiblanco, <i>Tree forcings and sharps<\/i>\r\n<br><br>\r\nThe Levy-Solovay theorem shows that any measurable cardinal $\\kappa$ is preserved by all forcings of size strictly less than $\\kappa$, and there are similar results for many other large cardinals. Moreover, many global consequences of large cardinals are preserved by certain forcings.\r\nFor instance, a <i>sharp<\/i> for a set of ordinals $x$ states the existence of a non-trivial elementary embedding $j\\colon L[x]\\rightarrow L[x]$ with $\\mathrm{crit}(j)>\\sup(x)$.\r\nIt is well-known that the existence of sharps for all sets of ordinals is preserved by all forcings.\r\nIn this project, we study interesting consequences of large cardinals in $H(\\omega_1)$ and prove that they are preserved under certain proper forcings.\r\nIn particular, we study the existence of sharps for all sets of ordinals in $H(\\omega_1)$, equivalently analytic determinacy.\r\nWe show that this condition is preserved by various tree forcings, for instance Mathias forcing, Sacks forcing and Laver forcing.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c42');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Philipp L\u00fccke, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c44')\"><i>The infinite productivity of Knaster properties<\/i><\/a>\r\n<div id=\"absbox-c44\" class=\"abstract_box\">\r\nPhilipp L\u00fccke, <i>The infinite productivity of Knaster properties<\/i>\r\n<br><br>\r\nGiven an uncountable regular cardinal $\\kappa$, we say that a partial order $\\mathbb{P}$ is <i>$\\kappa$-Knaster<\/i> if every set of $\\kappa$-many\r\nconditions in $\\mathbb{P}$ contains a subset of cardinality $\\kappa$ consisting of pairwise compatible conditions. This strengthening of the $\\kappa$-chain condition is typically used because of its nice product behavior: finite support products of $\\kappa$-Knaster partial orders are $\\kappa$-Knaster, and the product of a $\\kappa$-Knaster partial order with a partial order satisfying the $\\kappa$-chain condition satisfies the $\\kappa$-chain condition. Moreover, if $\\kappa$ is weakly compact, then the class of $\\kappa$-Knaster partial orders is closed under $\\nu$-support products for every $\\nu&lt;\\kappa$. This raises the question whether it is possible that the class of $\\kappa$-Knaster partial orders is closed under countable support products and $\\kappa$ is not weakly compact. I will present results that show that the axioms of $\\mathrm{ZFC}$ do not answer this question. This is partially joint work with Sean Cox (VCU Richmond).\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c44');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Proof Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Marija Boricic, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c46')\"><i>Natural deduction probabilized<\/i><\/a>\r\n<div id=\"absbox-c46\" class=\"abstract_box\">\r\nMarija Boricic, <i>Natural deduction probabilized<\/i>\r\n<br><br>\r\nBy combining Gentzen's and Prawitz's approach to deductive systems and Carnap\u2013Popper\u2013type probability logic semantics, we\r\nintroduce a probabilistic version of inference rules of natural deduction $\\mathbf{NK}$, denoted by $\\mathbf{NKprob}$. Probabilized\r\nnatural deduction systems have already been considered (see [2], [3] and [4]). For each propositional formula $A$ and each\r\n$a,b\\in I$, where $I$ is a finite subset of reals $[0,1]$ containing $0$ and $1$, closed under addition, the expression $A[a,b]$\r\nis probabilized formula in $\\mathbf{NKprob}$. The meaning of $A[a,b]$ is that 'the probability $c$ of truthfulness of a sentence\r\n$A$ belongs to the interval $[a,b]$'. Our system contains at least two inference rules for each connective, one introducing, and\r\nthe other one eliminating the connective. For example, the following rules are treating the introduction and elimination of\r\ndisjunction:\r\n$$\\frac{A[a,b]\\quad B[c,d]}{(A\\vee B)[\\max(a,c),b+d]}(I\\vee)\\qquad\\frac{A[a,b]\\quad (A\\vee B)[c,d]}{B[c-b,d]}(E\\vee)$$ Also,\r\nthere are specific rules treating inconsistency:\r\n$$ \\frac{\\begin{matrix} \\dfrac{\\underline{[A[c_1,c_1]]}}{A\\emptyset}   \\dfrac{\\underline{[A[c_2,c_2]]}}{A\\emptyset} \\dots\r\n\\dfrac{\\underline{[A[c_m,c_m]]}}{A\\emptyset}\r\n\\end{matrix} } {A\\emptyset}(I\\emptyset) \\qquad\\frac{A\\emptyset}{B[a,b]}(E\\emptyset)$$\r\nfor any propositional formulae $A$ and $B$, and any $a, b \\in I=\\{c_1, c_2, \\dots c_m\\}$, where $A\\emptyset$ is\r\n$A[a,b]$, for $a>b$.\r\nLet $\\text{For}$ be the set of all propositional formulae. Then any mapping $p:\\text{For}\\to I$ will be an\r\n$\\text{\\bf NKprob}$\u2013model if it satisfies the following conditions: (i) $p(\\top)=1$ and $p(\\bot)=0$; (ii)\r\nif $p(A\\wedge B)=0$, then $p(A\\vee B)=p(A)+p(B)$; (iii) if $A\\leftrightarrow B$ in classical logic, then\r\n$p(A)=p(B)$. We prove that our probabilistic natural deduction system $\\mathbf{NKprob}$ is sound and complete with\r\nrespect to this kind of models (see [1]).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nM. Bori\u010di\u0107, Inference rules for probability logic, Publications de\r\nl'Institut Math\u00e9matique  (to appear)\r\n<br>\r\n[2]\r\nA. M. Frisch, P. Haddawy, Anytime deduction for probabilistic logic,\r\nArtificial Intelligence, vol.&nbsp;69 (1993), pp.&nbsp;93\u2013122.\r\n<br>\r\n[3]\r\nT. Hailperin, Probability logic, Notre Dame Journal of Formal Logic, vol.&nbsp;25\r\n(1984), pp.&nbsp;198\u2013212.\r\n<br>\r\n[4]\r\nC. G. Wagner, Modus tollens probabilized, British Journal for the Philosophy\r\nof Science, vol.&nbsp;54(4) (2004), pp.&nbsp;747\u2013753.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c46');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Nobu-Yuki Suzuki, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c48')\"><i>Relations among some weak variants of existence and disjunction properties in intermediate predicate logics<\/i><\/a>\r\n<div id=\"absbox-c48\" class=\"abstract_box\">\r\nNobu-Yuki Suzuki, <i>Relations among some weak variants of existence and disjunction properties in intermediate predicate logics<\/i>\r\n<br><br>\r\nWe discuss relationships among the existence property (EP),\r\ndisjunction property (DP),\r\nand their weak variants in the setting of intermediate predicate logics.\r\nThese weak variants were treated in [1]\r\nfor constructing intermediate logics having EP but lacking DP.\r\nThis revealed that EP and DP are independent in intermediate logics.\r\nWe show the relationships among these properties and their combinations.\r\nThe properties we discuss are as follows.\r\nA(n intermediate predicate) logic $\\mathbf{L}$ is said to have EP,\r\nif $\\mathbf{L} \\vdash \\exists xA(x)$\r\nimplies that there is a variable $v$\r\nsuch that $\\mathbf{L} \\vdash A(v)$.\r\nA logic $\\mathbf{L}$ is said to have DP,\r\nif $\\mathbf{L} \\vdash A \\lor B$\r\nimplies that $\\mathbf{L} \\vdash A$ or $\\mathbf{L} \\vdash B$.\r\nA logic $\\mathbf{L}$ is said to have\r\nthe <i>sentential existence property<\/i>\r\nif for every <i>sentence<\/i> $\\exists xA(x)$,\r\n$\\mathbf{L} \\vdash \\exists xA(x)$\r\nimplies that there exists a <i>fresh<\/i>\r\nvariable\r\n$v$ such that\r\n$\\mathbf{L} \\vdash A(v)$.\r\nA logic $\\mathbf{L}$\r\nis said to have the <i>weak existence property<\/i>,\r\nif for every formula $\\exists xA(x)$\r\nand every finite non-empty $\\{v_1, \\dots, v_n\\}$ of individual variables\r\nsuch that $FV(\\exists xA(x)) \\subseteq \\{v_1, \\dots, v_n\\}$,\r\n$\\mathbf{L} \\vdash \\exists xA(x)$ implies\r\n$\\mathbf{L} \\vdash A(v_1) \\vee \\cdots \\vee A(v_n)$.\r\nMoreover, we introduced a very weak\r\nDP called $Z$-<i>normality<\/i>.\r\nThe $Z$-normality is\r\nimportant in the consideration of\r\nthe relation between EP and DP; every $Z$-normal logic with EP has DP.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSuzuki, N.-Y.,\r\nA negative solution to Ono's problem P52:\r\nExistence and disjunction properties in\r\nintermediate predicate logics,\r\nto appear.\r\n<br>\r\n[2]\r\nSuzuki, N.-Y.,\r\nSome weak variants of the existence and disjunction properties\r\nin intermediate predicate logics,\r\nsubmitted.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c48');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Bahareh Afshari, Stefan Hetzl and Graham Leigh, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c50')\"><i>Structural representation of Herbrand's theorem<\/i><\/a>\r\n<div id=\"absbox-c50\" class=\"abstract_box\">\r\nBahareh Afshari, Stefan Hetzl and Graham Leigh, <i>Structural representation of Herbrand's theorem<\/i>\r\n<br><br>\r\nWe present recent results on the deepening connection between proof theory and formal language theory. To each first-order proof with cuts of complexity at most $\\Pi_n$\/$\\Sigma_n$, we associate a typed (non-deterministic) tree grammar of order $n$ (equivalently, an order $n$ recursion scheme)  that abstracts the computation of Herbrand sets obtained through Gentzen-style cut elimination.\r\nApart from  offering a means to compute Herbrand expansions directly from proofs with cuts, these grammars provide a structural counterpart to Herbrand's theorem that opens the door to tackling a number of questions in proof-theory such as proof equivalence, proof compression and proof complexity, which will be discussed.\r\nThe grammars presented naturally generalise the rigid regular and context-free tree grammars introduced in  [2] and [1] that correspond to (respectively) proofs with $\\Pi_1$\/$\\Sigma_1$ and  $\\Pi_2$\/$\\Sigma_2$ cuts.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nBahareh Afshari, Stefan Hetzl, and Graham E. Leigh,\r\nHerbrand disjunctions, cut elimination and context-free tree grammars,\r\n13th International Conference on Typed Lambda Calculi and Applications\r\n(Warsaw, Poland),\r\n(Thorsten Altenkirch, editor),\r\nvol.&nbsp;38,\r\nSchloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik,\r\n2015,\r\npp.&nbsp;1\u201316.\r\n<br>\r\n[2]\r\nStefan Hetzl,\r\nApplying tree languages in proof theory,\r\nLanguage and Automata Theory and Applications: 6th International Conference\r\n(A Coru\u00f1a, Spain),\r\n(Adrian-Horia Dediu and Carlos Mart\u00edn-Vide, editors),\r\nvol.&nbsp;7183,\r\nSpringer Berlin Heidelberg,\r\n2012,\r\npp.&nbsp;301\u2013312.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c50');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Graham Leigh, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c52')\"><i>The simple truth<\/i><\/a>\r\n<div id=\"absbox-c52\" class=\"abstract_box\">\r\nGraham Leigh, <i>The simple truth<\/i>\r\n<br><br>\r\nWhat is implicit in the acceptance of the Tarskian truth biconditionals? In this talk I will present recent results that characterise the proof- and truth-theoretic content of iterated reflection principles over disquotational theories of truth. In particular, I confirm the conjecture that, modulo reflection, all there is to compositional and Kripke\u2013Feferman truth is captured by simple and natural collections of local truth and falsity biconditionals.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c52');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Kentaro Sato, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c54')\"><i>Monotone induction can be prolonged by exponential<\/i><\/a>\r\n<div id=\"absbox-c54\" class=\"abstract_box\">\r\nKentaro Sato, <i>Monotone induction can be prolonged by exponential<\/i>\r\n<br><br>\r\nThe usual formulation of induction is the implication from $A(0)$ and $\\forall n(A(n)\\,{\\to}\\,A(n{+}1))$ to $\\forall xA(x)$.\r\nA less popular formulation, called cumulative induction, is formulated by the implication from $\\forall n((\\forall k\\,{&lt;}\\,n)B(k)\\,{\\to}\\,B(n))$ to $\\forall xB(x)$.\r\nPractically, the difference is: in proofs by cumulative induction, we can use all $B(k)$ for $k$ below $n$ in order to show $B(n)$,\r\nwhereas in those by usual induction we can use only $A(n)$ to show $A(n{+}1)$.\r\nHowever, if we consider these schemata for classes of formulae closed under bounded quantifiers, the difference is, usually, not so important,\r\nfor we can replace $A(n)$ by $(\\forall k\\,{&lt;}\\,n)B(k)$.\r\nNow a natural question is: how about transfinite induction? The trick increases the complexity, as the quantifier $\\forall\\xi\\,{\\prec}\\,\\alpha$ is not bounded.\r\nIn this talk, I consider a transfinite analogue of the usual formulation of induction, the implication\r\nfrom \\[\\forall\\alpha((\\forall\\xi\\,{\\prec}\\,\\alpha)(\\exists\\eta\\,{\\prec}\\,\\alpha)\r\n(\\eta\\,{\\succeq}\\,\\xi\\,{\\land}\\,A(\\eta))\\,{\\to}\\,A(\\alpha))\\] to $\\forall\\alpha A(\\alpha),$\r\nand compare it with the usual formulation of transfinite induction, which is a straightforward generalization of the cumulative one.\r\nUnder the monotoneness assumption $\\xi\\,{\\prec}\\,\\eta\\land A(\\eta)\\,{\\to}\\,A(\\xi)$,\r\nboth are equivalent.\r\nI will show that the supremum of those ordinals along which this \"cofinal induction\" for $\\Delta^0_0$ formulae is provable in ${\\bf I}\\boldsymbol{\\Sigma}_n$ is\r\n$\\omega_{n+2}$ (including $n\\,{=}\\,0$) where $\\omega_0\\,{=}\\,1$, $\\omega_{k+1}\\,{=}\\,\\omega^{\\omega_k}$.\r\nThis is longer than that for the usual transfinite induction for $\\Delta_0$ formulae, by exponential base 2, or, for $n\\,{\\geq}\\,1$, equivalently base $\\omega$.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c54');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory: Homogeneous Structures <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">David Bradley-Williams, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c56')\"><i>Limits of betweenness relations<\/i><\/a>\r\n<div id=\"absbox-c56\" class=\"abstract_box\">\r\nDavid Bradley-Williams, <i>Limits of betweenness relations<\/i>\r\n<br><br>\r\nBetweenness relations on semilinear orderings, among other tree-like relational stuctures, were studied extensively by Adeleke and Neumann in [3]. Such tree-like structures were also investigated though a host of examples constructed by Cameron in [6]. Adeleke and Macpherson [2] then built on this knowledge to determine that when a transitive permutation group is a Jordan group, it preserves one from a list of various kinds of relational structures. Some of these structures are quite well understood as relational structures, but there are cases in which the invariant structure appears only as an exotic infinite `limit' of more familiar structures. In this talk we will discuss what `limits of betweenness relations' are and how they have been constructed (by Adeleke [1], Bhatacharjee and Macpherson [4] and myself [5]).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nS. A. Adeleke,\r\nOn irregular infinite Jordan groups,\r\nCommunications in Algebra,\r\nvol.&nbsp;41 (2013), no.&nbsp;4, pp.&nbsp;1514\u20131546.\r\n<br>\r\n[2]\r\nS. A. Adeleke and D. Macpherson,\r\nClassification of infinite primitive Jordan permutation groups,\r\nProceedings of the London Mathematical Society,\r\nvol.&nbsp;72 (1996), no.&nbsp;3, pp.&nbsp;63\u2013123.\r\n<br>\r\n[3]\r\nS. A. Adeleke and P. M. Neumann,\r\nRelations related to betweenness: their structure and automorphisms,\r\nMemoirs of the American Mathematical Society,\r\nThe American Mathematical Society,\r\n\\textbf{623}, 1998.\r\n<br>\r\n[4]\r\nM. Bhattacharjee and D. Macpherson,\r\nJordan groups and limits of betweenness relations,\r\nJournal of Group Theory,\r\nvol.&nbsp;9 (2006), no.&nbsp;1, pp.&nbsp;59\u201394.\r\n<br>\r\n[5]\r\nD. Bradley-Williams,\r\nJordan groups and homogeneous structures,\r\nPhD Thesis,\r\nUniversity of Leeds,\r\n2015.\r\n<br>\r\n[6]\r\nP. Cameron,\r\nSome treelike objects,\r\nThe Quarterly Journal of Mathematics,\r\nvol.&nbsp;38 (1987), no.&nbsp;2, pp.&nbsp;155\u2013183.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c56');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Lovkush Agarwal and Michael Kompatscher, <i>Continuum-many maximal-closed subgroups for Sym(N) via the Classification of the Reducts of the Henson Digraphs<\/i><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Thomas Coleman, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c59')\"><i>Permutation monoids and MB-homogeneous structures<\/i><\/a>\r\n<div id=\"absbox-c59\" class=\"abstract_box\">\r\nThomas Coleman, <i>Permutation monoids and MB-homogeneous structures<\/i>\r\n<br><br>\r\nGroup embeddable monoids, by their nature, can be represented as a submonoid of permutations contained in some symmetric group Sym$(X)$. As every finite group embeddable monoid is a group, we consider infinite submonoids $B$ of the infinite symmetric group Sym$(\\mathbb{N})$ to avoid triviality; such a $B$ is an <i>infinite permutation monoid<\/i>. Natural examples of these occur via the <i>bimorphism monoid<\/i> Bi$(\\mathcal{A})$ of a structure; that is, the collection of bijective endomorphisms of $\\mathcal{A}$. It follows that every automorphism of $\\mathcal{A}$ is a bimorphism of $\\mathcal{A}$ but in general the converse is not true; and so we have that Aut$(\\mathcal{A})\\subseteq$ Bi$(\\mathcal{A})\\subseteq$ Sym$(A)$, where $A$ is the domain of $\\mathcal{A}$.\r\nRecent work in this field by Cameron and Ne\u0161et\u0159il [1] and Lockett and Truss [2] generalizes the idea of homogeneity to several notions of <i>homomorphism-homogeneity<\/i>. One such example is the property of <i>MB-homogeneity<\/i>: a structure $\\mathcal{A}$ is MB-homogeneous if every monomorphism between finite substructures of $\\mathcal{A}$ extends to a bimorphism of $\\mathcal{A}$. Lockett and Truss completely classified homomorphism-homogeneous countable posets in [2].\r\nIn this talk, connections between permutation monoids and bimorphism monoids of structures are explored in order to develop a notion of oligomorphicity for infinite permutation monoids. In addition to this, a version of Fra\u00efss\u00e9's theorem is shown for MB-homogeneous structures, extending work of [1]. Finally, we construct $2^{\\aleph_0}$ non-isomorphic examples of MB-homogeneous graphs and take steps towards a classification result. This is joint work with David Evans and Robert Gray during the course of my PhD studies.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nP. J. Cameron, J. Ne\u0161et\u0159il,\r\nHomomorphism-homogeneous relational structures,\r\nCombinatorics, Probability and Computing,\r\nvol.&nbsp;15 (2006), no.&nbsp;(1-2), pp.&nbsp;91-103.\r\n<br>\r\n[2]\r\nD. C Lockett, J. K. Truss,\r\nSome more notions of homomorphism-homogeneity,\r\nDiscrete Mathematics,\r\nvol.&nbsp;336 (2014), pp.&nbsp;69-79.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c59');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Daoud Siniora, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c61')\"><i>A dense locally-finite subgroup of the automorphism group of a free homogeneous structure<\/i><\/a>\r\n<div id=\"absbox-c61\" class=\"abstract_box\">\r\nDaoud Siniora, <i>A dense locally-finite subgroup of the automorphism group of a free homogeneous structure<\/i>\r\n<br><br>\r\nConsider an amalgamation class of finite relational structures which has the free amalgamation property. We show that the automorphism group of its Fraisse limit has a dense locally-finite subgroup.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c61');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Dario Garcia, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c63')\"><i>On variations of unimodularity and measurability<\/i><\/a>\r\n<div id=\"absbox-c63\" class=\"abstract_box\">\r\nDario Garcia, <i>On variations of unimodularity and measurability.<\/i>\r\n<br><br>\r\nUnimodularity was defined by Hrushovski in [3] where he proved that a unimodular strongly minimal set is one-based, thus generalising Zilber's result that a locally finite strongly minimal set is locally modular. Recently, Hrushovski has re-visited unimodularity in the context of pseudofinite structures, aiming to develop an intersection theory for definable pseudofinite sets. It was claimed in [3] that unimodularity was equivalent to an <i>a priori<\/i> different notion called <i>functional unimodularity<\/i> in [1] and [2].\r\n<br>\r\nPillay and Kestner [4] have distinguished two types of functional unimodularity: one for definable sets and one for type-definable sets. They also showed that for strongly minimal theories, unimodularity is equivalent to functional unimodularity for arbitrary types, and is also equivalent to the structures being <i>measurable<\/i> in the sense of [5].\r\n<br>\r\nIn this talk we introduce yet another variant called <i>correspondence unimodularity<\/i> and present a study of the relationship between these different concepts. The main results state that unimodularity is equivalent to both correspondence unimodularity and to functional unimodularity for complete types, and for $\\omega$-stable theories unimodularity is also equivalent to both correspondence and functional unimodularity for partial types.\r\n<br>\r\nFinally, we showed that all variants of unimodularity coincide for strongly minimal theories, and more generally for non multi-dimensional theories where the dimensions are associated to strongly minimal types.\r\n<br>\r\nThis is joint work with Frank Wagner (Universit\u00e9 Claude Bernard - Lyon 1).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nR. Elwes,\r\nAsymptotic classes of finite structures,\r\nJournal of Symbolic Logic,\r\nvol. 72 (2007), no. 2, pp. 418\u2013438.\r\n<br>\r\n[2]\r\nR. Ewes, E. Jaligot, D. Macpherson, M. Ryten,\r\nGroups in simple and pseudofinite theories,\r\nProceedings of the London Mathematical Society,\r\nvol. 103 (2011), no. 6, pp. 1049\u20131082.\r\n<br>\r\n[3]\r\nE. Hrushovski.,\r\nUnimodular minimal structures.,\r\nJournal of the London Mathematical Society,\r\nvol. 46 (1992), no. 3, pp. 385\u2013396.\r\n<br>\r\n[4]\r\nC. Kestner, A. Pillay,\r\nRemarks on unimodularity,\r\nJournal of Symbolic Logic.,\r\nvol.&nbsp;76 (2011), no.&nbsp;4, pp.&nbsp;1453\u20131458.\r\n<br>\r\n[5]\r\nD. Macpherson, C. Steinhorn.,\r\nDefinability in classes of finite structures,\r\nFinite and Algorithmic Model Theory.\r\n(Javie Esparza, Christian Michaux and Charles Steinhorn, editors),\r\nLondon Mathematical Society,\r\nCambridge University Press,\r\n2011,\r\npp.&nbsp;140\u2013176.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c63');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Sylvy Anscombe, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c65')\"><i>Generalised measurable structures with the Tree Property<\/i><\/a>\r\n<div id=\"absbox-c65\" class=\"abstract_box\">\r\nSylvy Anscombe, <i>Generalised measurable structures with the Tree Property<\/i>\r\n<br><br>\r\nIn [1], Chatzidakis, van den Dries, and Macintyre gave an asymptotic description of the number of points in definable sets in finite fields,\r\nbuilding on earlier celebrated work of Lang and Weil, [2], for varieties.\r\nLater, in [3], Macpherson and Steinhorn turned these results into the definition of a `one-dimensional asymptotic class'.\r\nThis is a class $\\mathcal{C}$ of finite structures in which, given a definable set $X$, there is a real number $r$ and a natural number $d$ such that the number of points in $X$ in a structure $M\\in\\mathcal{C}$ is approximately equal to $r\\,|M|^{d}$.\r\nMoreover, the pair $(r,d)$ may be chosen somewhat uniformly, if $X$ is allowed to vary through a definable family.\r\nTaking ultraproducts of such classes, yields a `measurable structure':\r\nan infinite structure equipped with a function\r\n\\begin{align*}\r\n\\mathrm{Def}(M)&\\longrightarrow\\mathbb{R}_{\\geq0}\\times\\mathbb{N}\r\n<br>\r\nX&\\longmapsto(r,d)\r\n\\end{align*}\r\nsatisfying certain natural axioms\r\nwhich correspond to the intuition that $r$ is the `measure' and $d$ is the `dimension' of $X$.\r\nIn particular, the fact that dimension takes values in the natural numbers implies that any measurable structures is supersimple, of finite rank.\r\nIn joint work between the speaker and Macpherson, Steinhorn, and Wolf, we have broadened this framework to allow more exotic measures and dimensions.\r\nWe call such structures `generalised measurable'.\r\nIn this talk we describe several key examples of generalised measurable structures, including the generic triangle-free graph which has the Tree Property of the First Kind.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nZoe Chatzidakis, Lou van den Dries, and Angus Macintyre\",\r\nDefinable sets over finite fields\",\r\nJournal fur die Reine und Angewandte Mathematik,\r\nvol.&nbsp;427 (1992), pp.&nbsp;107\u2013136.\r\n<br>\r\n[2]\r\nSerge Lang and Andre Weil,\r\nNumber of points of varieties in Finite Fields\",\r\nAmerican Journal of Mathematics,\r\nvol.&nbsp;76 (1954), pp.&nbsp;819\u2013827.\r\n<br>\r\n[3]\r\nDugald Macpherson and Charles Steinhorn,\r\nOne-dimensional asymptotic classes of finite structures\",\r\nTransactions of the American Mathematical Society,\r\nvol.&nbsp;360 (2008), pp.&nbsp;411\u2013448.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c65');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Alireza Mofidi, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c67')\"><i>Some symbolic dynamical views in model theory<\/i><\/a>\r\n<div id=\"absbox-c67\" class=\"abstract_box\">\r\nAlireza Mofidi, <i>Some symbolic dynamical views in model theory<\/i>\r\n<br><br>\r\nThe topological dynamical aspects of model theory in particular stability theory has been recently investigated in several papers such as [2], [3] and [4].\r\nWe will have a symbolic dynamical and ergodic theoretical point of view, two other essential point of views in the theory of dynamical systems, to\r\nthe action of automorphisms and definable groups on certain model theoretic objects, such as stone spaces, models, etc, in particular in the presence of invariant measures.\r\nNote that applications of measures as a technique in stability theory are extensively studied in several papers such as [1].\r\nWe borrow the notion of symbolic representation from dynamical systems theory and develop it in the context of model theory. In particular, the symbolic representations of actions on spaces of types will be under consideration.\r\nWe generalize this notion in a way that deals with some standard definition of products of types (studied in for example [3])\r\nas well as product of measures.\r\nIn the particular case of the action of the group $\\mathbb{Z}$ (which includes the case of action of single automorphisms), we characterize some stability theoretic hierarchies in particular NIP.\r\nMoreover, we make connections between the symbolic representations of spaces of types\r\nand certain mathematical notions such as the rotation number of circle homeomorphisms, Bohr sets and some additive number theoretic definitions.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nE. Hrushovski and A. Pillay,\r\nOn NIP and invariant measures,\r\nJournal of the European Mathematical Society,\r\nvol.13 (2011), pp.1005\u20131061.\r\n<br>\r\n[2]\r\nL. Newelski,\r\nTopological dynamic of definable group actions,\r\nJournal of Symbolic Logic,\r\n74 (2009), pp. 50-72.\r\n<br>\r\n[3]\r\nL. Newelski,\r\nModel theoretic aspects of the Ellis semigroup,\r\nIsrael J. Math,\r\n190(2012), 477-507.\r\n<br>\r\n[4]\r\nA. Pillay,\r\nTopological dynamics and definable groups,\r\nJ. Symbolic Logic,\r\n78 (2013),  no. 2, 657-666.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c67');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Alexandre Ivanov, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c69')\"><i>Sofic metric groups and continuous logic<\/i><\/a>\r\n<div id=\"absbox-c69\" class=\"abstract_box\">\r\nAlexandre Ivanov, <i>Sofic metric groups and continuous logic<\/i>\r\n<br><br>\r\nLet us consider the class $\\mathcal{G}$ of all continuous\r\nstructures which are metric groups $(G,d)$ with\r\nbi-invariant metrics $d\\le 1$.\r\nLet $\\mathcal{G}_{sof} \\subset \\mathcal{G}$\r\nbe the subclass of  all closed metric\r\nsubgroups of metric ultraproducts of\r\nfinite symmetric groups with Hamming metrics.\r\nWe call metric groups from $\\mathcal{G}_{sof}$\r\nsofic metric groups.\r\nThe class $\\mathcal{G}_{w.sof}$\r\nof weakly sofic continuous metric groups,\r\nconsists of continuous metric groups $(G,d)$ which embed\r\ninto metric ultraproducts of finite metric groups with\r\ninvariant metrics bounded by 1.\r\nIn a similar way we define the classes\r\n$\\mathcal{G}_{l.sof}$ and  $\\mathcal{G}_{hyplin}$\r\nof continuous metric groups which are\r\nlinear sofic and hyperlinear as metric groups.\r\nWe study relationship among the classes\r\nof the collection\r\n$$\r\n\\{ \\mathcal{G} ,  \\mathcal{G}_{sof} , \\mathcal{G}_{w.sof} ,  \\mathcal{G}_{hyplin},\r\n\\mathcal{G}_{l.sof} \\} .\r\n$$\r\nAll of them are axiomatizable in continuous logic.\r\nIt is clear that $\\mathcal{G}_{sof} \\subseteq  \\mathcal{G}_{w.sof} \\subseteq \\mathcal{G}$.\r\nMoreover, by some arguments of Arzhantseva and P\u00e4unescu\r\n$\\mathcal{G}_{l.sof} \\subseteq \\mathcal{G}_{w.sof}$.\r\nWe show that the class\r\n$\\mathcal{G}_{w.sof} \\setminus (\\mathcal{G}_{sof} \\cup  \\mathcal{G}_{hyplin} \\cup \\mathcal{G}_{l.sof})$\r\nis not empty.\r\nWe emphasize that in these classes groups are\r\nconsidered together with metrics.\r\nThus Gromov's problem if any group is sofic\r\n[1] still remains open.\r\nWe also show that the question of all possible\r\ninclusions above is reduced to\r\ndiscrete members of these classes.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nV.Pestov,\r\nHyperlinear and sofic groups: a brief guide,\r\nBulletin of Symbolic Logic,\r\nvol.&nbsp;14 (2008), no.&nbsp;4, pp.&nbsp;449\u2013480.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c69');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Juan de Vicente, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c71')\"><i>Locally C-Nash groups<\/i><\/a>\r\n<div id=\"absbox-c71\" class=\"abstract_box\">\r\nJuan de Vicente, <i>Locally $\\mathbb{C}$-Nash groups<\/i>\r\n<br><br>\r\nLocally $\\mathbb{C}$-Nash groups are analytic groups which also carry a semialgebraic structure \u2013- seing $\\mathbb{C}$ as $\\mathbb{R}^2$.\r\nFor example,  the universal coverings of algebraic groups are locally C-Nash groups.\r\nThe definition of the latter is based on the concept of $\\mathbb{C}$-Nash map, which has been studied \u2013- with may\r\nbe other names \u2013- by different authors (see e.g. [1],[2]).\r\nIn this talk we give a classification of abelian locally $\\mathbb{C}$-Nash groups of dimension one and two.\r\n(Joint work with E. Baro and M. Otero.)\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nJ. Adamus and S. Randriambololona,\r\nTameness of holomorphic closure dimension in a semialgebraic set,\r\nMathematische Annalen,\r\nvol.355 (2013), no.3, pp.985\u20131005.\r\n<br>\r\n[2]\r\nY. Peterzil and S. Starchenko,\r\nComplex analytic geometry in a nonstandard setting,\r\nModel theory with applications to algebra and analysis. Vol. 1\r\n(Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie, editors),\r\nCambridge Univ. Press,\r\n2008,\r\npp. 117\u2013165.\r\n<br>\r\n[3]\r\nE. Baro, J. de Vicente, M. Otero,\r\nLocally $\\mathbb{C}$-Nash groups,\r\nIn preparation.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c71');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Computability Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Nurlan Kogabaev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c73')\"><i>Freely generated projective planes with finite computable dimension<\/i><\/a>\r\n<div id=\"absbox-c73\" class=\"abstract_box\">\r\nNurlan Kogabaev, <i>Freely generated projective planes with finite computable dimension<\/i>\r\n<br><br>\r\nIn [1] it was proved that every countable free projective plane has computable dimension either $1$ or $\\omega$. Futhermore, such a plane is computably categorical if and only if it has finite rank.\r\nHence the natural question arises: can we extend the above results to the case of freely generated projective planes? In particular, we are interested in the existence question of computably categorical freely generated projective plane of infinite rank. We also investigate the realizability of finite computable dimension $n>1$ in the class of freely generated projective planes.\r\nIn [2] it was shown that the class of symmetric irreflexive graphs is <i>complete<\/i> in the following computable-model-theoretic sense: for every countable structure $\\mathcal{A}$, there exists a countable symmetric irreflexive graph $\\mathcal{G}$ which has the same degree spectrum as $\\mathcal{A}$, the same $\\mathbf{d}$-computable dimension as $\\mathcal{A}$ (for each degree $\\mathbf{d}$), the same computable dimension as $\\mathcal{A}$ under expansion by a constant, and which realizes every degree spectrum $\\mathrm{DgSp}_{\\mathcal{A}}(R)$ (for every relation $R$ on $\\mathcal{A}$) as the degree spectrum of some relation on $\\mathcal{G}$.\r\nIn the present paper we construct an effective coding of symmetric irreflexive graphs into freely generated projective planes preserving most computable-model-theoretic properties and obtain the following result.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nThe class of freely generated projective planes is complete with respect to degree spectra of nontrivial structures, $\\mathbf{d}$-computable dimensions, expansion by constants, and degree spectra of relations.\r\nIn particular, for every natural $n\\geqslant 1$ there exists a freely generated projective plane of infinite rank with computable dimension $n$.\r\n<br><br>\r\nThis work was supported by RFBR (grant 14-01-00376-a) and by the Grants Council under RF President for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nN.T.Kogabaev,\r\nThe class of projective planes is noncomputable,\r\nAlgebra and Logic,\r\nvol.&nbsp;47 (2008), no.&nbsp;4, pp.&nbsp;242\u2013257.\r\n<br>\r\n[2]\r\nD.R.Hirschfeldt, B.Khoussainov, R.A.Shore, A.M.Slinko,\r\nDegree spectra and computable dimensions in algebraic structures,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;115 (2002), no.&nbsp;1-3, pp.&nbsp;71\u2013113.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c73');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Ruslan Kornev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c75')\"><i>Reducibilities of computable metrics on the real line<\/i><\/a>\r\n<div id=\"absbox-c75\" class=\"abstract_box\">\r\nRuslan Kornev, <i>Reducibilities of computable metrics on the real line<\/i>\r\n<br><br>\r\nWe are concerned with a natural computable-model-theoretic question for uncountable metric spaces: do there exist computable metric spaces, equivalent in some standard sense but not computably equivalent? Pour-El and Richards&nbsp;[2] considered different countable dense substructures of a given Banach space up to computable isometries; see also more recent papers&nbsp;[3] and&nbsp;[4]. In contrast, our goal is to construct computably inequivalent metrics on a separable space with fixed dense subset. In this talk, two notions of computable reducibility of metrics are discussed.\r\nOne of them is induced by reducibility of Cauchy representations. Namely, let $\\rho$ and $\\rho'$ be complete metrics on a separable space $X$, let $W$ be a countable dense subset of $X$, enumerated by integers. We say that the metric $\\rho$ is computably reducible to the metric $\\rho'$ (and denote this as $\\rho\\leq_c\\rho'$) if the Cauchy representation $\\delta_\\rho$ of effective metric space $(X, \\rho, W)$ is computably reducible to the representation $\\delta_{\\rho'}$ of the space $(X, \\rho', W)$; precise definitions can be found in&nbsp;[1].\r\nIt is possible to characterize $\\leq_c$ in the following terms: $\\rho\\leq_c\\rho'$ iff the identity homeomorphism $\\text{id}_X$ is $(\\delta_\\rho,\\delta_{\\rho'})$-computable. Based on this, we introduce weak reducibility of metrics: we say $\\rho\\leq_{ch}\\rho'$ if there exists a $(\\delta_\\rho,\\delta_{\\rho'})$-computable autohomeomorphism of $X$. Clearly, $c$-reducibility implies $ch$-reducibility.\r\nThe results obtained are mainly related to the case $X=\\mathbb{R}$ with the standard real line topology and $W=\\mathbb{Q}$.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nAll convex (i.e., admitting midpoints) computable metrics on the reals are $c$-equivalent.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nAny countable tree $T$ can be isomorphically embedded into the ordering of computable metrics on the reals under $c$-reducibility.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nThere exists a countable sequence of computable metrics on $\\mathbb{R}$ which are not $ch$-reducible to each other. Informally, copies of the real line, equipped with these metrics, are pairwise homeomorphic, but not computably homeomorphic.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nK.&nbsp;Weihrauch,\r\nComputable Analysis. An Introduction,\r\nSpringer-Verlag,\r\nBerlin\/Heidelberg,\r\n2000.\r\n<br>\r\n[2]\r\nM.&nbsp;B.&nbsp;Pour-El, J.&nbsp;I.&nbsp;Richards, Computability in Analysis and Physics, Springer-Verlag, Berlin, 1989.\r\n<br>\r\n[3]\r\nZ.&nbsp;Iljazovi\u0107, Isometries and Computability Structures, Journal of Universal Computer Science, vol.&nbsp;16 (2010), no.&nbsp;18, pp.&nbsp;2569\u20132596.\r\n<br>\r\n[4]\r\nA.&nbsp;G.&nbsp;Melnikov, Computably Isometric Spaces, Journal of Symbolic Logic, vol.&nbsp;78 (2013), no.&nbsp;4, pp.&nbsp;1055\u20131085.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c75');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Nikolay Bazhenov, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c77')\"><i>Effective categoricity for polymodal algebras<\/i><\/a>\r\n<div id=\"absbox-c77\" class=\"abstract_box\">\r\nNikolay Bazhenov, <i>Effective categoricity for polymodal algebras<\/i>\r\n<br><br>\r\nLet $\\mathbf{d}$ be a Turing degree. A computable structure $\\mathcal{A}$ is $\\mathbf{d}$-computably categorical if for every computable structure $\\mathcal{B}$ isomorphic to $\\mathcal{A}$, there is a $\\mathbf{d}$-computable isomorphism from $\\mathcal{A}$ onto $\\mathcal{B}$. The categoricity spectrum of $\\mathcal{A}$ is the set\r\n$$\r\nCatSpec(\\mathcal{A}) = \\{ \\mathbf{d} \\,\\colon \\mathcal{A} \\text{ is } \\mathbf{d} \\text{-computably categorical} \\}.\r\n$$\r\nThe results of&nbsp;[1] imply that not every categoricity spectrum is the categoricity spectrum of a Boolean algebra. A natural question that arises is the following: how does expanding the language of Boolean algebras affect categoricity spectra and other related properties?\r\nHirschfeldt, Khoussainov, Shore, and Slinko&nbsp;[2] introduced the notion of a class which is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. For brevity, we call such classes HKSS-comp\\-lete. If a class $K$ is HKSS-complete, then for every computable structure $\\mathcal{S}$, there is a structure $\\mathcal{A}_{\\mathcal{S}}\\in K$ with the property $CatSpec(\\mathcal{A}_{\\mathcal{S}}) = CatSpec(\\mathcal{S})$.\r\nKhoussainov and Kowalski&nbsp;[3] proved that the class of Boolean algebras with operators is HKSS-complete. They also asked whether the similar result is true for polymodal algebras. Here we give the positive answer to the question:\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nThe class of Boolean algebras with four distinguished modalities is HKSS-complete.\r\n<br><br>\r\nIn particular, this result implies that every categoricity spectrum is the categoricity spectrum of some polymodal algebra.\r\nThis work was supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nN.A. Bazhenov,\r\n$\\Delta^0_2$-categoricity of Boolean algebras,\r\nJournal of Mathematical Sciences,\r\nvol.&nbsp;203 (2014), no.&nbsp;4, pp.&nbsp;444\u2013454.\r\n<br>\r\n[2]\r\nD.R. Hirschfeldt, B. Khoussainov, R.A. Shore, A.M. Slinko,\r\nDegree spectra and computable dimensions in algebraic structures,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;115 (2002), no.&nbsp;1\u20133, pp.&nbsp;71\u2013113.\r\n<br>\r\n[3]\r\nB. Khoussainov, T. Kowalski,\r\nComputable isomorphisms of Boolean algebras with operators,\r\nStudia Logica,\r\nvol.&nbsp;100 (2012), no.&nbsp;3, pp.&nbsp;481\u2013496.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c77');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Sergey Goncharov, Nikolay Bazhenov and Margarita Marchuk, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c79')\"><i>Autostability relative to strong constructivizations of computable 2-step nilpotent groups<\/i><\/a>\r\n<div id=\"absbox-c79\" class=\"abstract_box\">\r\nSergey Goncharov, Nikolay Bazhenov and Margarita Marchuk, <i>Autostability relative to strong constructivizations of computable 2-step nilpotent groups<\/i>\r\n<br><br>\r\nFor a class K of structures, closed under isomorphism, the index set is the set I(K) of all indices for computable members of K in a universal computable numbering of all computable structures for a fixed computable language. We study the complexity of the index set of class of computable structures, which are autostable relative to strong constructivizations.\r\nA computable model $\\mathcal{M}$ is called strongly constructivizable if there exists a decidable model $\\mathcal{N}$ such that $\\mathcal{N}$ is isomorphic to $\\mathcal{M}$. A strongly constructivizable model $\\mathcal{M}$ is autostable relative to strong constructivizations if for any decidable copies $\\mathcal{N}_0$ and $\\mathcal{N}_1$ of the model $\\mathcal{M}$, there is a computable isomorphism $f : \\mathcal{N}_0 \\rightarrow \\mathcal{N}_1$.\r\nUsing the result of [1] for rings and a coding of rings into groups due to Mal'cev [2]\r\nwe prove the following theorem.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nThe index set $SCAut(Gr)$ of computable 2-step nilpotent groups, which are autostable relative to strong constructivizations is $m$-complete $\\Sigma^{0}_3(\\emptyset^{\\omega})$.\r\n<br><br>\r\nThis work was supported by RFBR (grant 14-01-00376) and  by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1). The second author was supported by RFBR (research project No. 16-31-60058 mol\\_a\\_dk).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nS. S. Goncharov, N. A Bazhenov., M. I. Marchuk,\r\nThe index set of Boolean algebras autostable relative to strong constructivizations,\r\nSiberian Mathematical Journal,\r\nvol.&nbsp;56 (2015), no.&nbsp;3, pp.&nbsp;393\u2013404.\r\n<br>\r\n[2]\r\nA. Mal\u00edcev,\r\nOn a correspondence between rings and groups,\r\nAmerican Mathematical Society Translations,\r\nseries&nbsp;2, vol.&nbsp;45 (1965), pp.&nbsp;221\u2013232.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c79');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Birzhan Kalmurzaev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c81')\"><i>Note on cardinality of Rogers semilattice<\/i><\/a>\r\n<div id=\"absbox-c81\" class=\"abstract_box\">\r\nBirzhan Kalmurzaev, <i>Note on cardinality of Rogers semilattices<\/i>\r\n<br><br>\r\nIt is easy to show that\r\n<ul>\r\n<li>For every family $\\mathcal{S}$ on $n$-c.e. sets, if the Rogers semilattice $\\mathcal{R}_m^{-1}(\\mathcal{S})$ is infinite for some $m\\geq n$ then $\\mathcal{R}_m^{-1}(\\mathcal{S})$ is infinite for all $k\\geq m$.\r\n<li>For every two-element family of $n$-c.e. sets $\\mathcal{S}=\\{A,B\\}$, the Rogers samilattices $\\mathcal{R}_m^{-1}(\\mathcal{S})$ is infinite if $m>2n$. If $n$ is even, then this statement is true for $m=2n$.\r\n<\/ul>\r\nAll known Rogers semilattices of the family in the hierarchy of Ershov are either one-element or infinite.\r\nTheorem ([1]).\r\nFor every nonzero $n\\in \\omega \\cup \\{\\omega\\}$, and for every ordinal notation $a$ of a nonzero ordinal, there exists a $\\Sigma_a^{-1}$-computable family $\\mathcal{A}$ of exactly $n$ sets such that $|\\mathcal{R}_a^{-1}(\\mathcal{A})|=1$.\r\nMain result:\r\nTheorem.\r\nFor every nonzero $n\\in \\omega$, there exist $n$-c.e. sets $A$ and $B$ such that\r\n$$|\\mathcal{R}_n^{-1}(A,B)|=|\\mathcal{R}_{n+1}^{-1}(A,B)|=\\ldots=|\\mathcal{R}_{2n}^{-1}(A,B)|=1 &nbsp; \\text{if}&nbsp; n&nbsp; \\text{is odd},$$\r\n$$|\\mathcal{R}_n^{-1}(A,B)|=|\\mathcal{R}_{n+1}^{-1}(A,B)|=\\ldots=|\\mathcal{R}_{2n-1}^{-1}(A,B)|=1&nbsp;  \\text{if}&nbsp; n&nbsp; \\text{is even}.$$\r\nCorollary.\r\nFor every nonzero $n\\in \\omega$ and for every $n&lt;m\\leq2n$, there exist $n$-c.e. sets $A$ and $B$ such that\r\n$$1=|\\mathcal{R}_n^{-1}(A,B)|=\\ldots=|\\mathcal{R}_{m-1}^{-1}(A,B)|&lt;|\\mathcal{R}_{m}^{-1}(A,B)|.$$\r\nQuestion.\r\nDoes there exist a family of sets in some level of the hierarchy of Ershov whose Rogers semilattice consists of $2, 3, \\ldots$ elements?\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSerikzhan A. Badaev, Mustafa Manat, Andrea Sorbi,\r\nRogers semilattices of families of two embedded sets in the Ershov hierarchy,\r\nMathematical Logic Quarterly,\r\nvol.&nbsp;58 (2012), no.&nbsp;4-5, pp.&nbsp;366\u2013376.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c81');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Categorical Logic and Type Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Dimitris Tsementzis, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c83')\"><i>A Syntactic Characterization of Morita Equivalence<\/i><\/a>\r\n<div id=\"absbox-c83\" class=\"abstract_box\">\r\nDimitris Tsementzis, <i>A Syntactic Characterization of Morita Equivalence<\/i>\r\n<br><br>\r\nWe characterize Johnstone's [2] topos-theoretic notion of Morita equivalence of theories in terms of an extended notion of a common definitional extension developed by Barrett and Halvorson [3].\r\nWe thus provide a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos.\r\nOur investigation may be understood as an inversion of the work of Awodey-Forsell [1] and Makkai [5]. There the question is asked: what extra structure do we need to impose on the category of models of a theory in order to recover the theory up to logical equivalence?\r\nHere, we answer the following: if we identify a theory with its category of models, then from a syntactic point of view what can we recover it up to?\r\nWe will first describe in detail the proof in the coherent fragment of (many-sorted) first-order logic.\r\nThen we will describe how to extend our method to regular, cartesian, geometric and full first-order theories.\r\nFinally, as an application, we provide a new proof (originally in [4]) that $\\omega$-categoricity for complete geometric theories is invariant under Morita equivalence.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nS. Awoday, H. Forsell,\r\nFirst-Order Logical Duality,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;164 (2013), no.&nbsp;3, pp.&nbsp;319\u2013348.\r\n<br>\r\n[2]\r\nP. Johnstone,\r\nSketches of an Elephant: A Topos Theory Compendium,\r\nOxford University Press,\r\n2003.\r\n<br>\r\n[3]\r\nT. Barrett and H. Halvorson,\r\nMorita Equivalence,\r\nAvailable at <tt>http:\/\/arxiv.org\/abs\/1506.04675<\/tt>\r\n<br>\r\n[4]\r\nO. Caramello,\r\nAtomic Toposes and Countable Categoricity,\r\nApplied Categorical Structures,\r\nvol.&nbsp;4 (2012), pp.&nbsp;379\u2013391.\r\n<br>\r\n[5]\r\nM. Makkai,\r\nStone Duality for First-Order Logic,\r\nAdvances in Mathematics,\r\nvol.&nbsp;65 (1987), no.&nbsp;2 pp.&nbsp;97\u2013170.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c83');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Ian Orton and Andrew Pitts, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c85')\"><i>Axioms for Modelling Cubical Type Theory in a Topos<\/i><\/a>\r\n<div id=\"absbox-c85\" class=\"abstract_box\">\r\nIan Orton and Andrew Pitts, <i>Axioms for modelling cubical type theory in a topos<\/i>\r\n<br><br>\r\n  The homotopical approach to intensional type theory views proofs of\r\n  equality as paths. We explore what is required of an interval-like\r\n  object $\\mathtt{I}$ in a topos to give a model of type theory in which\r\n  elements of identity types are morphisms from $\\mathtt{I}$.  Cohen, Coquand,\r\n  Huber and M\u00f6rtberg give such a model using a particular category\r\n  of presheaves [1].  We investigate the extent to which their model\r\n  construction can be expressed in the internal type theory of any\r\n  topos and identify a collection of quite weak axioms for this\r\n  purpose. This clarifies the definition and properties of the notion\r\n  of Kan filling that lies at the heart of their constructive\r\n  interpretation of Voevodsky's univalence axiom. Furthermore, since\r\n  our axioms can be satisfied in a number of different ways, we show\r\n  that there is a range of topos-theoretic models of homotopy type\r\n  theory in this style. This work presented in this talk is described in more detail in [2]\r\n<br><br><b>References<\/b>\r\n<br>[1]\r\n C. Cohen, T. Coquand, S. Huber and\r\n                  A. M\u00f6rtberg,\r\n Cubical Type Theory: a Constructive Interpretation\r\n                  of the Univalence Axiom,\r\n Preprint,\r\nDec, 2015.\r\n<br>[2]\r\n I. Orton and A. Pitts,\r\n Axioms for Modelling Cubical Type Theory in a Topos,\r\n Preprint,\r\nApril, 2016.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c85');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Eric Faber, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c87')\"><i>Relative computability in realisability toposes<\/i><\/a>\r\n<div id=\"absbox-c87\" class=\"abstract_box\">\r\nEric Faber, <i>Relative computability in realisability toposes<\/i>\r\n<br><br>\r\nWhen the effective topos was first devised in [3], Martin Hyland cunningly revealed the\r\nremarkably rich structure of its lattice of local operators: this admits an embedding of the Turing\r\ndegrees. Consequently, a Turing degree represented by a function $f: \\mathbb{N} \\rightarrow\r\n\\mathbb{N}$ corresponds to a subtopos $\\mathit{Eff}_f$ of the effective topos. In [1],\r\nJaap van Oosten and the author show a partial converse to this result; namely that restricted to the\r\ncategory of realisability toposes, the subtoposes of $\\mathit{Eff}$ are precisely those of the form\r\n$\\mathit{Eff}_f$ for a <i>partial<\/i> function $f$.  Here, a realisability topos $\\text{rt}(A)$ is a\r\ntopos constructed from a tripos on a partial combinatory algebra (pca) $A$.\r\nThe partial converse seems particular to the effective topos, but Hylands observation can be\r\nextended to realisability toposes. For any pca $A$ and partial map $f: A \\rightarrow A$, there is a\r\ncorresponding subtopos $\\text{rt}(A)_f$ of $\\text{rt}(A)$, which is a realisability topos, and for\r\ntotal functions $f$ this is the largest subtopos in which $f$ is computable,\r\nor realised.\r\nIn [2], we exhibit a technique to force a <i>functional<\/i> $F: A^A \\rightarrow A$ to be\r\ninternally realised. The resulting class of subtoposes $\\text{rt}(A)_F$ relative to a functional $F$\r\nhappens to be contained in the class of subtoposes $\\text{rt}(A)_f$ relative to a partial function\r\n$f$. In the effective topos, the technique can be used to force internal functionals on the natural\r\nnumbers object. This theory is related to the theory on effective operations.\r\nWith this work we aim to extract the computational content from local operators, which one can now\r\nregard as \"generalised oracles\". Examples of local operators not captured by\r\nthe above show that there is still a lot to be understood.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nFaber, E. and van Oosten, J.,\r\nMore on geometric morphisms between realizability toposes,\r\nTheory and Applications of Categories,\r\nvol.&nbsp;29 (2014), pp.&nbsp;874\u2013895.\r\n<br>\r\n[2]\r\n\u2015\r\nEffective operations of type 2 in PCAs,\r\nComputability,\r\nvol.&nbsp;5 (2016), no.&nbsp;2, pp.&nbsp;127\u2013146.\r\n<br>\r\n[3]\r\nHyland, J.M.E.,\r\nThe effective topos,\r\nThe L.E.J. Brouwer Centenary Symposium\r\n(Noordwijkerhout, 1981),\r\nStudies in Logic and the Foundations of Mathematics,\r\nvol.&nbsp;110,\r\nNorth-Holland,\r\nAmsterdam-New York,\r\n1982,\r\npp.&nbsp;165\u2013216.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c87');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Jacopo Emmenegger and Erik Palmgren, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c89')\"><i>Exact completion and constructive theories of sets<\/i><\/a>\r\n<div id=\"absbox-c89\" class=\"abstract_box\">\r\nJacopo Emmenegger and Erik Palmgren, <i>Exact completion and constructive theories of sets<\/i>\r\n<br><br>\r\nIn&nbsp;[2] Palmgren proposed a constructive and predicative version of\r\nLawvere's Elementary Theory of the Category of Sets (ETCS), called CETCS,\r\nwhich provides a structuralist foundation for constructive mathematics in the style of Bishop.\r\nAs shown in&nbsp;[2], a CETCS category is precisely a well-pointed locally cartesian closed pretopos\r\nwith a natural number object and enough projectives.\r\nIn both the intended models of CETCS categories, namely setoids in Martin-L\u00f6f type theory and sets in Aczel's CZF,\r\nsets can be seen as quotients of projective sets and can thus be regarded as exact completions.\r\nWe generalise this approach to CETCS categories and characterise them in terms of weaker properties of their projective covers.\r\nIn particular, it seems that Carboni and Rosolini's characterisation of local closure&nbsp;[1]\r\nin terms of weak closure of the maximal projective cover cannot be applied in this case,\r\nbecause splitting of idempotents is undecidable in Martin-L\u00f6f type theory.\r\nWe solve this issue providing an alternative characterisation of local closure that applies to any projective cover.\r\nWe apply this characterisation to the category of small types in Martin-L\u00f6f type theory with a universe and basic type formers,\r\nobtaining as a consequence that the category of setoids is a CETCS category.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\n{A. Carboni and G. Rosolini},\r\n{Locally cartesian closed exact completions},\r\n{Journal of Pure and Applied Algebra},\r\nvol.&nbsp;154 (2000), no.&nbsp;1\u20133, pp.&nbsp;103\u2013116.\r\n<br>\r\n[2]\r\n{E. Palmgren},\r\n{Constructivist and structuralist foundations: Bishop's and Lawvere's theories of sets},\r\n{Annals of Pure and Applied Logic},\r\nvol.&nbsp;163 (2012), no.&nbsp;10, pp.&nbsp;1384\u20131399.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c89');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Maria Emilia Maietti, Fabio Pasquali and Giuseppe Rosolini, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c91')\"><i>When the tripos-to-topos construction factors through the elementary quotient completion<\/i><\/a>\r\n<div id=\"absbox-c91\" class=\"abstract_box\">\r\nMaria Emilia Maietti, Fabio Pasquali and Giuseppe Rosolini, <i>When the tripos-to-topos construction factors through the elementary quotient completion<\/i>\r\n<br><br>\r\nMaietti and Rosolini in [4] generalized the notion of exact completion on a weakly lex category in [3]\r\nto that of elementary quotient completion of a Lawvere's elementary doctrine.\r\nFrom the logic point of view a Lawvere's elementary doctrine, denoted\r\nwith  $(\\mathbb{C},P)$,  can be seen as a many sorted conjunctive logic $P$\r\nwith equality depending on sorts which are objects of a category $\\mathbb{C}$.  The elementary quotient completion of $(\\mathbb{C},P)$, denoted by $(\\mathbb{C}_q,P_q)$,  is a way to obtain a new elementary doctrine whose base is\r\nclosed under effective quotients of equivalence relations expressed in the logic $P_q$.\r\n<br>\r\nIn this talk we focus on triposes, which are a special class of Lawvere's elementary  doctrines  introduced in [1] to build elementary toposes by mean of the so-called tripos-to-topos construction.\r\nWe characterize when the tripos-to-topos construction\r\nfactors through an elementary quotient completion by using a result in [5].\r\n<br>\r\nOur main result is that an  elementary topos arising from a tripos $(\\mathbb{C},P)$ is the elementary quotient completion of the free full comprehension\r\nof the starting tripos $(\\mathbb{C},P)$ if and only if  the  tripos $(\\mathbb{C},P)$ validates a form of rule\r\nof choice inspired by Hilbert epsilon operator and introduced in [2].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nJ.M.E. Hyland, P.T. Johnstone, A.M.Pitts,\r\nTripos theory,\r\nMathematical Proceedings of the Cambridge Philosophical Society,\r\nvol.&nbsp;88 (1980), no.&nbsp;2, pp.&nbsp;205\u2013232.\r\n<br>\r\n[2]\r\nF. Pasquali,\r\nHilbert's {$\\epsilon$}-operator in doctrines,\r\nIfCoLog Journal of Logics and their Applications,\r\nTo appear.\r\n<br>\r\n[3]\r\nA. Carboni, R. Celia Magno,\r\nThe free exact category on a left exact one,\r\nJournal of the Australian Mathematical Society,\r\nvol.&nbsp;33 (1982), no.&nbsp;3, pp.&nbsp;295\u2013301.\r\n<br>\r\n[4]\r\nM.E.Maietti, G. Rosolini,\r\nElementary quotient completion,\r\nTheory and applications of categories,\r\nvol.&nbsp;27 (2013), no.&nbsp;17, pp.&nbsp;445\u2013463.\r\n<br>\r\n[5]\r\nM. E. Maietti, G. Rosolini,\r\nRelating quotient completions via categorical logic,\r\nIn Dieter Probst and Peter Schuster (eds.), \"Concepts of Proof in Mathematics, Philosophy, and Computer Science\". Ontos Mathematical Logic. Walter de Gruyter, Berlin.,\r\nTo appear.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c91');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Intuitionistic Logic and Theories <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Anupam Das, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c93')\"><i>Intuitionistic bounded arithmetic and monotone proof complexity<\/i><\/a>\r\n<div id=\"absbox-c93\" class=\"abstract_box\">\r\nAnupam Das, <i>Intuitionistic bounded arithmetic and monotone proof complexity<\/i>\r\n<br><br>\r\nThis work concerns the relationship between weak theories of arithmetic and the complexity of proofs in propositional logic. We introduce a hierarchy of intuitionistic theories of bounded arithmetic, based on Buss' second-order (classical) theories $U^1_2$ and $V^1_2$ [2] as developed in [3], by controling the type-level of induction formulae, i.e. the nesting to the left of an implication symbol.\r\nWe calibrate a Paris-Wilkie style propositional translation from bounded arithmetic with a Brouwer-Heyting-Kolmogorov interpretation of implication as transformation of proofs. As a result, we obtain a general translation from arbitrary proofs of $\\Pi^0_1$ sentences in intuitionistic bounded arithmetic to monotone propositional proofs, i.e. proofs free of negation. In the case of type-level 1 the complexity of the transformation is quasipolynomial-time. In order to deal with second-order existential quantifiers we prove a sort of witnessing theorem, reducing the interaction between induction and comprehension to a form of iterated comprehension, thereby sidestepping the need for extension variables that may introduce complications with respect to negation.\r\nFinally we show that this amount of negation, type-level 1, is sufficient to prove a converse result, the soundness of monotone propositional proofs, thereby establishing a full correspondence.\r\nThis talk is based on work appearing in [1].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAnupam Das,\r\nFrom positive and intuitionistic bounded arithmetic to monotone proof complexity,\r\nLICS '16\r\nAccepted.\r\n2016.\r\n<br>\r\n[2]\r\nSamuel R. Buss.\r\nBounded arithmetic,\r\nBibliopolis,\r\nNaples,\r\n1986.\r\n<br>\r\n[3]\r\nJan Kraj\u00ed\u010dek.\r\nBounded arithmetic, propositional logic, and complexity theory,\r\nCambridge University Press,\r\nNew York,\r\n1995.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c93');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Robert Lubarsky, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c95')\"><i>D-Fan and c-Fan<\/i><\/a>\r\n<div id=\"absbox-c95\" class=\"abstract_box\">\r\nRobert Lubarsky, <i>D-Fan and c-Fan<\/i>\r\n<br><br>\r\nBrouwer's Fan Theorem \u2013 that every binary tree with no infinite\r\npath is finite \u2013 is an important principle in constructive\r\nmathematics. Various fragments of the Fan Theorem, which limit the\r\ntrees to which it applies, have been of interest over the years,\r\nbecause they are equivalent to basic principles of analysis. These\r\nweakenings of Fan are easily seen to be implicationally linear:\r\n\\begin{equation*}\r\n\\textrm{FAN}_{\\mathrm{full}} \\Rightarrow\r\n\\textrm{FAN}_{\\Pi^{0}_{1}} \\Rightarrow \\textrm{FAN}_{c}\r\n\\Rightarrow \\textrm{FAN}_{\\Delta}.\r\n\\end{equation*}\r\nThe obvious question is whether those implications are\r\nstrict. Some of them were shown over the years to be strict, by a\r\nvariety of methods; in [4], they were all shown to be\r\nstrict, by a fairly uniform method. Here I provide a new proof\r\nthat Decidable Fan ($\\textrm{FAN}_{\\Delta}$) does not imply c-Fan\r\n($\\textrm{FAN}_{c}$). The argument is a mixture of realizability,\r\nHeyting-valued models, and Kripke models. It remains possible, yet\r\nstill unknown, that the same method will separate the other\r\nimplications.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1] Michael Beeson,\r\nFoundations of Constructive Mathematics,\r\nSpringer, 1985\r\n<br>\r\n[2] Josef Berger,\r\nA separation result for varieties of Brouwer's Fan\r\nTheorem, in\r\nProceedings of the 10th Asian Logic Conference (ALC 10),\r\nKobe University in Kobe, Hyogo, Japan, September 1-6, 2008 (Arai et al., eds.),\r\nWorld Scientific, 2010, p. 85 -  92\r\n<br>\r\n[3] Michael Fourman and J.M.E. Hyland,\r\nSheaf models for analysis, in\r\nApplications of Sheaves, Lecture Notes in Mathematics 753\r\n(Fourman, Mulvey and Scott, eds.), Springer, 1979,  p.  280 - 301\r\n<br>\r\n[4]  Robert Lubarsky and Hannes Diener,\r\nSeparating the Fan Theorem and Its Weakenings,\r\nJournal of Symbolic Logic, 79 (2014), pp.\r\n792-813, doi: 10.1017\/jsl.2014.9\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c95');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Tatsuji Kawai and Matthew de Brecht, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c97')\"><i>Interactions between powerlocales and Scott topology on locally compact locales<\/i><\/a>\r\n<div id=\"absbox-c97\" class=\"abstract_box\">\r\nTatsuji Kawai and Matthew de Brecht, <i>Interactions between powerlocales and Scott topology on locally compact locales<\/i>\r\n<br><br>\r\nWe study interactions between three powerlocales\r\non a locally compact locale $X$:\r\nupper powerlocale $\\mathrm{P_U}(X)$;\r\nlower powerlocale $\\mathrm{P_L}(X)$;\r\nand Scott topology $\\mathbb{S}^{X}$,\r\nwhere $\\mathbb{S}^{X}$ is the exponential over the Sierpinski locale $\\mathbb{S}$.\r\nIt is well known that upper and lower powerlocales\r\ncommute&nbsp;[1],\r\nand that composition of upper and lower powerlocales coincides with\r\ntaking Scott topology twice&nbsp;[3], i.e.\r\n$\\mathrm{P_U}(\\mathrm{P_L}(X)) \\cong \\mathrm{P_L}(\\mathrm{P_U}(X)) \\cong\r\n\\mathbb{S}^{\\mathbb{S}^{X}}$.\r\nIn this talk, we show that three powerlocales commute in a mixed way,\r\ni.e. $\\mathrm{P_U}(\\mathbb{S}^{X}) \\cong \\mathbb{S}^{\\mathrm{P_L}(X)}$ and $\\mathrm{P_L}(\\mathbb{S}^{X})\r\n\\cong \\mathbb{S}^{\\mathrm{P_U}(X)}$.\r\nThis gives us a complete picture of how these powerlocales\r\non locally compact locales interact with each other.\r\nThe result is obtained in the setting of formal topology&nbsp;[2].\r\nThis work was supported by JSPS Core-to-Core Program, A. Advanced Research\r\nNetworks. The first author was supported by JSPS KAKENHI Grant Number\r\n15K15940.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nP.&nbsp;T.&nbsp;Johnstone and S.&nbsp;Vickers,\r\nPreframe presentations present,\r\nCategory Theory,\r\n(A.&nbsp;Carboni, M.&nbsp;Pedicchio, and G.&nbsp;Rosolini, editors),\r\nLecture Notes in Mathematics,\r\nvol.&nbsp;1488,\r\nSpringer,\r\nBerlin Heidelberg,\r\n1991,\r\npp.&nbsp;193\u2013212.\r\n<br>\r\n[2]\r\nG.&nbsp;Sambin,\r\nIntuitionistic formal spaces \u2013- a first communication,\r\nMathematical Logic and its Applications,\r\nvol. 305, (D.&nbsp;Skordev, editor),\r\nPlenum Press, New York, 1987, pp.&nbsp;187\u2013204.\r\n<br>\r\n[3]\r\nS.&nbsp;Vickers,\r\nThe double powerlocale and exponentiation: A case study in geometric\r\nlogic,\r\nTheory and Applications of Categories,\r\nvol.&nbsp;12 (2004), no.&nbsp;13, pp.&nbsp;272\u2013422.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c97');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Nonstandard Analysis and Arithmetic <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Tin Lok Wong, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c99')\"><i>A new construction of models of the Weak Koenig Lemma<\/i><\/a>\r\n<div id=\"absbox-c99\" class=\"abstract_box\">\r\nTin Lok Wong, <i>A new construction of models of the Weak K\\\"onig Lemma<\/i>\r\n<br><br>\r\nThe subsystem&nbsp;$\\mathrm{WKL}_0$ of second-order arithmetic\r\noccupies a prominent position in reverse mathematics.\r\nNonstandard models of&nbsp;$\\mathrm{WKL}_0$ are typically constructed\r\nusing forcing arguments in which conditions are trees.\r\nRecently, Enayat and I&nbsp;[1]\r\nfound an alternative construction\r\nbased on the <i>Arithmetized Completeness Theorem<\/i>&nbsp;(ACT),\r\nwhich is a formalization of G\u00f6del's Completeness Theorem in arithmetic.\r\nIn the talk,\r\nI will demonstrate how to enhance our method\r\nto perform more delicate constructions\r\nin Simpson\u2013Tanaka\u2013Yamazaki&nbsp;[2].\r\nIn particular,\r\nI will present a new proof of the conservativity of&nbsp;$\\mathrm{WKL}_0$ over&nbsp;$\\mathrm{RCA}_0$\r\nfor sentences of the form\r\n$\\forall X \\exists!Y \\theta(X,Y)$,\r\nwhere $\\theta(X,Y)$ is an arithmetical formula.\r\nThe heart of our construction is a forcing argument\r\nin which conditions are theories.\r\nThis research is joint with Ali Enayat,\r\nUniversity of Gothenburg, Sweden.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAli Enayat and Tin Lok Wong,\r\nModel theory of&nbsp;$\\mathrm{WKL}_0^*$,\r\nPreprint available at\r\n<tt>http:\/\/www.logic.univie.ac.at\/~wongt9\/papers\/ew-wkl0star.pdf<\/tt>.\r\n<br>\r\n[2]\r\nStephen G. Simpson, Kazuyuki Tanaka, and Takeshi Yamazaki,\r\nSome conservation results on Weak K\u00f6nig's Lemma,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;118 (2002), no.&nbsp;1\u20132, pp.&nbsp;87\u2013114.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c99');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Bruno Dinis and Fernando Ferreira, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c101')\"><i>Interpreting weak K\u00f6nig's lemma in nonstandard theories of arithmetic<\/i><\/a>\r\n<div id=\"absbox-c101\" class=\"abstract_box\">\r\nBruno Dinis and Fernando Ferreira, <i>Interpreting weak K\u00f6nig's lemma in nonstandard theories of arithmetic<\/i>\r\n<br><br>\r\nA number of nonstandard versions of G\u00f6del's system T (introduced in [4]) were recently developed. These systems are based on Nelson's IST version of nonstandard analysis [5]. Not without surprise, these developments pointed out relations between constructive notions and nonstandard ones. We show that weak K\u00f6nig's lemma can be interpreted in the nonstandard theories presented in [3] and in [1]. These interpretations provide yet another route for proving Friedman's conservation theorem of ${\\sf WKL}_0$ over ${\\sf RCA}_0$.\r\nThis work was done in collaboration with Fernando Ferreira [2].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nB.&nbsp;van den Berg, E.&nbsp;Briseid, and P.&nbsp;Safarik,\r\nA functional interpretation for nonstandard arithmetic,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;163(2012), pp. 1962\u20131994.\r\n<br>\r\n[2]\r\nB.&nbsp;Dinis, F.&nbsp;Ferreira,\r\nInterpreting weak K\u00f6nig's lemma in nonstandard theories of arithmetic.\r\nManuscript in preparation.\r\n<br>\r\n[3]\r\nF.&nbsp;Ferreira and J.&nbsp;Gaspar,\r\nNonstandardness and the bounded functional interpretation,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;166(2015), pp. 701\u2013712.\r\n<br>\r\n[4]\r\nK.&nbsp;G\u00f6del,\r\n\u00dcber eine bisher noch nicht ben\u00fctzte Erweiterung des finiten Standpunktes,\r\nDialectica,\r\nvol.&nbsp;12(1958), pp. 280\u2013287.\r\n<br>\r\n[5]\r\nE.&nbsp;Nelson,\r\nInternal set theory: A new approach\r\nto nonstandard analysis,\r\nBulletin of the American Mathematical Society,\r\nvol.&nbsp;83(1977), pp. 1165\u20131198.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c101');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Emanuele Bottazzi, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c103')\"><i>A nonstandard generalization of the space of distributions and the Schwartz impossibility theorem<\/i><\/a>\r\n<div id=\"absbox-c103\" class=\"abstract_box\">\r\nEmanuele Bottazzi, <i>A nonstandard generalization of the space of distributions and the Schwartz impossibility theorem<\/i>\r\n<br><br>\r\nIn 1954, L. Schwartz proved that there is no differential algebra $(A, +, \\otimes, D)$ in which the real distributions $\\mathcal{D}'$ can be embedded and the following conditions are satisfied:\r\n<ul>\r\n<li>$\\otimes$ extends the product over $C^0$ functions;\r\n<li>$D$ extends the distributional derivative $\\partial$;\r\n<li>the product rule holds: $D(u\\otimes v) = (Du)\\otimes v + u\\otimes(Dv)$.\r\n<\/ul>\r\nHowever, $\\mathcal{D}'$ can be embedded in a space of functions of nonstandard analysis in a way that conditions 1.\u20133. are satisfied with only an infinitesimal perturbation.\r\nLet $^\\ast\\mathbb{R}$ be a set of hyperreals of nonstandard analysis, let $\\varepsilon\\in\\,\\! ^\\ast\\mathbb{R}$ be a positive infinitesimal, and let $(^\\ast\\mathbb{R}^\\Lambda, +, \\cdot)$ be the vector space of $^\\ast\\mathbb{R}$-valued functions defined on $\\Lambda = \\{ n\\varepsilon : n \\in ^\\ast{\\mathbb{Z}} \\}$, with pointwise sum and product.\r\nLet $\\mathfrak{D}$ be the finite difference operator defined by\r\n$$\r\n\\mathfrak{D} u(x) = \\frac{u(x+\\varepsilon)-u(x)}{\\varepsilon}\r\n$$\r\nfor all $u \\in \\,\\! ^\\ast\\mathbb{R}^\\Lambda$.\r\nWe prove that there is an embedding $\\iota : \\mathcal{D}' \\rightarrow\\, ^\\ast\\mathbb{R}^\\Lambda$ and a projection $\\pi:\\, ^\\ast\\mathbb{R}^\\Lambda \\rightarrow \\mathcal{D}'$ satisfying:\r\n<br><br>\r\n&nbsp;0. for all $T \\in \\mathcal{D}'$, $\\pi(\\iota(T)) = T$;\r\n<br>\r\n&nbsp;1. the product over $^\\ast\\mathbb{R}^\\Lambda$ extends the product over $\\iota(C^0(\\mathbb{R}))$;\r\n<br>\r\n&nbsp;2'. $\\mathfrak{D}$ extends the distributional derivative $\\partial$ in the sense that, for all $T \\in \\mathcal{D}'$,\r\n$$\r\n\\pi (\\mathfrak{D}(\\iota (T))) = \\partial T;\r\n$$\r\n<br>\r\n&nbsp;3'. for all $u, v \\in\\,\\! ^\\ast\\mathbb{R}^\\Lambda$, the following discrete product rule holds:\r\n$$\r\n\\mathfrak{D}(u v)(x) = (\\mathfrak{D}u(x)) v(x) + u(x+\\varepsilon)(\\mathfrak{D}v(x)).\r\n$$\r\n<br><br>\r\nThus, $(^\\ast\\mathbb{R}^\\Lambda, +, \\cdot, \\mathfrak{D})$ provides a non-trivial generalization of the space of distributions.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c103');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Imme van den Berg, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c105')\"><i>Complete arithmetical solids and nonstandard analysis<\/i><\/a>\r\n<div id=\"absbox-c105\" class=\"abstract_box\">\r\nImme van den Berg, <i>Complete arithmetical solids and nonstandard analysis<\/i>\r\n<br><br>\r\nA <i>neutrix<\/i> is a convex additive subgroup of a nonstandard model of the\r\nreal numbers. Obvious neutrices are \u00a3, the external set of of\r\nlimited numbers and $\\oslash $, the external set of infinitesimals, as\r\ngroups they are not isomorphic. The set of non-isomorphic neutrices is at\r\nleast countable [1]. An <i>external number<\/i> $\r\n\\alpha $ is the sum $\\alpha =a+A$ of a nonstandard real number $a$ and a\r\nneutrix $A$. Due to the stability by some shifts, external numbers may be\r\nseen as mathematical models of vague transitions of Sorites type, orders of\r\nmagnitude, or errors of measurement [3].\r\nThe external numbers form a completely regular commutative semigroup (union\r\nof groups) for addition and multiplication. The distributive law holds up to\r\na neutrix. The order relation is total and respects the operations [3][2].\r\nDedekind completeness is valid in a model which is sufficiently saturated\r\nfor Nelson's reduction algorithm [4] to hold: definable\r\nhalflines either are cofinal with an external number, or there is an\r\nexternal number just beyond.\r\nThe structure is Archimedean for nonstandard natural numbers.\r\nIn a joint work with B.Dinis, University of Lisbon, we present a first-order\r\naxiomatics for the external numbers in the language $\\{+,\\cdot ,\\leq \\}$,\r\nusing algebraic, analytic and arithmetical axioms. A model is called a <i>\r\ncomplete arithmetical solid<\/i>. Its precise numbers ($A=\\{0\\}$) must be\r\ncontained in a nonstandard model for the real number system.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nI.P. van den Berg,\r\nA decomposition theorem for neutrices,\r\nAnnals of Pure and Applied Logic,\r\nvol. 161 (2010), no. 7, pp. 851\u2013865.\r\n<br>\r\n[2]\r\nB. Dinis, I. P. van den Berg,\r\nAlgebraic\tproperties of external numbers,\r\nJournal of Logic and Analysis,\r\nvol. 3:9 (2011), pp. 1\u201330.\r\n<br>\r\n[3]\r\nF. Koudjeti, I.P. van den Berg,\r\nNeutrices, external numbers and external calculus,\r\nNonstandard Analysis in Practice\r\n(F. Diener and M. Diener, editors),\r\nSpringer Universitext,\r\nBerlin-Heidelberg,\r\n1995,\r\npp. 145\u2013170.\r\n<br>\r\n[4]\r\nE. Nelson,\r\nThe syntax of nonstandard analysis,\r\nAnnals of Pure and Applied Logic,\r\nvol. 38 (1988), no. 2, pp. 123\u2013134.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c105');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\"><br><\/td><td style=\"vertical-align: top;\"><br><\/td><\/tr>\r\n\r\n\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"contrib4\"><\/a>Thursday 4th August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Set Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Neil Barton, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c107')\"><i>(Sub)systems of second-order set theory<\/i><\/a>\r\n<div id=\"absbox-c107\" class=\"abstract_box\">\r\nNeil Barton, <i>(Sub)systems of second-order set theory<\/i>\r\n<br><br>\r\nMuch of set-theoretic practice concerns questions that are, at first blush, second-order in content. The study of the construction of inner models (such as Woodin's Ultimate-$L$ conjecture and the construction of the Steel Core Model), the investigation of embedding principles (for example large cardinals and Kunen's Theorem that there is no $j: V \\longrightarrow V$), and examination of certain kinds of maximality principle (such as Friedman's Inner Model Hypothesis), are all naturally understood as concerning second-order classes rather than sets.\r\nUnderstandably, given the pleasant metalogical properties of first-order $\\mathbf{ZFC}$, many set theorists work hard to render their second-order interests in first-order terms. However, increasingly set theorists have become engaged in questions that are greater than first-order (good examples being the results concerning embeddings in [8], the study of open determinacy for class games in [5], and the consistency of principles in [7]).\r\nIn the philosophical literature, there is a debate concerning how to characterise proper classes within the framework of there being a unique, maximal proper class model of set theory. Traditionally, talk of proper classes in set theory was understood as shorthand for statements definable in terms of first-order formulae with parameters. However, in the last forty years, philosophical conceptions of proper classes have been proposed which aim to capture this essentially second-order character of set-theoretic practice. In particular, [4] and [3] develop a paraphrase in terms of plural quantification, where [2] provides a mereological conception of proper classes.\r\nIn this paper, we examine what can be extracted from particular philosophical conceptions of classes, focussing on the plural conception. First (\\S1), we provide some motivating considerations for the choice of the plural paraphrase. In particular, we argue that the plural paraphrase meshes better with the foundational role many have seen for set theory. Next (\\S2), we note that this conception of classes has been viewed to motivate one of two class theories, either (1.) $MK$ (as in [4]) or (2.) $NBG$ (on the basis of recent work by [1]). We argue that this is a false dichotomy; just as in the case of subsystems of second-order arithmetic, we should expect there to be various philosophical motivations for different strengths of class theories both intermediate between $NBG$ and $MK$, and above $MK$. Finally (\\S3), we examine some of the relevant technical literature, and draw some philosophical conclusions. We argue that naturalistic considerations motivate the use of some non-definable class talk. In particular, we argue for two conclusions (1.) $\\Pi^1_1$-comprehension for classes is motivated by its having many independently justified consequences made clear in the work of [5], and (2.) given a stronger naturalism we can justify the use of strong choice principles for classes extending $MK$ on the basis of work in [6]. We conclude that a detailed philosophical and mathematical study of (sub)systems of second-order set theory is in order, including some which extend $MK$.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nS. Florio and \\O. Linnebo,\r\nOn the Innocence and Determinacy of Plural Quantification,\r\nNo\u00fbs,\r\nvol.&nbsp;49 (2015), no.&nbsp;1, pp.&nbsp;1\u201319.\r\n<br>\r\n[2]\r\nL. Horsten and P. Welch,\r\nReflecting on absolute infinity,\r\nJournal of Philosophy,\r\nForthcoming.\r\n<br>\r\n[3]\r\nG. Uzquiano,\r\nPlural quantification and classes,\r\nPhilosophia Mathematica,\r\nvol.&nbsp;11 (2003), no.&nbsp;1, pp.&nbsp;67\u201381.\r\n<br>\r\n[4]\r\nG. Boolos,\r\nTo be is to be a value of a variable (or to be some values of some variables),\r\nJournal of Philosophy,\r\nvol.&nbsp;81 (1984), no.&nbsp;8, pp.&nbsp;430\u2013449.\r\n<br>\r\n[5]\r\nV. Gitman and J. Hamkins,\r\nOpen determinacy for class games,\r\narXiv:1509.01099 [math.LO],\r\n<br>\r\n[6]\r\nV. Gitman and J. Hamkins and T. Johnstone,\r\nKelley-Morse set theory and choice principles for classes,\r\nUnpublished,\r\n<br>\r\n[7]\r\nS. Friedman,\r\nInternal consistency and the inner model hypothesis,\r\nBulletin of Symbolic Logic,\r\nvol.&nbsp;12 (2006), no.&nbsp;4, pp.&nbsp;591\u2013600.\r\n<br>\r\n[8]\r\nJ. Vickers and P. Welch,\r\nOn elementary embeddings from an inner model to the universe,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;66 (2006), no.&nbsp;3, pp.&nbsp;1090\u20131116.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c107');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Jaykov Foukzon, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c109')\"><i>Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible Cardinals<\/i><\/a>\r\n<div id=\"absbox-c109\" class=\"abstract_box\">\r\nJaykov Foukzon, <i>Inconsistent countable set in second order ZFC and unexistence of the strongly inaccessible cardinals<\/i>\r\n<br><br>\r\nIn this article we derived an important example of the inconsistent countable\r\nset in second order $ZFC$ $(ZFC_{2})$ with the full second-order semantic. Main results is:\r\n<br>\r\n(i) $\\lnot Con(ZFC_{2})$.\r\n<br>\r\n(ii) Let $\\kappa$ be an inaccessible cardinal and $H_{\\kappa}$ is a set of all sets having hereditary size less then $\\kappa$, then $\\lnot\r\nCon(ZFC+(V=H_{\\kappa})).$\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c109');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Michael Lieberman and Jiri Rosicky, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c111')\"><i>Abstract tameness from large cardinals, via accessible categories<\/i><\/a>\r\n<div id=\"absbox-c111\" class=\"abstract_box\">\r\nMichael Lieberman and Jiri Rosicky, <i>Abstract tameness from large cardinals, via accessible categories<\/i>\r\n<br><br>\r\nTameness of types in an abstract elementary class\u2013the principle that distinct types over large models can be distinguished by restrictions to a submodel of fixed small size\u2013appears as a necessary condition in nearly all of the major classification-theoretic results for AECs.  By a theorem of [1], tameness of AECs is known to follow from the existence of a proper class of (almost) strongly compact cardinals.  The authors reprove this result in [3] by entirely different methods, namely by reducing tameness to the accessibility of the (powerful) image of a certain accessible functor.  We show that this method extends naturally to prove that the metric analogue of tameness holds for metric AECs (analyzed as in [4]) under the same cardinal assumption.  While this fact is already known, having been proven by different means in [2], we note that our analysis provides a template for the analysis of tameness more broadly.  As in the metric case, we may consider notions of tameness intrinsic to the ambient category of objects over which an abstract class' structures are built\u2013the characterization via accessible images provides a natural, unified framework in which to address these generalized notions.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nW. Boney,\r\nTameness from large cardinal axioms,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;79 (2014), no.&nbsp;4, pp.&nbsp;1092\u20131119.\r\n<br>\r\n[2]\r\nW. Boney and P. Zambrano,\r\nAround the set-theoretical consistency of d-tameness of abstract elementary classes,\r\narXiv:1508.05529.\r\n<br>\r\n[3]\r\nM. Lieberman and J. Rosick\\'y,\r\nClassification theory for accessible categories,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;81 (2016), no.&nbsp;1, pp.&nbsp;151\u2013165.\r\n<br>\r\n[4]\r\n\u2015,\r\nMetric abstract elementary classes as accessible categories,\r\narXiv:1504.02660.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c111');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Ali Enayat, Paul Gorbow and Zachiri McKenzie, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c113')\"><i>Feferman's Forays into the Foundations of Category Theory: Accommodating unrestricted categories<\/i><\/a>\r\n<div id=\"absbox-c113\" class=\"abstract_box\">\r\nAli Enayat, Paul Gorbow and Zachiri McKenzie, <i>Feferman's forays into the foundations of category theory<\/i>\r\n<br><br>\r\nThis talk is primarily concerned with assessing a set-theoretical system, $S^*$, for the foundations of category theory suggested by Solomon Feferman. $S^*$ is an extension of NFU, and may be seen as an attempt to accommodate unrestricted categories such as the category of all groups (without any small\/large restrictions), while still obtaining the benefits of ZFC on part of the domain. A substantial part of the paper is devoted to establishing an improved upper bound on the consistency strength of $S^*$. The assessment of $S^*$ as a foundation of category theory is framed by the following general desiderata (R) and (S). (R) asks for the unrestricted existence of the category of all groups, the category of all categories, the category of all functors between two categories, etc., along with natural implementability of ordinary mathematics and category theory. (S) asks for a certain relative distinction between large and small sets, and the requirement that they both enjoy the full benefits of the ZFC axioms. $S^*$ satisfies (R) simply because it is an extension of NFU. By means of a recursive construction utilizing the notion of strongly cantorian sets, we argue that it also satisfies (S). Moreover, this construction yields a lower bound on the consistency strength of $S^*$. We also exhibit a basic positive result for category theory internal to NFU that provides motivation for studying NFU-based foundations of category theory.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c113');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Jeffrey Bergfalk, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c115')\"><i>Homological Characterizations of Small Cardinals<\/i><\/a>\r\n<div id=\"absbox-c115\" class=\"abstract_box\">\r\nJeffrey Bergfalk, <i>Homological characterizations of ``small'' cardinals<\/i>\r\n<br><br>\r\nWe consider a number of homological invariants of the \"small\" cardinals $\\omega_n$. We show, for example, that $\\omega_n$ is the least ordinal whose $n$th sheaf cohomology group is nonzero. The best-understood case is that of $n=1$: here the nonzero group is that of nontrivial coherent families of functions on $\\omega_1$. We consider what the (ZFC) existence of analogous, higher-dimensional families corresponding to higher nonzero homology groups begins to tell us about the combinatorics of higher $\\omega_n$.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c115');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Proof Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Jan Walker, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c117')\"><i>Finitism, truth, and reflection<\/i><\/a>\r\n<div id=\"absbox-c117\" class=\"abstract_box\">\r\nJan Walker, <i>Finitism, truth, and reflection<\/i>\r\n<br><br>\r\nLet $\\mathsf{S}$ be a theory in a language $\\mathcal{L}$. Define $\\mathsf{Ref}(\\mathsf{S})$\r\nas the theory in $\\mathcal{L} \\cup \\{ T \\}$ which is obtained by adding to $\\mathsf{S}$: first,\r\nfor all formulae containing $T$, the corresponding substitution instances of axiom schemas\r\nand rules of $\\mathsf{S}$; second, axioms to the effect that $T$ denotes partial truth in the\r\nsense of Saul Kripke [2]. In a seminal article, Solomon Feferman [1]\r\nproposed to identify the statements in $\\mathcal{L}$ which are derivable in $\\mathsf{Ref}(\\mathsf{S})$\r\nas <i>the statements in $\\mathcal{L}$ one ought to accept if one has accepted $\\mathsf{S}$<\/i>.\r\nMy talk is inspired by Feferman's approach, but takes a different direction. It starts off with\r\nthe assumption that <i>only theorems of finitist arithmetic ought to be accepted<\/i> and aims\r\nat answering <i>truth-theoretic<\/i> questions such as: How to adapt (the axioms for) Kripke's\r\naccount of truth to what is acceptable from the finitist perspective? How to adapt other prominent\r\nconceptions of truth? What features do finitist conceptions of truth have in common? In the\r\nsequel of my talk, I will turn to a proof-theoretic analysis of (Turing-like) progressions in which\r\nreflection (or formal soundness) principles are successively added to theories in a language of\r\n<i>iterated<\/i> finitist truth.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSolomon Feferman,\r\nReflecting on incompleteness,\r\nThe Journal of Symbolic Logic,\r\nvol.&nbsp;56 (1991), no.&nbsp;1, pp.&nbsp;1\u201349.\r\n<br>\r\n[2]\r\nSaul Kripke,\r\nOutline of a theory of truth,\r\nThe Journal of Philosophy,\r\nvol.&nbsp;72 (1975), no.&nbsp;19, pp.&nbsp;690\u2013716.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c117');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Anton Freund, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c119')\"><i>The slow reflection hierarchy<\/i><\/a>\r\n<div id=\"absbox-c119\" class=\"abstract_box\">\r\nAnton Freund, <i>The slow reflection hierarchy<\/i>\r\n<br><br>\r\nWe generalize the notion of slow consistency, due to S.-D. Friedman, Rathjen and Weiermann [3], to obtain slow (uniform) reflection statements of arbitrary arithmetical complexity. The resulting hierarchy over Peano Arithmetic is incomparable with the hierarchy of usual reflection principles: No single one of the usual reflection statements implies all slow reflection statements. Interestingly, though, slow reflection is much weaker when it comes to $\\Pi_1$-consequences: Any $\\Pi_1$-formula that is provable in the slow reflection hierarchy already follows from the (usual) consistency of Peano Arithmetic. For detailed proofs we refer to [2]. Henk and Pakhomov [4] independently establish similar results. A more computational viewpoint is adopted in [1], where we determine the provably total functions of slow $\\Sigma_1$-reflection.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAnton Freund,\r\nProof lengths for instances of the Paris-Harrington Principle,\r\narXiv:1601.08185 (preprint).\r\n<br>\r\n[2]\r\n\u2015\r\nSlow reflection,\r\narXiv:1601.08214 (preprint).\r\n<br>\r\n[3]\r\nSy-David Friedman, Michael Rathjen, Andreas Weiermann,\r\nSlow consistency,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;164 (2013), no.&nbsp;3, pp.&nbsp;382\u2013393.\r\n<br>\r\n[4]\r\nPaula Henk, Fedor Pakhomov,\r\nSlow and ordinary provability for Peano Arithmetic,\r\narXiv:1602.01822.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c119');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Andrei Sipos, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c121')\"><i>Proof mining and positive-bounded logic<\/i><\/a>\r\n<div id=\"absbox-c121\" class=\"abstract_box\">\r\nAndrei Sipos, <i>Proof mining and positive-bounded logic<\/i>\r\n<br><br>\r\nProof mining is a research program introduced by U. Kohlenbach in the 1990s ([3] is a comprehensive reference), which aims to obtain explicit quantitative information (witnesses and bounds) from proofs of an apparently ineffective nature. This paradigm in applied logic has successfully led so far to obtaining some previously unknown effective bounds, primarily in nonlinear analysis and ergodic theory. A large number of these are guaranteed to exist by a series of logical metatheorems which cover general classes of bounded or unbounded metric structures.\r\nIn order to apply these metatheorems, the structures are typically formalized in higher-order systems of arithmetic and analysis, using appropriate codings of real numbers and related operations. The classes for which metatheorems have already been proven include normed spaces and hyperbolic spaces. Recently, G\u00fcnzel and Kohlenbach [1] have shown that, in principle, one could obtain metatheorems for a plethora of classes of structures, provided that they are formalized in positive-bounded logic (in the sense of Henson and Iovino [2]) and that some preparation of the axioms is undertaken beforehand. We aim to show how this process may be carried out in some additional classes suggested by the two authors above. We illustrate it with some concrete applications.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nDaniel G\u00fcnzel and Ulrich Kohlenbach,\r\nLogical metatheorems for abstract spaces axiomatized in positive bounded logic,\r\nAdvances in Mathematics,\r\nvol. 290, pp. 503-551 (2016).\r\n<br>\r\n[2]\r\nC. Ward Henson and Jose Iovino,\r\nUltraproducts in Analysis,\r\nin Analysis and Logic, London Math. Soc. Lecture Note Ser., vol. 262,\r\nCambridge Univ. Press,\r\npp. 1\u2013115 (2002).\r\n<br>\r\n[3]\r\nUlrich Kohlenbach,\r\nApplied proof theory: Proof interpretations and their use in mathematics,\r\nSpringer Monographs in Mathematics,\r\nSpringer-Verlag,\r\nBerlin-Heidelberg (2008).\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c121');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Leroy Chew, Olaf Beyersdorff and Ilario Bonacina, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c123')\"><i>Lower bounds: from circuits to QBF proof systems<\/i><\/a>\r\n<div id=\"absbox-c123\" class=\"abstract_box\">\r\nLeroy Chew, Olaf Beyersdorff and Ilario Bonacina, <i>Lower bounds: from circuits to QBF proof systems<\/i>\r\n<br><br>\r\nA general and long-standing belief in the proof complexity community asserts that there is a close connection between progress in lower bounds for Boolean circuits and progress in proof size lower bounds for strong propositional proof systems. Although there are famous examples where a transfer from ideas and techniques from circuit complexity to proof complexity has been effective [4], a formal connection between the two areas has never been established so far. Here we provide such a formal relation between lower bounds for circuit classes and lower bounds for $\\mathsf{Frege}$ systems for quantified Boolean formulas (QBF).\r\nStarting from a propositional proof system $P$ we exhibit a general method how to obtain a QBF proof system $P+\\forall$red, which is inspired by the transition from resolution to Q-resolution. For us the most important case is a new and natural hierarchy of QBF $\\mathcal{C}$-$\\mathsf{Frege}$ systems $\\mathcal{C}$-$\\mathsf{Frege+}\\forall\\mathsf{red}$ that parallels the well-studied propositional hierarchy of $\\mathcal{C}$-$\\mathsf{Frege}$ systems, where lines in proofs are restricted to a circuit class $\\mathcal{C}$.\r\nBuilding on earlier work for resolution [1] we establish a lower bound technique via strategy extraction that transfers arbitrary lower bounds for the circuit class $\\mathcal{C}$ to lower bounds in $\\mathcal{C}$-$\\mathsf{Frege+}\\forall\\mathsf{red}$.\r\nBy using the full spectrum of state-of-the-art circuit lower bounds [3, 5, 6, 2], our new lower bound method leads to very strong lower bounds for QBF \\FREGE systems:\r\n<ul>\r\n<li>exponential lower bounds and separations for\r\n$\\mathsf{AC}^0[p]$-$\\mathsf{Frege+}\\forall\\mathsf{red}$ for all primes $p$;\r\n<li>an exponential separation of $\\mathsf{AC}^0[p]$-$\\mathsf{Frege+}\\forall\\mathsf{red}$ from $\\mathsf{TC}^0[p]$-$\\mathsf{Frege+}\\forall\\mathsf{red}$;\r\n<li>an exponential separation of the hierarchy of constant-depth systems $\\mathsf{AC}^0_d$-$\\mathsf{Frege+}\\forall\\mathsf{red}$ by formulas of depth independent of $d$.\r\n<\/ul>\r\nIn the propositional case, all these results correspond to major open problems.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nOlaf Beyersdorff, Leroy Chew, and Mikol\u00e1s Janota.\r\nProof complexity of resolution-based QBF calculi.\r\nIn <i>32nd International Symposium on Theoretical Aspects of\r\nComputer Science (STACS 2015)<\/i>, pages 76\u201389, 2015a.\r\n<br>\r\n[2]\r\nRavi&nbsp;B. Boppana and Michael Sipser.\r\nHandbook of theoretical computer science (vol. {A}).\r\nchapter The Complexity of Finite Functions, pages 757\u2013804. MIT\r\nPress, Cambridge, MA, USA, 1990.\r\n<br>\r\n[3]\r\nJohan H\u00e5stad.\r\nAlmost optimal lower bounds for small depth circuits.\r\nIn <i>Proc. 18th STOC<\/i>, pages 6\u201320. ACM Press, 1986.\r\n<br>\r\n[4]\r\nJan Kraj\u00ed\u010dek.\r\nInterpolation theorems, lower bounds for proof systems and\r\nindependence results for bounded arithmetic.\r\n<i>The Journal of Symbolic Logic<\/i>, 62 (2):\r\n457\u2013486, 1997.\r\n<br>\r\n[5]\r\nAlexander&nbsp;A. Razborov.\r\nLower bounds for the size of circuits of bounded depth with basis $\\{ \\&, \\oplus \\}$.\r\n<i>Math. Notes Acad. Sci. USSR<\/i>, 41 (4):\r\n333\u2013338, 1987.\r\n<br>\r\n[6]\r\nRoman Smolensky.\r\nAlgebraic methods in the theory of lower bounds for Boolean circuit\r\ncomplexity.\r\nIn <i>Proc. of 19th ACM STOC<\/i>, pages 77\u201382, 1987.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c123');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Carlo Nicolai, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c125')\"><i>Transfinite induction and reflection for some systems of truth in basic De Morgan logic<\/i><\/a>\r\n<div id=\"absbox-c125\" class=\"abstract_box\">\r\nCarlo Nicolai, <i>Transfinite induction and reflection for some systems of truth in basic De Morgan logic<\/i>\r\n<br><br>\r\nSemantic paradoxes force us to question our na\u00efve intuition about truth, encompassed in the schema `$\\phi$' is true if and only if $\\phi$.\r\nThey may be approached by giving up the na\u00efve truth schema and retain classical logic; alternatively, one may retain our na\u00efve intuition and abandon classical logic. It has been argued that the latter approach severely cripples our capability of employing mathematical \u2013 or more generally extra-semantic \u2013 patterns of reasoning in semantics [3]. This is usually motivated by the different amount of transfinite induction that is provable in the classical axiomatization of the fixed point construction of [2] \u2013 due to [1] and known as $\\mathsf{KF}$  \u2013 and the corresponding nonclassical system $\\mathsf{PKF}$ formulated in a logic $\\mathsf{B}$ featuring only introduction and elimination rules for positive connectives (and their negations) [4]. In the talk we consider several subsystems or extensions of $\\mathsf{PKF}$: we focus on (i) a basic disquotational theory extending $\\mathsf{B}$ with arithmetical axioms and the basic principles `if $\\phi$, then `$\\phi$' is true', and `if `$\\phi$' is true, then $\\phi$' for all sentences in the language with the truth predicate; (ii) a variant of $\\mathsf{PKF}$ with no semantic induction; (iii) the extension of $\\mathsf{PKF}$ with a rule of transfinite induction up to $\\varepsilon_0$ for the entire vocabulary. We first measure the strength of the  systems by comparing them to the sentences provably true in $\\mathsf{KF}$ and variants thereof. We then consider a strategy for employing \u2013 suitably adjusted \u2013 reflection principles to climb up, in a rather natural way, from (i) to (iii) via (ii). We conclude by examining the role of reflection and induction in the debate between advocates of the classical and the nonclassical approaches sketched above.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSolomon Feferman,\r\nReflecting on incompleteness,\r\nThe Journal of Symbolic Logic,\r\nvol.&nbsp;56, no.&nbsp;1 (1991), 1\u201349.\r\n<br>\r\n[2]\r\nSaul Kripke,\r\nOutline of a Theory of Truth,\r\nJournal of Philosophy, no. 72 (1975): 690\u2013712.\r\n<br>\r\n[3]\r\nVolker Halbach,\r\nAxiomatic Theories of Truth. Revised Edition,\r\nCambridge University Press,\r\n2014.\r\n<br>\r\n[4]\r\nVolker Halbach and Leon Horsten,\r\nAxiomatizing Kripke's theory of Truth,\r\nThe Journal of Symbolic Logic,\r\nvol.&nbsp;71, no.&nbsp;2 (2006),  677\u2013712.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c125');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Michael Kompatscher and Trung Van Pham, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c127')\"><i>Constraint satisfaction problems over the random poset<\/i><\/a>\r\n<div id=\"absbox-c127\" class=\"abstract_box\">\r\nMichael Kompatscher and Trung Van Pham, <i>Constraint satisfaction problems of reducts of the random poset<\/i>\r\n<br><br>\r\nWe present a dichotomy result for computational problems of the form Poset-SAT($\\Phi$). In these problems the input consists of variables and constraints about them which have to be taken from $\\Phi$, a fixed finite set of quantifier-free formulas in the language of orders. The decision problem is whether the variables can be assigned elements of a partial order so that all constraints are satisfied. We show that Poset-SAT($\\Phi$) is NP-complete or solvable in polynomial time.\r\nAll problems Poset-SAT($\\Phi$) can be stated as constraint satisfaction problems of reducts of the random poset, the Fra\u00efss\u00e8-limit of all finite partial orders. By studying their polymorphism clones with the help of methods developed by Bodirsky and Pinsker, we show the following result: Either the polymorphism clone contains a weak nu-term (modulo endomorphism) and the CSP is in P, or the problem is NP-complete.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c127');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Dana Bartosova, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c129')\"><i>Combinatorics of ultrafilters on automorphism groups<\/i><\/a>\r\n<div id=\"absbox-c129\" class=\"abstract_box\">\r\nDana Bartosova, <i>Combinatorics of ultrafilters on automorphism groups<\/i>\r\n<br><br>\r\nFor a topological group $G,$ an ambit is a compact pointed space $(X,x_0)$ with a (jointly) continuous action of $G$ on $X$ with the orbit $Gx_0=\\{gx_0:g\\in G\\}$ dense in $X.$ The greatest ambit of $G$, denoted by $S(G),$ is an ambit that has every ambit as its quotient preserving the distringuished points. We will study $S(G)$ for $G$ an automorphism group of a countable first order structure as a space of ultrafilters, describe how the multiplication in $G$ extends to $S(G)$ and show a couple of results about combinatorics and algebra in $S(G).$\r\nThis is partially a joint work with Andrew Zucker (CMU).\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c129');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Bektur Baizhanov, Olzhas Umbetbayev and Tatyana Zambarnaya, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c131')\"><i>The properties of linear orders defined on the classes of convex equivalence of 1-formulas<\/i><\/a>\r\n<div id=\"absbox-c131\" class=\"abstract_box\">\r\nBektur Baizhanov, Olzhas Umbetbayev and Tatyana Zambarnaya, <i>The properties of linear orders defined on the classes of convex equivalence of 1-formulas<\/i>\r\n<br><br>\r\nIn the report we consider small countable  theories  with an\r\n$\\emptyset$-definable binary  relation of linear order. Let $A$ be\r\na finite subset of a countable saturated  model $N$, and $ H(x)$ and\r\n$\\Theta (x)$ be $A$-definable $1$-formulas such that $H(N) \\subset\r\n\\Theta(N) $.\r\nDefine\r\n$E _{H,\\Theta }(x,y):=H(x)\\land\r\nH(y)\\land (x &lt; y\\to\\forall z((x &lt; z&lt; y\\land\\Theta (z))\\to\r\nH(z)))\\land (y&lt;x\\to\\forall z((y &lt; z &lt; x\\land\\Theta (z))\\to\r\nH(z))$.\r\n$E_{H,\\Theta}(x,y)$ is an $A$-definable relation of equivalence\r\non $H(N)$ such that any  $E_{H,\\Theta}$-class is convex in\r\n$\\Theta(N)$.\r\nWe say that an ordered theory $T$ has the property of finiteness\r\nof discrete chains convex equivalences (FDCCE)  if for every two\r\none-formulas $H(x)$ and $\\Theta (x)$ such that $H(N)\\subset \\Theta(N)$,\r\nfor any $k$ ($1&lt;k &lt; \\omega$) every discrete chain of convex\r\n$E_{H,\\Theta}$-classes is finite.\r\nWe say that the set of $A$-definable one-formulas $C\\subset F_1(A)$\r\nis a $BH-algebra$ if it is closed under the following logical operations:\r\n$\\land$, $\\neg$, $\\lor$, $\\triangleleft^i_k$ ($0&lt;i&lt;k$, $1&lt;k &lt;\\omega$).\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nLet $T$ be a small ordered theory with FDCCE, $A$ be a finite\r\nsubset of a countable saturated model $N$ of the theory $T$. Then for every\r\nfinite set of $A$-definable one-formulas $\\{\\phi_1(x), \\dots ,\r\n\\phi_n(x)\\}$, $n&lt;\\omega$ the $BH$-algebra generated by this set is finite.\r\n<br><br>\r\nAn ordered theory $T$ is a theory of a pure order if it is in a\r\nlanguage $L=\\{=,&lt;\\}$.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nLet $T$ be a small theory of a pure order.  Then $T$ is\r\n$\\omega$-categorical if and only if it has FDCCE.\r\n<br><br>\r\n<b>Corollary<\/b>.\r\nLet $T$ be a non-$\\omega$-categorical small theory of a pure order.\r\nThen there is $\\emptyset$-definable $1$-formula $\\phi(x)$  such that\r\nfor some elements $\\alpha,\\beta \\in \\phi(N)$  ($\\alpha &lt;\\beta$),\r\n$(\\alpha,\\beta)\\cap \\phi(N)$ is an infinite discrete chain.\r\n<br><br>\r\n<b>Corollary<\/b>.\r\nLet $T$ be a countable complete ordered theory in a language $L$\r\nand $T_0\\subset T$ be a complete theory in a\r\nlanguage $L_0: =\\{=,&lt;\r\n\\}\\subset L$.  If $T_0$ is non-$\\omega$-categorical then\r\n$I(T,\\omega)=2^{\\omega}.$\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nM. Rubin,\r\nTheories of linear order,\r\nIsrael Journal of Mathematics,\r\nvol.&nbsp;17 (1974), pp.&nbsp;392\u2013443.\r\n<br>\r\n[2]\r\nS. Shelah,\r\nEnd extensions and numbers of countable models,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;43 (1978), pp.&nbsp;550\u2013562.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c131');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Natasha Dobrinen, Claude Laflamme and Norbert Sauer, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c133')\"><i>Rainbow Ramsey simple structures<\/i><\/a>\r\n<div id=\"absbox-c133\" class=\"abstract_box\">\r\nNatasha Dobrinen, Claude Laflamme and Norbert Sauer, <i>Rainbow Ramsey simple structures<\/i>\r\n<br><br>\r\nWe prove that the Rado graph $\\mathcal{R}$ has the rainbow Ramsey property. This means that given any finite graph $G$, any finite number $k$, and any coloring of the copies of $G$ in the Rado graph, $\\mathcal{R}$, where each color appears no more than $k$ times, there is a subgraph $\\mathcal{R}'\\le \\mathcal{R}$ which is also a Rado graph in which the copies of $G$ use each color at most once. More generally, we show that a class of binary relational structures generalizing the Rado graph are rainbow Ramsey. By compactness, it follows that for all finite graphs $B$ and $C$ and any finite number $k$, there is a graph $A$ so that for every coloring of the copies of $C$ in $A$ such that each color is used at most k times, there is a copy  $B'$ of $B$ in $A$ in which each copy of $C$ has a different color. This is joint work with Claude Laflamme and Norbert Sauer.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c133');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory: Stability Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Sebastien Vasey, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c135')\"><i>A proof of Shelah's eventual categoricity conjecture in universal classes<\/i><\/a>\r\n<div id=\"absbox-c135\" class=\"abstract_box\">\r\nSebastien Vasey, <i>A proof of Shelah's eventual categoricity conjecture in universal classes<\/i>\r\n<br><br>\r\nAbstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an $\\mathbb{L}_{\\infty, \\omega}$ theory, as well as numerous algebraic examples. They were introduced by Saharon Shelah forty years ago. Shelah focused on generalizations of Morley's categoricity theorem and conjectured the following eventual version: An AEC categorical in a high-enough cardinal is categorical on a tail of cardinals. I will present my proof of the conjecture for <i>universal classes<\/i>. They are a special case of AECs (studied by Shelah in a milestone 1987 paper [1]) corresponding to classes of models of a universal $\\mathbb{L}_{\\infty, \\omega}$ theory.\r\nAn instance of our result is:\r\n<br><br>\r\n<b>Theorem.<\/b>\r\nIf $\\psi$ is a universal $\\mathbb{L}_{\\omega_1, \\omega}$ sentence that is categorical in <i>some<\/i> $\\lambda \\ge \\beth_{\\beth_{\\omega_1}}$, then $\\psi$ is categorical in <i>all<\/i> $\\lambda' \\ge \\beth_{\\beth_{\\omega_1}}$.\r\n<br><br>\r\nThe proof combines Shelah's earlier work on universal classes with a study of AECs that have amalgamation and are <i>tame<\/i> (a locality property isolated by Grossberg and VanDieren which says roughly that orbital types are determined by their small restrictions).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSaharon Shelah,\r\nUniversal classes,\r\nClassification Theory\r\n(Proceedings of the U.S.-Israel Workshop on Model Theory in Mathematical Logic held in Chicago, Dec. 15-19, 1985),\r\n(John T. Baldwin, editor),\r\nvol.&nbsp;1292,\r\nSpringer Berlin Heidelberg,\r\n1987,\r\npp.&nbsp;264\u2013418.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c135');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Lubna Shaheen, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c137')\"><i>On $\\mathbb{F}_1$ geometry and model theory<\/i><\/a>\r\n<div id=\"absbox-c137\" class=\"abstract_box\">\r\nLubna Shaheen, <i>On $\\mathbb{F}_1$-geometry and model theory.<\/i>\r\n<br><br>\r\n(Joint work with Boris Zilber and John Alexander Cruz Morales)\r\n<br>\r\nThe geometric motivation of the object $\\mathbb{F}_1$,\" the field with one element\" came from the work of Jack Tits in 1956,  where he explained how one can define the Chevalley group of characteristic un to obtain  some interesting geometries such that the symmetric groups happens to be the Weyl group of the corresponding Lie groups.\r\nIn the last two decades there have been a lot of development, with motivation coming from Arakelov theory and some ideas  relating the notion to the Riemann zeta function.\r\nWe interpret fields  of characteristic $1$ and algebras over fields of characteristic $1$ as {\\bf multiplicative monoids\r\nwith a shadow addition}  by which we mean structures of the form $(R; \\cdot, 0,1, D_1^{m_1,p_1},D_2^{m_2,p_2},\\ldots)$ where $(R;\\cdot,0,1)$ is a multiplicative commutative  monoid with $0$\r\nand $D_k^{m_k,p_k}$ is an $m_k$-ary relation, $p_k$ a prime number or $0$, which we will write in a more suggestive form as a divisor equality\r\n$$ n_1x_1+\\ldots+n_mx_m\\equiv 0\\, \\mbox{mod}\\, p$$\r\nThe initial object in the category of $\\mathbb{F}_1$-algebras is $\\mathbb{F}_1=\\{0,1\\}$ the field with one element.\r\nWe define the cyclotomic extensions $\\mathbb{F}_{1^{n}}$ of $\\mathbb{F}_1$ as the $\\mathbb{F}_{1}$-algebra   where $R= 0 \\cup \\mu_{n}.$ We set $\\mathbb{F}_1^{alg}=\\bigcup_{n\\in \\mathbb{N}} \\mathbb{F}_{1^n}$ (with the universe $\\mu_{\\infty}\\cup \\{ 0\\}$) and  formulate a conjecture that such an object is $\\omega$-stable.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c137');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Gwyneth Harrison-Shermoen, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c139')\"><i>Independence, via limits<\/i><\/a>\r\n<div id=\"absbox-c139\" class=\"abstract_box\">\r\nGwyneth Harrison-Shermoen, <i>Independence, via limits<\/i>\r\n<br><br>\r\nThe concept of an independence relation (a ternary relation among sets, satisfying certain properties) generalises that of linear independence in vector spaces and algebraic independence in fields, and gives us a way to determine what behaviour is \"generic\" in a given theory. Kim and Pillay showed [1] that if a theory has an abstract independence relation satisfying an extra property (the \"independence theorem over a model\"), then the theory is simple and the independence relation is non-forking independence. There are, however, non-simple theories with relations that satisfy quite a few of the desired properties for a notion of independence. Given a large model M of some theory T, and following work of N. Granger [2] (on two-sorted theories of infinite-dimensional vector spaces over an algebraically closed field and with a bilinear form), we describe a method of lifting independence relations from the (tame) theories of substructures of M to a reasonably well-behaved notion of independence in M.\r\nThis talk is based on work from the author's PhD thesis [3].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nByunghan Kim and Anand Pillay,\r\nSimple theories,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;88 (1997), no.&nbsp;2\u20133, pp.&nbsp;149\u2013164.\r\n<br>\r\n[2]\r\nNicolas Granger,\r\nStability, simplicity, and the model theory of bilinear forms,\r\nPhD thesis,\r\nUniversity of Manchester,\r\n1999.\r\n<br>\r\n[3]\r\nGwyneth Harrison-Shermoen,\r\nIndependence relations in theories with the tree property,\r\nPhD thesis,\r\nUniversity of California, Berkeley,\r\n2013.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c139');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Tim Zander, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c141')\"><i>Higher amalgamation in stable theories<\/i><\/a>\r\n<div id=\"absbox-c141\" class=\"abstract_box\">\r\nTim Zander, <i>Higher amalgamation in stable theories<\/i>\r\n<br><br>\r\nAmalgamating types is an essential tool in stable and simple theories.\r\nFor example the reason to consider imaginaries in stable theories was\r\nexactly done to obtain uniqueness of $2$-Amalgamation over algebraic\r\nclosed sets.\r\nThe Independence Theorem in simple theory is nothing more than\r\n$3$-Amalgamation over models. But $n$-Amalgamation for $n>3$ can fail in\r\nstable theories. But if one expands the theory by certain finite covers (see\r\n(called generalised imaginaries)\r\nwe can still obtain $n$-Amalgamation over the empty set for every $n$ (see 4.11 in [1]).\r\nNow if forking is easy enough (SU-rank $1$) this directly translates to\r\nAmalgamation over parameters. In general this does not need to hold.\r\nThis is part of my unpublished PhD-research.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nHrushovski, Ehud,\r\nGroupoids, imaginaries and internal covers,\r\nTurkish Journal of Mathematics,\r\n36(2), pp.173-198.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c141');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Computability Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Andrea Sorbi and Uri Andrews, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c143')\"><i>Uniformly effectively inseparable equivalence relations<\/i><\/a>\r\n<div id=\"absbox-c143\" class=\"abstract_box\">\r\nAndrea Sorbi and Uri Andrews, <i>Uniformly effectively inseparable equivalence relations<\/i>\r\n<br><br>\r\nA computably enumerable equivalence relation (ceer) $R$ is called\r\n<i>uniformly effectively inseparable (u.e.i.)<\/i> if it is nontrivial and\r\nthere is a computable function $p(a,b)$ such that the partial computable\r\nfunction $\\varphi_{p(a,b)}$ witnesses effective inseparability of the pair\r\n$([a]_R, [b]_R)$, whenever $a$ and $b$ are non-$R$-equivalent. It is shown in\r\n[1] that any u.e.i. ceer $R$ is <i>universal<\/i>, i.e. for every\r\nceer $S$ there exists a computable function $f$ such that, for all $x,y$, $x\r\n\\mathrel{R} y$ if and only if $f(x) \\mathrel{S} f(y)$. Despite universality,\r\nand unlike Smullyan's classical theorem establishing computable isomorphism\r\nof any two pairs of effectively inseparable c.e. sets, the u.e.i. ceers do\r\nnot fall into a unique computable isomorphism type. The previously known\r\nlargest class of u.e.i. ceers was the class of the <i>uniformly finitely\r\nprecomplete (u.f.p.)<\/i> ceers, which can be characterized as those ceers which\r\nare computably isomorphic to nontrivial c.e. extensions of the relation\r\n$\\sim_{PA}$ of provable equivalence in Peano Arithmetic. Answering a question\r\nin [1], we show:\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nThere exist u.e.i. ceers that are not u.f.p.\r\n<br><br>\r\nAmong the u.f.p. ceers, $\\sim_{PA}$ itself can be characterized (up to\r\ncomputable isomorphim) as the unique ceer $R$ which has a diagonal function,\r\ni.e. a computable function $d$ such that, for all $x$, $x$ and $d(x)$ are\r\nnon-$R$-equivalent, [2], or equivalently (up\r\nto computable isomorphim) as the unique ceer $R$ which has a strong diagonal\r\nfunction, i.e. a computable function $d$ such that for every finite set $D$,\r\n$d(D)$ is non-$R$-equivalent to any element in $D$. We show:\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nEvery u.e.i. ceer $R$ with a strong diagonal function is\r\ncomputably isomorphic to $\\sim_{PA}$.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]  U.&nbsp;Andrews, S.&nbsp;Lempp, J.&nbsp;S. Miller, K.&nbsp;M. Ng, L.&nbsp;San&nbsp;Mauro,\r\nand A.&nbsp;Sorbi.\r\nUniversal computably enumerable equivalence relations.\r\nJournal Symbolic Logic, 79(1):60\u201388, March 2014.\r\n<br>\r\n[2]  C.&nbsp;Bernardi and F.&nbsp;Montagna.\r\nEquivalence relations induced by extensional formulae:\r\nClassifications by means of a new fixed point property.\r\nFundamenta Mathematicae, 124:221\u2013232, 1984.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c143');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Serikzhan Badaev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c145')\"><i>A chain of weakly precomplete computably enumerable equivalence relations<\/i><\/a>\r\n<div id=\"absbox-c145\" class=\"abstract_box\">\r\nSerikzhan Badaev, <i>A Chain of weekly precomplete computably enumerable equivalence relations<\/i>\r\n<br><br>\r\nA computably enumerable equivalence relation (ceer) $E$ on $\\omega$ is weakly\r\nprecomplete if and only if $E$ has no computable diagonal function. The class\r\nof weakly precomplete ceers include the important classes of precomplete\r\nceers and uniformly finitely precomplete (u.f.p.) ceers.\r\nIn [1], it was shown that there are infinitely many computable\r\nisomorphism types of universal weakly precomplete (in fact u.f.p.) ceers; and\r\nthere are infinitely many computable isomorphism types of non-universal\r\nweakly precomplete ceers.\r\nWe consider ceers relatively to the following well known reduction: a ceer\r\n$R$ is said to be  reducible to a ceer $S$, if there is a computable function\r\n$f$ such that, for all $x$ and $y$, $x R y \\iff f (x) S f(y)$. We construct\r\nan infinite $\\omega$-chain of non-equivalent weakly precomplete ceers under\r\nthis reduction.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSerikzhan Badaev and Andrea Sorbi,\r\nWeakly precomplete computably enumerable equivalence relations,\r\nMathematical Logic Quarterly,\r\nvol.&nbsp;62 (2016), no.&nbsp;1, pp.&nbsp;111\u2013127.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c145');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Micha\u0142 Tomasz Godziszewski, Marek Czarnecki and Dariusz Kaloci\u0144ski, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c147')\"><i>Learnability in the Limit meets the Church Thesis<\/i><\/a>\r\n<div id=\"absbox-c147\" class=\"abstract_box\">\r\nMicha\u0142 Tomasz Godziszewski, Marek Czarnecki and Dariusz Kaloci\u0144ski, <i>Learnability in the Limit meets the Church Thesis<\/i>\r\n<br><br>\r\nWe consider the notion of intuitive learnability and its relation to intuitive computability. We briefly discuss the Church's Thesis.\r\nWe formulate the Learnability Thesis.\r\nFurther\r\nwe analyse the\r\nproof of the Church's Thesis presented by M. Mostowski.\r\nWe indicate which assumptions of the Mostowski's argument implicitly include that the Church's Thesis holds. The impossibility of this kind of argument is strengthened\r\nby showing that the Learnability Thesis does not imply the Church's Thesis.\r\nSpecifically, we show a <i>natural<\/i> interpretation of intuitive computability under which intuitively learnable sets are exactly algorithmically learnable but intuitively computable sets form a proper superset of recursive sets.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c147');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Luca San Mauro, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c149')\"><i>Complexity of Relations via Computable Reducibility<\/i><\/a>\r\n<div id=\"absbox-c149\" class=\"abstract_box\">\r\nLuca San Mauro, <i>Complexity of Relations via Computable Reducibility<\/i>\r\n<br><br>\r\nComputable reducibility provides a natural way of ranking\r\nbinary relations on $\\omega$ according to their complexity.\r\nLet $R$ and $S$ be two binary relations, we say that $R$ is <i>computably reducible<\/i> to $S$ iff there is a computable function $f$ such that, for all $x,y \\in \\omega$, the following holds:\r\n\\[\r\nxRy \\Leftrightarrow f(x)Sf(y).\r\n\\]\r\nComputable reducibility has been object of study for decades, being mostly applied to the case of equivalence relations. In particular, a prominent problem in the area has been that of characterizing <i>universal<\/i> equivalence relations, i.e. relations to which all others relations, of a given complexity, can be computably reduced.\r\nIn this talk, we address the problem of universality for a more general context than that of equivalence relations. First, we prove that, contrary to the case of equivalence relations and preorders, for each level of the arithmetical hierarchy there is a universal binary relation. Then, we define natural examples of universal $\\Sigma^0_n$ binary relations and of universal $\\Pi^0_n$ binary relations.\r\nMore precisely, let $U^{\\in}_n$ be the following $\\Sigma^0_n$ binary relation,\r\n\\[\r\nxU^{\\in}_n y \\Leftrightarrow x\\in W^{\\emptyset^{(n-1)}}_y,\r\n\\]\r\nand, for $n>2$, let $U^\\subseteq_n$ be the following binary relation\r\n\\[\r\nxU^\\subseteq_n y \\Leftrightarrow W_x \\subseteq W^{(n-2)}_y.\r\n\\]. We show that:\r\n<ul>\r\n<li>For all $n$, $U^\\in_n$ is a universal $\\Sigma^0_n$ binary relation;\r\n<li>There exists a total computable function $f$ such that the following binary relation\r\n\\[\r\nxU^\\subseteq_2 y \\Leftrightarrow W_x \\subseteq W_{f(y)}\r\n\\]\r\nis a universal $\\Pi^0_2$ binary relation;\r\n<li>For $n>2$, $U^\\subseteq_n$ is a universal $\\Pi^0_n$ binary relation.\r\n<\/ul>\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nUri Andrews, Steffen Lempp, Joseph S. Miller, Keng Meng Ng, Luca San Mauro, Andrea Sorbi,\r\nUniversal computably enumerable equivalence relations,\r\nThe Journal of Symbolic Logic,\r\nvol.&nbsp;79 (2014), no.&nbsp;1, pp.&nbsp;60\u201388.\r\n<br>\r\n[2]\r\nEgor Ianovski, Russell Miller, Keng Meng Ng, Andr\u00e9 Nies,\r\nComplexity of equivalence relations and preorders from computability theory,\r\nThe Journal of Symbolic Logic,\r\nvol.&nbsp;79 (2015), no.&nbsp;3, pp.&nbsp;859\u2013881.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c149');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Categorical Logic and Type Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Andrew Swan, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c151')\"><i>Identity types in Algebraic Model Structures<\/i><\/a>\r\n<div id=\"absbox-c151\" class=\"abstract_box\">\r\nAndrew Swan, <i>Identity types in algebraic model structures<\/i>\r\n<br><br>\r\nThe original Bezem-Coquand-Huber cubical set model promised to give a\r\nconstructive model of homotopy type theory. However, in its original\r\nform there was a notable shortcoming: one of the definitional\r\nequalities usually included in type theory, the J computation rule was\r\nabsent. One way to fix this is to use an alternative definition of\r\nidentity type in which we keep track more carefully of degenerate\r\npaths. The new identity type has a nice presentation in the setting of\r\nalgebraic model structures. To model identity types what we need is\r\nvery good path objects: a factorisation of each diagonal as a trivial\r\ncofibration followed by a fibration. I'll show a general way to create\r\nvery good path objects from path objects in the weaker sense of\r\nfactorisations of diagonals as weak equivalence followed by fibration,\r\nand that under certain reasonable conditions on an ams this can be\r\ncarried out \"stably and functorially\" allowing us to satisfy the\r\nGarner-van den Berg notion of \"stable functorial choice of diagonal\r\nfactorisation.\"  These can then be used to model identity types.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c151');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Steve Awodey, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c153')\"><i>Natural models of type theory<\/i><\/a>\r\n<div id=\"absbox-c153\" class=\"abstract_box\">\r\nSteve Awodey, <i>Natural models of type theory<\/i>\r\n<br><br>\r\nThe notion of a <i>natural model<\/i> of type theory provides an entirely algebraic description of a system of dependent type theory with an operation of context extension, and can serve as a flexible notion of a model of type theory.  We give the main definition, originally stated in [1], make the algebraic character explicit, and then sketch the proof that this notion is equivalent to that of a <i>category with families<\/i> in the sense of Dybjer [2].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nS.&nbsp;Awodey,\r\nNatural models of homotopy type theory,\r\narXiv,\r\n1406.3219v2.\r\n<br>\r\n[2]\r\nP.&nbsp;Dybjer,\r\nInternal type theory,\r\nLNCS,\r\nvol.&nbsp;1158 (1996), pp.&nbsp;120\u2013134.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c153');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Clive Newstead, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c155')\"><i>Categories of natural models of type theory<\/i><\/a>\r\n<div id=\"absbox-c155\" class=\"abstract_box\">\r\nClive Newstead, <i>Categories of natural models of type theory<\/i>\r\n<br><br>\r\nNatural models of type theory (see [1]) provide an algebraic setting for the interpretation of dependent type theory. First, we define homomorphisms of natural models of type theory with a basic type and context extension, and prove that the syntactic category of contexts is initial in this category. We then extend our construction to allow for dependent sums and products. Time permitting, we prove that, in this latter case, the polynomial functor associated with a given natural model is a polynomial monad. This is joint work with Steve Awodey.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSteve Awodey,\r\nNatural models of homotopy type theory,\r\neprint arXiv:1406.3219v2\r\n(2015)\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c155');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Egbert Rijke and Ulrik Buchholtz, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c157')\"><i>The real projective spaces in HoTT<\/i><\/a>\r\n<div id=\"absbox-c157\" class=\"abstract_box\">\r\nEgbert Rijke and Ulrik Buchholtz, <i>The real projective spaces in HoTT<\/i>\r\n<br><br>\r\nWe construct the real projective spaces as certain higher inductive types in Homotopy Type Theory. The classical definition of $\\mathbb{R}\\mathrm{P}^n$, as the quotient space where the antipodal points of the $n$-sphere are identified, does not translate directly to Homotopy Type Theory. Instead, we define $\\mathbb{R}\\mathrm{P}^n$ by induction on $n$ simultaneously with its tautological bundle. As the base case, $\\mathbb{R}\\mathrm{P}^{-1}$ is taken to be the empty type. In the inductive step, $\\mathbb{R}\\mathrm{P}^{n+1}$ is taken to be the mapping cone of the projection map of the tautological bundle. It is then possible to define the tautological bundle on $\\mathbb{R}\\mathrm{P}^{n+1}$ using the universal property of $\\mathbb{R}\\mathrm{P}^{n+1}$ and the univalence axiom.\r\nWith this definition, one can use the descent theorem to show that the total space of the tautological bundle of $\\mathbb{R}\\mathrm{P}^n$ is the $n$-sphere. Hence one can retrieve the description of $\\mathbb{R}\\mathrm{P}^{n+1}$ as $\\mathbb{R}\\mathrm{P}^n$ with an $(n+1)$-disk attached to it. The infinite dimensional real projective space $\\mathbb{R}\\mathrm{P}^\\infty$, defined as the sequential colimit of $\\mathbb{R}\\mathrm{P}^n$ with the canonical inclusion maps, is equivalent to $K(\\mathbb{Z}\/2\\mathbb{Z},1)$, and the tautological bundles of the finite dimensional real projective spaces factor through $\\mathbb{R}\\mathrm{P}^{n+1}$ as expected. Indeed, the infinite dimensional projective space classifies the $0$-sphere bundles \u2013 which one can think of as synthetic line bundles \u2013 and using the join connectivity theorem one can then show that the tautological bundle of $\\mathbb{R}\\mathrm{P}^n$ is an $(n-1)$-connected map into $\\mathbb{R}\\mathrm{P}^\\infty$.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c157');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Peter Aczel, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c159')\"><i>On Type Theory and the Philosophy of Mathematics<\/i><\/a>\r\n<div id=\"absbox-c159\" class=\"abstract_box\">\r\nPeter Aczel, <i>On Type Theory and the Philosophy of Mathematics<\/i>\r\n<br><br>\r\nThis talk is intended to advocate the basics of Per Martin-Lof's\r\ndependent type theory as a general purpose tool in the philosophy of\r\nmathematics that is an improvement on the traditional uses of first order\r\nlogic.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c159');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Philosophical Logic <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Joan Casas-Roma, Antonia Huertas and M. Elena Rodr\u00edguez, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c161')\"><i>Towards a semantics for Dynamic Imagination Logic<\/i><\/a>\r\n<div id=\"absbox-c161\" class=\"abstract_box\">\r\nJoan Casas-Roma, Antonia Huertas and M. Elena Rodr\u00edguez, <i>Towards a semantics for Dynamic Imagination Logic<\/i>\r\n<br><br>\r\nImagining alternative ways the reality could be is something we do almost everyday in our lives: when we wonder how things would be if I was to win the lottery or if I were working in another company, when we read a book that describes a reality different than ours, when we \"log in\" to a virtual world and we incarnate our <i>alter-ego<\/i>, or even when we consider more recent technological developments like virtual reality: while wearing a pair of VR goggles, we start experiencing a new, different reality in which we accept some things to be different, in which we are someone else, and in which we embrace new rules that we know they would be impossible in our reality but which are, nonetheless, possible in that alternative one.\r\nWe define and present the syntax and semantics of the Dynamic Imagination Logic: this system models an epistemic agent who is able to deal with issues involving imagination in a dynamic way, and perform actions such as imagining \"whether $\\varphi$ could be the case, provided everything else stays the same\", or even imagining \"how different things should be in order for $\\varphi$ to be the case\". These actions are processed in a dynamic way that expands the model by creating new and different \"spaces\" that represent alternative realities the agent thinks about; moreover, the model is expanded in such a way that every imagination step can be fully traced to see what the agent imagined and from where, which alternative realities are generated by each imagination step, and what would the agent come to know or ignore in each of these alternatives.\r\n<br><br>\r\n<i>\\textbf{Acknowledgements:} This work has been supported by a doctoral grant from the Universitat Oberta de Catalunya (UOC) and the funding for the project \"Hybrid Intensional Logic\" (with reference FFI2013-47126-P) given by the Spanish Ministry of Economy and Competitiveness.<\/i>\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c161');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Hao-Cheng Fu, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c163')\"><i>On the problem of higher order vagueness<\/i><\/a>\r\n<div id=\"absbox-c163\" class=\"abstract_box\">\r\nHao-Cheng Fu, <i>On the problem of higher order vagueness<\/i>\r\n<br><br>\r\nIt is well-known that the higher order vagueness is an arguable phenomenon while we want to construct an adequate semantics for vagueness. Timothy Williamson has taken the phenomenon of higher order vagueness to refute some attempts to revise the semantics of classical logic. In other words, Williamson claimed that best way to dissolve the problem of higher order vagueness is the epistemic theory rather than trying to amend the logical system such as three-valued, many-valued, continuum valued and supervaluationist's theories. When we consider Williamson's strategy to refute these theories, it seems there is a more basic problem which we need to contemplate. Is the phenomenon of higher order vagueness real or just verbal? So, I must say that we have to distinguish two different problems about the confusion of higher order vagueness. Of course the first one is to ask whether the phenomenon is real and the other one is to inquire whether the phenomenon does matter or not? I shall argue that the phenomenon of higher order vagueness is real but we needn't to be afraid because the phenomenon would not threaten supervaluationism as Williamson contended.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c163');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Edoardo Rivello, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c165')\"><i>A revision-theoretic general theory of definitions<\/i><\/a>\r\n<div id=\"absbox-c165\" class=\"abstract_box\">\r\nEdoardo Rivello, <i>A revision-theoretic general theory of definitions<\/i>\r\n<br><br>\r\nThe classical theory of definitions bans so-called <i>circular<\/i> definitions, namely, definitions of, say, a unary predicate $P$ based on stipulations of the form\r\n\\[ P(x) =_{\\mathsf{Df}} \\Phi(P, x),  \\]\r\nwhere $\\Phi$ is a formula of a fixed first-order language and the <i>definiendum<\/i> $P$ occurs into the <i>definiens<\/i> $\\Phi$.\r\nIn their seminal book <i>The Revision Theory of Truth<\/i> [1], Gupta and Belnap claim that \"<i>General<\/i> theories of definitions are possible within which circular definitions [...] make logical and semantic sense\" [p. IX]. In order to sustain their claim, they develop in this book one general theory of definitions (in some variants) based on <i>revision sequences<\/i>, namely, ordinal-length iterations of the operator which is induced by the (possibly circular) definition of the predicate.\r\nGupta-Belnap's approach to circular definitions has been criticised, among others, by Martin [2] and McGee [3]. Their criticisms point on the logical complexity of revision sequences, on their relations with ordinary mathematical practice, and on their merits relative to alternative approaches. In my talk I will present an alternative general theory of definitions, based on a combination of supervaluation and $\\omega$-length  revision, which aims to address some issues raised by Martin and McGee while preserving the philosophical and mathematical core of revision.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAnil Gupta and Nuel Belnap,\r\nThe Revision Theory of Truth,\r\nA Bradford Book,\r\nMIT Press,\r\n1993.\r\n<br>\r\n[2]\r\nDonald A. Martin,\r\nRevision and its rivals,\r\nPhilosophical Issues,\r\nvol.&nbsp;8 (1997), pp.&nbsp;387\u2013406.\r\n<br>\r\n[3]\r\nVann McGee,\r\nRevision,\r\nPhilosophical Issues,\r\nvol.&nbsp;8 (1997), pp.&nbsp;407\u2013418.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c165');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Simon Hewitt, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c167')\"><i>Indefinite extensibility and set-theoretic relativity<\/i><\/a>\r\n<div id=\"absbox-c167\" class=\"abstract_box\">\r\nSimon Hewitt, <i>Indefinite extensibility and set-theoretic relativity<\/i>\r\n<br><br>\r\nFollowing Dummett, several authors have responded to the set-theoretic paradoxes by claiming that the concept <i>set<\/i> is indefinitely extensible [1]. One way of understanding this is in terms of ontological extensibility: however many sets there are, it is always possible that there be more. Recent work by Gabriel Uzquiano provides an alternative characterisation of indefinite extensibility, laying out an account of set-formation within a fixed domain modal logic [2]. The modality is interpreted  linguistically, in terms of possible expansions of the extensions of predicates. We review Uzquiano's proposal, and make clear the underlying philosophical picture of set-theoretic ontology. We argue that this picture sits uncomfortably with widespread convictions about the nature of sets. More seriously, we call into question whether the proponent of a linguistic account of indefinite extensibility is entitled to assume a sufficiently large cardinality of objects to recover set-theory.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nMichael Dummett,\r\nFrege: Philosophy of Mathematics,\r\nDuckworth,\r\n1991.\r\n<br>\r\n[2]\r\nGabriel Uzquiano,\r\nVarieties of Indefinite Extensibility,\r\nNotre Dame Journal of Formal Logic,\r\nvol.56 (2015), no.1, pp.147-166.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c167');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Farshad Badie and Hans G\u00f6tzsche, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c169')\"><i>Towards Logical Analysis of Occurrence Values in Truth-Functional Independent Occurrence Logic<\/i><\/a>\r\n<div id=\"absbox-c169\" class=\"abstract_box\">\r\nFarshad Badie and Hans G\u00f6tzsche, <i>Towards Logical Analysis of Occurrence Values in Truth-Functional Independent Occurrence Logic<\/i>\r\n<br><br>\r\nThe human beings never really understood how truth could be recognised as the centrepiece of philosophy. The idea of truth vs. falsity is based on the assumption that the truth-value of statements about things beyond actual settings can, indisputably, be determined (\u2018false\u2019 statements about settings are just counterfactuals).\r\nIn this discussion, we will rely on our alternative kind of logic: <i>Occurrence Logic<\/i> (Occ Log), which is not based on truth functionality, see [1]. The Occ Log $z&nbsp;^\\circ{>}&nbsp;y$ expression denotes the fact that \u2018$y$ occurs in case and only in case $z$ occurs\u2019. Note that '$z&nbsp;^\\circ{>}&nbsp;y$' does not by itself express any kind of truth-value semantics. We will see that the  <i>Occurrences<\/i> as the main building blocks of our approach are independent from truth-values, but they are strongly dependent on the occurrence values. The fact that '$y$ would only occur [and would only have an occurrence value] in case $z$ occurs [and has an occurrence value]', has been represented by Occ Log expression $z&nbsp;^\\circ{>}&nbsp;y$. We shall stress that what is in logic often called \u2018states of affairs\u2019 (including \u2018events\u2019) of the real world could be called <i>Local Universes<\/i> that are made of Entities and Properties. Focusing on the events $z$ and $y$ we can justifiably say that \u201cin case, and only in case, the local universe of $z$ differs from the local universe of $y$ regarding at least one but not all Entities and Properties, one of them can, potentially, be said to be a change of the other\u201d.\r\n\\begin{thebibliography}\r\n<br>\r\n[1]\r\nG\u00f6tzsche, Hans,\r\nDeviational Syntactic Structures. London \/ New Delhi \/ New York \/ Sydney: Bloomsbury Academic, 2013.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c169');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Models of Arithmetic <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Micha\u0142 Tomasz Godziszewski, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c171')\"><i>Short Elementary Cuts in Recursively Saturated Models of Arithmetic<\/i><\/a>\r\n<div id=\"absbox-c171\" class=\"abstract_box\">\r\nMicha\u0142 Tomasz Godziszewski, <i>Short elementary cuts in recursively saturated models of arithmetic<\/i>\r\n<br><br>\r\nWe study certain model-theoretic properties of countable recursively saturated models of arithmetic.\r\nOur primary inspiration for examining mathematical features of such structures, and recursively saturated in particular, is that every countable recursively saturated model of Peano Arithmetic supports a great variety of nonstandard satisfaction classes that can serve as models for formal theories of truth - those models allow to investigate the role of arithmetic induction in semantic considerations. In the other direction, nonstandard satisfaction classes are used as a tool in model theoretic constructions providing answers to questions in the model theory of formal arithmetic and often allow to solve problems that do not explicitly involve nonstandard semantics.\r\nDue to the fact that a for a countable model of arithmetic, it is equivalent to admit a full satisfaction class (i.e. satisfy the formal theory of compositional truth) and to be recursively saturated, the project can be thought of as an investigation into structure of possible interpretations for theory of compositional, arithmetical truth. It needs to be underlined that the purpose of our research is to examine model-theoretic but purely arithmetical properties of models admitting satisfaction classes. In particular, we study various substructures of recursively saturated models of PA, and we focus on the cofinal extensions of models of PA.\r\nFirst-order theories of pairs $(N,M)$, where $N\\models PA$ and $M$ is an elementary cofinal submodel of $M$ reveal great diversity and demand systematic study. The case of models admitting satisfaction classes is of particular interest in this respect: all countable recursively saturated models of PA have continuum many nonisomorphic cofinal submodels, and after acknowledging the variety of the abovementioned pairs for $N$ being countable recursively saturated, the next goal is to consider isomorphism types and first-order theories for pairs of models $(N, M)$ for a fixed countable recursively saturated model $N$ and a fixed isomorphism type of $M$. The method that has already been shown quite effective in this direction is the method of <i>gaps<\/i> (also called <i>skies<\/i>). We present briefly the gap terminology and explain why it is useful.\r\nSkolem terms, also called simply definable functions\\footnote{with a slight abuse of terminology that is unimportant to our investigations}, are paramter-free definable and PA-provably total functions. Let $\\mathcal{M}$ be a nonstandard model of arithmetic and let $\\mathcal{F}$ be some family of Skolem terms  $f : M \\rightarrow M$ such that $\\forall x,y \\in M \\: x &lt;y \\Rightarrow  x \\leq f(x) \\leq f(y)$. There is a partition of $M$ into sets, which we call $\\mathcal{F}$-gaps. For any $a \\in M$, $gap_{\\mathcal{F}}(a)$ is the smallest set $C \\subseteq M$ such that $a \\in C$ and:\r\n$ \\forall b \\in C \\: \\forall f \\in \\mathcal{F} \\: \\forall x \\in M\\: \\:  b \\leq x \\leq f(b) \\: \\vee \\: x \\leq b \\leq f(x)) \\Rightarrow x \\in C.$\r\nThis is a natural generalization of an idea of partitioning the universe of a nonstandard model into $\\mathbb{Z}$-blocks around each element (then, each such block is $gap_{\\mathcal{F}}(a)$ for some $a$, where $\\mathcal{F}$ consists only of the successor function $s$).\r\nThe gap of $a \\in M$, denoted by $gap(a)$, is the $\\mathcal{F}$-gap of $a$, where $\\mathcal{F}$ is the family of \\textbf{all} such definable functions, i.e.\r\n$\\mathcal{F} = \\{f : M \\rightarrow M: f \\: \\: \\text{is definable and} \\: \\: \\forall x,y \\in M \\: x &lt;y \\Rightarrow  x \\leq f(x) \\leq f(y)\\}.$\r\nEvery model $\\mathcal{M}$ has the least gap, the $gap(0)$. Let $A \\subseteq M$. Then, we denote $sup(A) = \\{x \\in M : \\exists y \\in A \\: x \\leq y\\}$. If for some $a \\in M$, $M =sup(gap(a))$, then we call $gap(a)$ the \\textbf{last gap of $\\mathcal{M}$}. A model with a last gap is called \\textbf{short}. If $\\mathcal{M} \\preceq_{cut} \\mathcal{N}$ (i.e. $\\mathcal{M}$ is an elementary cut of $\\mathcal{N}$), we say that $\\mathcal{M}$ is \\textbf{short elementary cut} of $\\mathcal{N}$ if $\\mathcal{M}$ is short - in other words, if by $Scl(a)$ we denote the set $\\{t(a): \\text{ $t$ is a Skolem term of PA}\\}$ $\\mathcal{M}$ is short if there is such an element $a \\in M$ that its Skolem closure in $\\mathcal{M}$ is cofinal in $\\mathcal{M}$, i.e. for all $x \\in M$ there is $b \\in Scl(a)$ such that $x &lt;_{\\mathcal{M}} b$. An elementary cut is \\textbf{coshort} if $\\mathcal{N} \\setminus \\mathcal{M}$ has the least gap, i.e. there is $a \\in N \\setminus M$ s.t. $M = inf(gap(a))$, where  $inf(A) = \\{x \\in M : \\forall y \\in A \\: x \\leq y\\}$.\r\nNow, to clarify the gap terminology, if we put: $\\mathcal{M}(a) = sup(Scl(a))$, and $\\mathcal{M}[a] = \\{b \\in M: \\: \\forall t \\in Scl(b) \\: t(b) &lt; a\\}$, then the set $[a) = \\mathcal{M}(a) \\setminus \\mathcal{M}[a]$ is exactly the $gap(a)$. It can be shown that $\\mathcal{M}(a)$ is the smallest elementary cut of $\\mathcal{M}$ containing $a$, and that $\\mathcal{M}[a]$ is empty if and only if everey elementary cut of $\\mathcal{M}$ contains $a$. Gap terminology is particularly useful in the study of recursively saturated models of PA (see e.g. [5] for a reference to many methods and properties).\r\nOne of the interesting and natural questions conerning <i>pairs<\/i> for countable recursively saturated models of arithmetic and its cofinal submodels is the following <i>big<\/i> question of our particular interest:\r\n<ul>\r\n<li>\r\n<i>Let $\\mathcal{M} \\models PA$ be a countable recursively saturated model and let $\\mathcal{K}, \\mathcal{K}'$ be elementary cuts of $\\mathcal{M}$. Suppose that $(\\mathcal{M}, \\mathcal{K}) \\equiv (\\mathcal{M}, \\mathcal{K}')$. Does it follow that $(\\mathcal{M}, \\mathcal{K}) \\cong (\\mathcal{M}, \\mathcal{K}')$? <\/i>\r\n<\/ul>\r\nAnother way to put it is: under what conditions, does the identity of theories of such pairs imply their isomoprhism? It is known that the answer to the <i>big<\/i> question above is negative, if $\\mathcal{K}$ and $\\mathcal{K}'$ in question are coshort, as shown by R. Kossak and J. Schmerl in [6]. However, it remains open (and is considered to be difficult) what is the answer for the case in which $\\mathcal{K}$ and $\\mathcal{K}'$ are short elementary cuts of $\\mathcal{M}$, i.e. are of the form $\\mathcal{M}(a)$ and $\\mathcal{M}(b)$ for some $a, b \\in M$.\r\nSince it is not hard to prove the equivalence that there exists an automorphism of such $\\mathcal{M}$ if and only if $tp(a) = tp(b)$ (in the purely arithmetical language $\\mathcal{L}$), where $tp(a) =\\{\\varphi(x): \\mathcal{M} \\models \\varphi(a)\\}$ is the set of formulae satisfied in $\\mathcal{M}$ by $a \\in M$, the natural way to proceed is to consider the definable sets in $(\\mathcal{M}, \\mathcal{M}(a))$ and complete types realized in the last gap of $\\mathcal{M}$. We might then first ask  under what circumstances there is an element $c \\in gap(a)$ such that $tp(c) \\in Def(\\mathcal{M}, \\mathcal{M}(a))$ for $\\mathcal{M}$ being a countable recursively saturated model of PA.\r\nUsing results of Smorynski from [12] and working with gaps and standard systems $SSy(\\mathcal{M})$  of $\\mathcal{M}$, i.e. the family of all subsets of $\\mathbb{N}$ that are coded in $\\mathcal{M}$\\footnote{It turns out that the standard system tells you a lot about the model; for example, any two countable recursively saturated models of the same completion of PA with the same standard system\r\nare isomorphic.} we show that\r\n<br><br>\r\n<b>Theorem<\/b> (Tin Lok Wong, MTG).\r\nLet $\\mathcal{M} \\models PA$ be a countable recursively saturated model and let $a, b \\in M$. Suppose that $(\\mathcal{M}, \\mathcal{M}(a)) \\equiv (\\mathcal{M}, \\mathcal{M}(b))$ (recall that $\\mathcal{M}(a)$ and $\\mathcal{M}(b)$ are short). If $SSy(\\mathcal{M}) \\subseteq Def(\\mathbb{N})$, then $(\\mathcal{M}, \\mathcal{M}(a)) \\cong (\\mathcal{M}, \\mathcal{M}(b))$.\r\n<br><br>\r\nAs the project is essentially <i>in progress<\/i>, we end with perspective paths for further work.\r\nThe conceptual import of the result is that taking a nonstandard model of compositional truth such that all its coded sets are already definable in the standard model, we are able to identify isomorphic cofinal short elementary cuts of the model just by looking on the arithmetical theory of both pairs considered.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nJ. Barwise, J. Schlipf,\r\nAn introduction to recursively saturated and resplendent models,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;41 (1976), pp.&nbsp;531\u2013536.\r\n<br>\r\n[2]\r\nA. Enayat, A. Visser,\r\nNew constructions of satisfaction classes,\r\nUnifying the Philosophy of Truth\r\n(T. Achourioti, H. Galinon, J. Martinez Fernandez and K. Fujimoto, editors),\r\nSpringer-Verlag,\r\nPublisher's address,\r\n2015,\r\npp.&nbsp;X\u2013XX.\r\n<br>\r\n[3]\r\nF. Engstr\u00f6m,\r\nSatisfaction classes in nonstandard models of first-order arithmetic,\r\nChalmers University of Technology and G\u00f6teborg University, G\u00f6teborg,\r\nvol.&nbsp;XX (2002), no.&nbsp;X, pp.&nbsp;XXX\u2013XXX.\r\n<br>\r\n[4]\r\nR. Kaye,\r\nModels of Peano Arithmetic,\r\nOxford University Press,\r\nOxford,\r\n1991.\r\n<br>\r\n[5]\r\nR. Kossak, J. Schmerl,\r\nThe structure of models of Peano Arithmetic,\r\nClarendon Press,\r\nOxford,\r\n2006.\r\n<br>\r\n[6]\r\nR. Kossak, J. Schmerl,\r\nOn Cofinal Submodels and Elementary Interstices,\r\nNotre Dame Journal of formal Logic,\r\nvol.&nbsp;53 (2012), pp.&nbsp;267\u2013287.\r\n<br>\r\n[7]\r\nH. Kotlarski, S. Krajewski, A. Lachlan,\r\nConstruction of satisfaction classes for nonstandard models,\r\nCanadian Mathematical Bulletin,\r\nvol.&nbsp;24 (1981), pp.&nbsp;283\u2013293.\r\n<br>\r\n[8]\r\nH. Kotlarski,\r\nFull satisfaction classes: a survey,\r\nNotre Dame Journal of Formal Logic,\r\nvol.&nbsp;32 (1991), pp.&nbsp;573\u2013579.\r\n<br>\r\n[9]\r\nS. Krajewski,\r\nNonstadard satisfaction classes,\r\nSet Theory and Hierarchy Theory\r\n(W. Marek, M. Srebrny and A. Zarach, editors),\r\nEditorial,\r\nHeidelberg,\r\n1976,\r\npp.&nbsp;X\u2013XX.\r\n<br>\r\n[10]\r\nA. Lachlan,\r\nFull satisfaction classes and recursive saturation,\r\nCanadian Mathematical Bulletin,\r\nvol.&nbsp;42 (1981), pp.&nbsp;295\u2013297.\r\n<br>\r\n[11]\r\nA. Robinson,\r\nOn languages based on non-standard arithmetic,\r\nNagoya Mathematical Journal,\r\nvol.&nbsp;22 (1963), pp.&nbsp;83\u2013107.\r\n<br>\r\n[12]\r\nC. Smorynski,\r\nCofinal extensions and nonstandard models of arithmetic,\r\nNotre Dame Journal of formal Logic,\r\nvol.&nbsp;2 (1981), pp.&nbsp;133\u2013144.\r\n<br>\r\n[13]\r\nC. Smorynski,\r\nElementary extensions of recursively saturated models of arithmetic,\r\nNotre Dame Journal of formal Logic,\r\nvol.&nbsp;2 (1981), pp.&nbsp;193\u2013203.\r\n<br>\r\n[14]\r\nC. Smorynski,\r\nRecursively saturated nonstandard models of arithmetic,\r\nJournal of Symbolic Logic,\r\nvol.&nbsp;46 (1981), pp.&nbsp;259\u2013286.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c171');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Jana Glivick\u00e1, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c173')\"><i>Skolem arithmetic, its extensions and their properties<\/i><\/a>\r\n<div id=\"absbox-c173\" class=\"abstract_box\">\r\nJana Glivick\u00e1, <i>Skolem arithmetic, its extensions and their properties<\/i>\r\n<br><br>\r\nSkolem arithmetic (SA) is the theory of the structure $(\\mathbb{N}-\\{0\\},\\cdot)$. This structure, its standard model, can be viewed as an infinite direct sum of the standard model $(\\mathbb{N},+)$ of Presburger arithmetic. SA is known to be decidable [3], with explicit complete axiomatization and simple elimination set [1]. We are interested in extensions of SA, in some enriched language, that have similar properties and thus are, in this sense, still very far from Peano arithmetic.\r\nTo obtain such extensions, we first prove a general theorem on substructures of infinite direct products. The setting is as follows: For every $p \\in \\mathbb{N}$, $\\mathcal{M}_p$ is a structure for language $\\mathcal{L}$ with constant symbol $0$. We impose certain technical conditions on $\\mathcal{L}$ and $\\mathcal{M}_p$. $\\mathcal{M}$ is an $\\mathcal{L}$-structure with $\\coprod_{p \\in \\mathbb{N}} \\mathcal{M}_p \\subseteq \\mathcal{M} \\subseteq \\prod_{p \\in \\mathbb{N}} \\mathcal{M}_p$, where $\\coprod$ denotes direct sum and $\\prod$ direct product.\r\nWe say that $\\mathcal{M}$ has uniform witnessing property if, for any $\\mathcal{L}$-formula $\\varphi(\\bar{x},y)$ such that $\\varphi(\\bar{0},0)$ is true in all $\\mathcal{M}_p$ and any $\\bar{a} \\in \\mathcal{M}$, the following holds true\r\n$$(\\forall p \\in \\mathbb{N}) \\mathcal{M}_p \\vDash \\exists y \\varphi(\\bar{a}(p),y) \\Rightarrow (\\exists b \\in \\mathcal{M})(\\forall p \\in \\mathbb{N}) \\mathcal{M}_p \\vDash \\varphi(\\bar{a}(p),b(p)).$$\r\nSuppose $\\mathcal{M}$ has the uniform witnessing property and for any $\\bar{a} \\in \\mathcal{M}$ there are infinitely many $p \\in \\mathbb{N}$ such that $\\bar{a}(p) = \\bar{0}$. We prove that $\\coprod_{p \\in \\mathbb{N}} \\mathcal{M}_p$ is then an elementary substructure of $\\mathcal{M}$. The two conditions are sufficient, but provably not necessary.\r\nIt is possible to express the uniform witnessing property in the language of SA. It allows us to \"lift\" results about extensions of Presburger arithmetic to extensions of SA. We show how to obtain natural complete axiomatizations of theories extending SA. Moreover, we show how to lift properties such as decidability, quantifier elimination, existence of simple elimination set and so on. A prominent example of such a lifting is the theory $SA^a$, Skolem arithmetic with a \"slice\" of exponentiation (function thought of as $f(x) = x^a$, where $a$ is nonstandard). Here, we use results on certain extensions of Presburger arithmetic called linear arithmetics [2].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1] P. Cegielski, Theorie elementaire de la multiplication des entiers naturels,Lecture notes in mathematics, vol.&nbsp;890, pp.&nbsp;44\u201389.\r\n<br>\r\n[2] P. Glivick\u00fd, Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis, PhD. thesis,Charles University in Prague, 2013.\r\n<br>\r\n[3] A. Mostowski, On direct product of theories, Journal of symbolic logic, vol.&nbsp;36, pp.&nbsp;1\u201331.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c173');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Michal Garlik, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c175')\"><i>Models of bounded arithmetic and a restricted ultrapower construction<\/i><\/a>\r\n<div id=\"absbox-c175\" class=\"abstract_box\">\r\nMichal Garlik, <i>Models of bounded arithmetic and a restricted ultrapower construction<\/i>\r\n<br><br>\r\nIt is well known that some problems in complexity theory can be reformulated as problems of constructions of expanded extensions of models of bounded arithmetic. Usually, these models are required to satisfy some form of bounded induction and at the same time not to introduce any new lengths. It seems that modifications of the ultrapower construction could make it easier to meet the last requirement. Our attempt in this direction is a construction by a restricted reduced power. The construction starts with a model $M$ of true arithmetic, a nonstandard element $n$ of $M$, and an index set $\\Omega$ which is a subset of numbers of length $n$. It uses straight-line programs defined in $M$ to build the universe of a new model $N$. The construction assumes a hypothesis that there is a formula $\\psi(x,y)$ such that for each straight-line program $f$ of size $m^s$ the formula $\\psi(x,f(x))$ is true in $M$ for at most $q$ fraction of the time when $x$ ranges over $\\Omega$. Using these parameters, which are moreover required to satisfy $qn^{m^i}&lt;1$ for each $i\\in \\mathbb{N}$, the construction proceeds by defining a filter on the powerset of $\\Omega$ and a suitable subset of straight-line programs of size $m^s$. This is done in countably many stages and the crucial step is to ensure induction. The resulting model $N$ then satisfies the theory $strict R^1_2$, agrees with $M$ on lengths and contains an element which falsifies $\\psi$.\r\nWe use the construction to separate theories $R^1_2(g)$ and $strict R^1_2(g)$ assuming that $g$ is a one-way permutation hard against polynomial-size circuits. We add further applications of the construction.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c175');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\"><br><\/td><td style=\"vertical-align: top;\"><br><\/td><\/tr>\r\n\r\n\r\n<tr><td style=\"vertical-align: top;\" colspan=\"2\"><p><b><a name=\"contrib5\"><\/a>Friday 5th August<\/b><\/p><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Set Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Murdoch Gabbay, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c177')\"><i>Consistency of Quine's NF using nominal techniques<\/i><\/a>\r\n<div id=\"absbox-c177\" class=\"abstract_box\">\r\nMurdoch Gabbay, <i>Consistency of Quine's NF using nominal techniques<\/i>\r\n<br><br>\r\n<i>Naive set theory<\/i> has one rule; <i>naive sets comprehension<\/i>:\r\n<ul><li>\r\nIf $\\phi$ is a predicate, then $\\{a\\mid \\phi(a)\\}$ is a set (the $a$ such that $\\phi$).\r\n<\/ul>\r\nThis is inconsistent by Bertrand Russell's famous observation of 1901 that\r\n$\\{a\\mid a{\\not\\in} a\\}\\in\\{a\\mid a{\\not\\in} a\\}$ if and only if $\\{a\\mid a{\\not\\in} a\\}\\not\\in\\{a\\mid a{\\not\\in} a\\}$.\r\nSolutions proposed included Zermelo-Fraenkel set theory, simple type theory, and Quine's New Foundations (NF).\r\nZermelo-Fraenkel set theory restricts comprehension so we can only form comprehension within an existing set.\r\nSimple type theory imposes a type system.\r\nQuine's NF weakens comprehension by restricting it to <i>stratifiable formulae<\/i>; formulae in which variables can be assigned `levels', which are natural numbers, such that if $a\\in b$ occurs in a formula and $a$ has level $n$, then $b$ must have level $n{+}1$.\r\nRussell's example is clearly ruled out because $a\\in a$ cannot be stratified.\r\nConsistency of NF has been an open problem since it was proposed by Quine in 1937.\r\nI will present a claimed proof of consistency of Quine's NF, based on ideas previously developed to extend duality theory to logics with quantifiers (the paper is available on my webpage and on arXiv).\r\nIn the paper:\r\n<ul>\r\n<li>\r\nStratifiability corresponds to a simple normalisability property on terms,\r\n<li>\r\nsubstitution corresponds to both a nominal algebra axiomatisation and a renormalisation procedure,\r\n<li>\r\nuniversal quantification corresponds to a colimit in nominal lattices, and to a form of the nominal new-quantifier (and existential quantification to its dual),\r\n<li>\r\nsets comprehension corresponds to nominal atoms-abstraction, and\r\n<li>\r\nextensionality corresponds to a non-evident transfinite construction which amounts to saturating an equality theory with all possible equalities to ensure that extensionality holds.\r\n<\/ul>\r\nThe end result is a points-based representation of NF in which predicates correspond to sets of points, and points are deductively closed sets of predicates subject to some interesting filter-style conditions.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c177');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Alessandro Vignati, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c179')\"><i>Forcing axioms and automorphisms of C*-algebras<\/i><\/a>\r\n<div id=\"absbox-c179\" class=\"abstract_box\">\r\nAlessandro Vignati, <i>Forcing axioms and automorphisms of C*-algebras<\/i>\r\n<br><br>\r\nIn the last decade Farah developed the study of the group of automorphisms of coronas C*-algebras. He and Coskey conjectured that under CH there are wild automorphisms of corona algebras, while under PFA the situation is conjectured to be quite rigid. McKenney and I recently verified the conjecture in the case of PFA for a large class of algebras, although very much remains open. After a brief introduction of the objects, we present the current situation of the conjecture, and then we focus on some of the difficulties stopping us from expanding our results.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c179');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Dorottya Sziraki, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c181')\"><i>A dichotomy for infinitely many $\\Sigma^0_2(\\kappa)$ relations on the $\\kappa$-Baire space<\/i><\/a>\r\n<div id=\"absbox-c181\" class=\"abstract_box\">\r\nDorottya Sziraki, <i>A dichotomy for infinitely many $\\mathbf{\\Sigma}^0_2(\\kappa)$ relations on the $\\kappa$-Baire space<\/i>\r\n<br><br>\r\nAs an initial step in studying dichotomies about independent sets with respect to certain (sets of) definable relations on the $\\kappa$-Baire space ${}^\\kappa\\kappa$,\r\nwe examine the case of a set $\\mathcal R$ of $\\kappa$ many $\\mathbf{\\Sigma}^0_2(\\kappa)$ relations (of arbitrary finite arity).\r\nBy considering games introduced by Jouko V\u00e4\u00e4n\u00e4nen [4] which allow trees to play, for the $\\kappa$-Baire space, a role analogous to that of Cantor-Bendixson ranks in the classical case\r\nand by restricting the allowed strategies for these games, we show that the $\\kappa$-version of a recent result of Martin Dole\u017eal and Wies\\l{}aw Kubi\u015b [1] (see also [2]) holds under $\\Diamond_\\kappa$.\r\nNamely, for a set $\\mathcal R$ of relations as above,\r\nif the $\\kappa$-Baire space has $\\mathcal R$-independent subsets of \"arbitrary Cantor-Bendixson rank\", then there exists a $\\kappa$-perfect $\\mathcal R$-independent subset.\r\nFor $\\kappa$ inaccessible, this is true already in ZFC.\r\nAs a corollary, we obtain (the slightly more general version of) a recent theorem of Jouko V\u00e4\u00e4n\u00e4nen and the author [3].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nM. Dole\u017eal, W. Kubi\u015b,\r\nPerfect independent sets with respect to infinitely many relations,\r\nsubmitted.\r\n<br>\r\n[2]\r\nW. Kubi\u015b,\r\nPerfect cliques and $G_\\delta$ colorings of Polish spaces,\r\nProceedings of the American Mathematical Society,\r\nvol.&nbsp;131 (2003), no.&nbsp;2, pp.&nbsp;619\u2013623.\r\n<br>\r\n[3]\r\nD. Szir\u00e1ki, J. V\u00e4\u00e4n\u00e4nen,\r\nA dichotomy for the generalized Baire space and elementary embeddability at uncountable cardinals,\r\nsubmitted.\r\n<br>\r\n[4]\r\nJ. V\u00e4\u00e4n\u00e4nen,\r\nA {Cantor-Bendixson} theorem for the space $\\omega_1^{\\omega_1}$,\r\nFundamenta Mathematicae,\r\nvol.&nbsp;137 (1991), no.&nbsp;3, pp.&nbsp;187\u2013199.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c181');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Proof Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Makoto Fujiwara, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c183')\"><i>Effective computability and constructive provability for existence sentences<\/i><\/a>\r\n<div id=\"absbox-c183\" class=\"abstract_box\">\r\nMakoto Fujiwara, <i>Effective computability and constructive provability for existence sentences<\/i>\r\n<br><br>\r\nAlong the line of [1], we investigate the relationship between effective computability and constructive provability for existence sentences.\r\nWe show the following.\r\nHere $\\mathsf{E\\mbox{-}PA^{\\omega}}$ (resp. $\\mathsf{E\\mbox{-}HA^{\\omega}}$) is the finite type extension of $\\mathsf{ PA}$ (resp. $\\mathsf{ HA}$), $\\mathsf{EL}$ is the system of elementary analysis, $AC$ (resp. $\\mathrm{AC_{00}}, \\mathrm{\\Pi^0_1\\text{-}AC_{00}} $) is the axiom of choice in all finite types (resp. the axiom of countable choice, that for purely universal formulas), $IP^\\omega_{ef}$ is the independence of premise for $\\exists$-free formulas, $ \\mathrm{\\Sigma^0_2\\text{-}DNS^0}$ is the fragment of double negation shift principle:\r\n$$\\forall \\alpha^1 (\\forall x^0 \\neg \\neg \\exists y^0 \\forall z^0 \\alpha (x,y,z)=0 \\to   \\neg \\neg \\forall x^0 \\exists y^0 \\forall z^0 \\alpha (x,y,z)=0  ).$$\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nFor every sentence $\\forall \\xi^1 (A(\\xi) \\rightarrow \\exists \\zeta^1 B (\\xi, \\zeta))$ where $A(\\xi)$ is $\\exists$-free and $B(\\xi, \\zeta)\\in \\mathrm{\\Gamma_1}$,\r\nif\r\n$$\\mathsf{E\\mbox{-}HA^{\\omega}} + AC+IP^\\omega_{ef}+  \\mathrm{\\Sigma^0_2\\text{-}DNS^0} \\vdash \\forall \\xi (A(\\xi) \\rightarrow \\exists \\zeta B (\\xi, \\zeta)),$$\r\nthen there exists a term $t^{1\\to 1}$ of $\\mathsf{E\\mbox{-}PA^{\\omega}}$ such that $$\\mathsf{E\\mbox{-}PA^{\\omega}} +\\mathrm{\\Pi^0_1\\text{-}AC_{00}} \\vdash \\forall \\xi (A(\\xi) \\rightarrow \\exists \\zeta B (\\xi, t\\xi)).$$\r\n<br><br>\r\nThe classes $\\mathcal{A}$, $\\mathcal{B}$ of formulas in $\\mathcal{L}(\\mathsf{E\\mbox{-}HA^{\\omega}}) $ are defined simultaneously by\r\n<ul>\r\n<li>\r\n$P$, $A_1\\wedge A_2$, $A_1\\vee A_2$, $\\forall x A_1$, $\\exists x A_1$, $B_1\\to A_1$ are in $\\mathcal{A}$;\r\n<li>\r\n$P$, $B_1\\wedge B_2$, $\\forall x B_1$, $A_1\\to B_1$ are in $\\mathcal{B}$;\r\n<\/ul>\r\nwhere $P$, $A_i$, $B_i$ range over prime formulas, formulas  in $\\mathcal{A}$, $\\mathcal{B}$ respectively.\r\n<br><br>\r\n<b>Theorem<\/b>.\r\nFor every sentence $\\forall \\xi^1 (A(\\xi) \\rightarrow \\exists \\zeta^1 B (\\xi, \\zeta))$ where $A(\\xi)\\in \\mathcal{A}$ and $B(\\xi, \\zeta)\\in \\mathcal{B}$,\r\nif there exists a term $t^{1\\to 1}$ of $\\mathsf{E\\mbox{-}PA^{\\omega}}$ such that $$\\mathsf{E\\mbox{-}PA^{\\omega}} +\\mathrm{\\Pi^0_1\\text{-}AC_{00}} \\vdash \\forall \\xi (A(\\xi) \\rightarrow \\exists \\zeta B (\\xi, t\\xi)),$$\r\nthen\r\n$$\\mathsf{EL}+\\mathrm{\\Pi^0_1\\text{-}AC_{00}} + \\mathrm{\\Sigma^0_2\\text{-}DNS^0} \\vdash \\forall \\xi (A(\\xi) \\rightarrow \\exists \\zeta B (\\xi, \\zeta)).$$\r\n<br><br>\r\n<b>Corollary<\/b>.\r\nFor every sentence $\\forall \\xi^1 (A(\\xi) \\rightarrow \\exists \\zeta^1 B (\\xi, \\zeta))$ where $A(\\xi)$ and $B(\\xi, \\zeta)$ are $\\exists$-free,\r\nthere exists a term $t^{1\\to 1}$ of $\\mathsf{E\\mbox{-}PA^{\\omega}}$ such that $$\\mathsf{E\\mbox{-}PA^{\\omega}} +\\mathrm{\\Pi^0_1\\text{-}AC_{00}} \\vdash \\forall \\xi (A(\\xi) \\rightarrow \\exists \\zeta B (\\xi, t\\xi))$$\r\nif and only if\r\n$$\\mathsf{EL}+\\mathrm{AC_{00}} + \\mathrm{\\Sigma^0_2\\text{-}DNS^0} \\vdash \\forall \\xi (A(\\xi) \\rightarrow \\exists \\zeta B (\\xi, \\zeta)).$$\r\n<br><br>\r\nThese also hold for $\\mathsf{\\widehat {E\\mbox{-}PA}^\\omega \\hspace{-.4em}\\upharpoonright}$, $\\mathsf{\\widehat {E\\mbox{-}HA}^\\omega \\hspace{-.4em}\\upharpoonright}$ and ${\\sf EL_0}$, which are the restrictions of corresponding systems to primitive recursion of type $0$ and quantifier-free induction, instead of $\\mathsf{E\\mbox{-}PA^{\\omega}}$, $\\mathsf{E\\mbox{-}HA^{\\omega}}$ and $\\mathsf{EL}$.\r\nNote that $\\mathsf{\\widehat {E\\mbox{-}PA}^\\omega \\hspace{-.4em}\\upharpoonright} +\\mathrm{\\Pi^0_1\\text{-}AC_{00}}$ contains arithmetical comprehension, which allows to develop most of ordinary mathematics.\r\nIt is known that most of Bishop's constructive mathematics can be formalized in  $\\mathsf{EL}+\\mathrm{AC_{00}}$.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nMakoto Fujiwara,\r\nIntuitionistic Provability versus Uniform Provability in $\\mathsf{RCA}$,\r\nLecture Notes in Computer Science,\r\nvol.&nbsp;9136 (2015), pp.&nbsp;186\u2013195.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c183');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Franco Parlamento and Flavio Previale, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c185')\"><i>The cut elimination and nonlengthening property for Gentzen's sequent calculus for first order logic with equality<\/i><\/a>\r\n<div id=\"absbox-c185\" class=\"abstract_box\">\r\nFranco Parlamento and Flavio Previale, <i>The Cut Elimination and Nonlengthening Property for Gentzen's Sequent Calculus for First Order Logic with Equality<\/i>\r\n<br><br>\r\nLeibniz's indiscernibility principle, in the framework of second order logic,\r\nleads to a sequent  calculus for  first order logic with equality,\r\nthat satisfies the cut elimination theorem, but cut free derivations may not  satisfy the subformula property. We note that  instead,  the path described by von Plato in his historical reconstruction of Gentzen's discovery of the sequent calculus  in [2], leads\r\nto a calculus that is fully satisfactory.\r\nIn addition to the reflexivity axiom $\\Rightarrow t=t$, it  has the following two left introduction rules for $=$:\r\n\\[\r\n\\frac{\\Gamma\\Rightarrow\\Delta, F\\{v\/r\\}}{\\Gamma, r=s\\Rightarrow\\Delta, F\\{v\/s\\}}=_1\\quad \\frac{ \\Gamma \\Rightarrow\\Delta, F\\{v\/s\\} }{\\Gamma, r=s\\Rightarrow\\Delta, F\\{v\/r\\}}=_2\r\n\\]\r\nOther satisfactory calculi can be  obtained by taking into account  the following other rules:\r\n\\[\r\n\\frac{\\Gamma, F\\{v\/r\\} \\Rightarrow\\Delta}{\\Gamma, F\\{v\/s\\},  r=s  \\Rightarrow\\Delta}=^l_1 \\quad \\frac{ \\Gamma , F\\{v\/s\\} \\Rightarrow\\Delta}{\\Gamma, F\\{v\/r\\},  r=s  \\Rightarrow\\Delta}=^l_2 \r\n\\]\r\nWe  give a very simple proof that cut elimination holds for a calculus obtained by adding to the reflexivity axiom some  of the above four rules if and only if it contains (at least)  $ =_1$ and $ =_2$ or $=_1$ and $ =_1^l$ or $ =_2$ and $ =_2^l$. The admissibility results\r\nthat are used, can be  refined and extended in order to show that\r\nif (and only if) all the above four rules are added, then every derivation can be trasformed into one that is  cut-free and satisfies the nonlenthening property of [1].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nA.V.  Lifschitz,\r\nSpecialization of the form of deduction in the predicate calculus  with equality and function symbols,\r\nThe Calculi of Symbolic Logic I\r\n( V.P. Orevkov, editor),\r\nProceedings of the Steklov Institute of Mathematics 98,\r\n1971,\r\npp.&nbsp;1\u201323.\r\n<br>\r\n[2]\r\nJ. von Plato,,\r\nGentzen's Proof Systems: Byproducts in a Work of Genius,\r\nThe Bulletin of Symbolic  Logic ,\r\nvol.&nbsp;18 (2012), no.&nbsp;3, pp.&nbsp;317\u2013367\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c185');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Bartosz Wcis\u0142o, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c187')\"><i>Models of positive truth<\/i><\/a>\r\n<div id=\"absbox-c187\" class=\"abstract_box\">\r\nBartosz Wcis\u0142o, <i>Models of positive truth<\/i>\r\n<br><br>\r\nWe study axiomatic theories of truth, i.e. extensions of theories such as Peano arithmetic $\\textrm{PA}$ or Zermelo-Fr\u00e4nkel set theory $\\textrm{ZFC}$ with a fresh predicate $T(x)$ with the intended reading '$x$ is a (code of a) true sentence' from the model-theoretic perspective. One of the most natural axiom systems, which we call $\\textrm{CT}^-$ consists of the Tarskian compositional clauses for the truth predicate. It is a classical theorem of Lachlan that all models of $\\textrm{PA}$ which admit an expansion to a model of $\\textrm{CT}^-$ are recursively saturated (see e.g. [2]). In effect not every model of $\\textrm{PA}$ admits an expansion to a model of $\\textrm{CT}^-$. In such situation we say that $\\textrm{CT}^-$ is not <i>model-theoretically<\/i> conservative over $\\textrm{PA}.$\r\nOne of the theories studied in the literature is $\\textrm{PT}^-$: a theory of truth whose axioms consist of positive compositional clauses, i.e. the usual compositional clauses for the conjunction, disjunction, the universal and the existential quantifier, but with no clause for the negation. Instead of it, clauses for double negations and de Morgan clauses for negated conjunctions, disjunctions, quantifiers and negated atomic sentences are introduced. A routine argument shows that unlike $\\textrm{CT}^-$ its positive counterpart sis model-theoretically conservative over $\\textrm{PA}.$\r\nIn our joint work with Cezary Cie\u015bli\\'nski and Mateusz \\L{}e\\l{}yk we study models of various natural extensions of $\\textrm{PT}^-$ proof-theoretically conservative over $\\textrm{PA}$ (which means that they do not prove any arithmetical sentences not provable in $\\textrm{PA}$ alone). In particular, we will focus on the principles which may be spelled out informally in the following way:\r\n<ul>\r\n<li>For every arithmetical formula $\\phi$ the set of $x$ such that $T(\\phi(x))$  holds is either empty or has the least element (the internal induction principle).\r\n<li>For every arithmetical formula $\\phi,$ if $\\phi$ is total, i.e. for every $x$ either $T (\\phi(x))$ or $T (\\neg \\phi(x))$ holds, then the set of $x$ such that $T (\\phi(x)$ is either empty or has the least element (internal induction for total formulae).\r\n<\/ul>\r\nIt turns out that adding any of these principles to $\\textrm{PT}^-$ result in a theory which is not model-theoretically conservative. Moreover, the models of the fist theory are recursively saturated, whereas all recursively saturated models (including the uncountable ones) admit an expansion to a model of the second theory.\r\nIf time allows, we will also discuss a variant of $\\textrm{PT}^-$ called $\\textrm{WPT}^-$ (weak $\\textrm{PT}^-$), whose axioms are modelled after weak Kleene logic rather than the strong one (e.g. a disjunction is true if and only if both disjuncts are true or false and at least one of them is true). The latter theory extended with the internal induction principle is still a model-theoretically conservative extension of $\\textrm{PA}.$\r\nWe will also discuss our current knowledge on the relationships between the models of theories of positive truth and the models of the theories based on the classical notion of truth.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nFischer, Martin,\r\nMinimal Truth and Interpretability,\r\nThe Review of Symbolic Logic,\r\nvol. 2 (2009), no. 4, pp.779\u2013815.\r\n<br>\r\n[2]\r\nHalbach, Volker,\r\nAxiomatic Theories of Truth,\r\nCambridge University Press,\r\n2011.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c187');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Mateusz \u0141e\u0142yk, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c189')\"><i>Principles equivalent to Delta0 Induction for the compositional truth predicate<\/i><\/a>\r\n<div id=\"absbox-c189\" class=\"abstract_box\">\r\nMateusz \u0141e\u0142yk, <i>Principles equivalent to $\\Delta_0-$induction for the compositional truth predicate<\/i>\r\n<br><br>\r\nOur talk concerns study of the proof-theoretic strength of compositional theories of truth. The theories we study are obtained via augmenting Peano arithmetic $\\textrm{PA}$ with a fresh unary predicate $T(x)$ with the intended reading \"$x$ is a (code of a) true (arithmetical) sentence\". Then we study various axiom systems trying to capture the most natural properties of the notion of arithmetical truth.\r\nOne of the basic conditions a satisfactory truth theory may be expected to meet is compositionality. Rather surprisingly, it turns out that if the compositionality is the only thing we demand from the truth predicate, then the resulting theory (usually denoted $\\textrm{CT}^-$) is conservative over $\\textrm{PA}$ (as shown by Enayat\u2013Visser (in [2]) and Leigh (in [4]), with a similar results being proved much earlier by Kotlarski, Krajewski and Lachlan). On the other hand a theory of compositional truth with full induction for formulae of the extended language (known as $\\textrm{CT}$) is obviously non-conservative over $\\textrm{PA},$ since we may show that all instances of the induction scheme are true (in the sense of our truth predicate) and then show by induction on the length of derivations in $\\textrm{PA}$ that every sentence provable in $\\textrm{PA}$ is true (reconstructing the usual proof of soundness of First Order Logic inside our theory). In our research we tried to understand which natural principles for the truth predicate make this notion non-conservative over the arithmetic.\r\nIt turned out that all the natural principles whose strength we were able to understand thus far are either conservative over $\\textrm{PA}$ or are equivalent over $\\textrm{CT}^-$ to the principle of global reflection over $\\textrm{PA}$. More precisely, we can show that over $\\textrm{CT}^-$ the following sets of axioms are equivalent:\r\n<ul>\r\n<li>\"Axioms of $\\textrm{PA}$ are true and sentences provable in First Order Logic from true premises are true as well.\" (principle of global reflection)\r\n<li>\"Sentences provable from true premises in First Order Logic are true.\"\r\n<li>\"Sentences provable from true premises in Propositional Logic are true.\"\r\n<li>\"Sentences valid in First Order Logic are true.\"\r\n<li>\"A disjunction is true if and only if one of its disjuncts is and axioms of $\\textrm{PA}$ are true.\"\r\n<li>$\\Delta_0$-induction scheme for sentences containing the truth predicate.\r\n<\/ul>\r\nThe fact that the theory in question admits the listed characterisations is even more striking taking into account that the principle \"Axioms of $\\textrm{PA}$ are true\" by itself, when added to the theory of compositional truth over $\\textrm{PA}$ is still conservative, as shown independently by Enayat\u2013Visser (unpublished) and Leigh (in [4]).\r\nSome important parts of the result presented above have been proved by Cie\u015bli\\'nski and Enayat (both in published papers (see [1]) and in unpublished notes). Our main contribution consists in showing that $\\textrm{CT}^-$ with $\\Delta_0$-induction for the truth predicate is enough for proving that the truth predicate commutes with blocks of quantifiers (of the same type) of arbitrary length. By proving this we fix a gap which was discovered in 2008 by Heck and Visser (independently) in an old proof by Kotlarski (see [3]). Since then the problem whether $\\textrm{CT}^-$ augmented with $\\Delta_0-$induction proves the principle of global reflection over $\\textrm{PA}$ remained (to our best knowledge) open.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nCezary Cie\u015bli\u0144ski,\r\nDeflationary Truth and Pathologies,\r\nJournal of Philosophical Logic,\r\nvol. 39 (2010), no. 3, pp. 325\u2013337.\r\n<br>\r\n[2]\r\nAli Enayat, Albert Visser,\r\nNew Constructions of Satisfaction Classes,\r\nUnifying the Philosophy of Truth\r\n(Theodora Achourioti, Henri Galinon, Jos\u00e9 Mart\u00ednez Fern\u00e1ndez, Kentaro Fujimoto, editors),\r\nSpringer,\r\nDordrecht,\r\n2015,\r\npp. 321\u2013337.\r\n<br>\r\n[3]\r\nHenryk Kotlarski,\r\nBounded Induction and Satisfaction Classes,\r\nMathematical Logic Quarterly,\r\nvol. 32 (1986), no. 31-34, pp. 531\u2013544.\r\n<br>\r\n[4]\r\nGraham Leigh,\r\nConservativity for theories of compositional truth via cut elimination,\r\nJournal of Symbolic Logic,\r\nvol. 80 (2015), no. 03, pp. 825\u2013865.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c189');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Modal Logic <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Susumu Yamasaki, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c191')\"><i>A semantics for multi-modal mu-calculus with interaction on Heyting algebra<\/i><\/a>\r\n<div id=\"absbox-c191\" class=\"abstract_box\">\r\nSusumu Yamasaki, <i>A semantics for multi-modal mu-calculus with interaction on Heyting algebra<\/i>\r\n<br><br>\r\nAs regards a human interaction to implementation in programming languages, abstraction of interactive states\r\nshould be formulated in computer science logic. Communications (for interaction) and ($\\lambda$-)terms motivate\r\nmulti-modality in logic of action ([2]).\r\nThe syntax of the logical formulas  is now given by BNF (Backus Naur Form) into some modification of modal mu-calculus:\r\n\\[\r\n\\begin{array}{l}\r\n\\varphi :: = \\mbox{tt} \\mid p \\mid \\neg \\varphi \\mid \\mathop\\sim \\varphi \\mid \\varphi \\vee \\varphi \\mid \\langle c \\rangle \\varphi\r\n\\mid \\mu x. \\varphi \\mid \\varphi\\rangle t \\rangle\r\n\\end{array}\r\n\\]\r\nwhere\r\na prefix modality $\\langle c \\rangle$ (for communications), a postfix one $\\rangle t \\rangle$\r\n(for terms), and a negation $\\mathop\\sim$ (denoting incapability of interaction) are taken, in addition to\r\ntruth $\\mbox{tt}$,  propositions $p$, the\r\nlogical negation $\\neg$ and a least fixed point operator $\\mu$.\r\nTo represent the meaning of  (a formula) $\\varphi$ at a state, receiving communication $c$ (requirement), and being followed by\r\nterm $t$ (effect), we here have the state sets  $[\\![\\varphi]\\!]_{pos}$, $[\\![\\varphi]\\!]_{inter}$ and $[\\![\\varphi]\\!]_{neg}$\r\nfor the formula (condition) to be positively, interactively and negatively modeled, respectively, in a transition system.\r\nThis is an extended semantics for logic of action, from the version of [3].\r\nThe state sets are applicable to a triplet of $\\langle c \\rangle \\varphi$, $\\varphi$ and $\\varphi \\langle t \\langle$, concerned with\r\ncommunication requirement and term effect. We then adopt a Heyting algebra  $H$ $=$ $(\\{ 0, 1\/2, 1 \\}, \\leq, \\bigvee, \\bigwedge, 0, 1)$, equipped with a binary operation $\\longrightarrow$ such that $c \\bigwedge a \\leq b$ iff $c \\leq a \\longrightarrow b$.\r\nWith $H$, we define a semantic function $Deno$: $\\Phi \\rightarrow S \\rightarrow \\{ 0, 1\/2, 1 \\}$, to see the positive, interactive,\r\nor negative states for the formula (condition) to be at, where $\\Phi$ and $S$ are the set of formulas and the set of states, respectively.\r\nA semiring structure ([1])  is related to, with multiplicative inverse, if the alternation of applying postfix modal operators\r\nis regarded as addition, and the composition is interpreted as multiplication. Kleene star can be included as\r\nin star semiring, with relevance to a state constraint system ([4]).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nM. Droste, W. Kuich and H. Vogler, editors.\r\nHandbook of Weighted Automata, Springer, 2009.\r\n<br>\r\n[2]\r\nM. Hennessy and R. Milner,\r\nAlgebraic laws for nondeterminism and concurrency,\r\nJournal of the ACM,\r\nvol.&nbsp;32 (1985), no.&nbsp;1, pp.137\u2013161.\r\n<br>\r\n[3]\r\nA. Kucera and J. Esparza,\r\nA logical viewpoint on process-algebraic quotients,\r\nJournal of Logic and Computation,\r\nvol.&nbsp;13 (2003), no.&nbsp;6, pp.863\u2013880.\r\n<br>\r\n[4]\r\nS. Yamasaki,\r\nState constraint system applicable to adjusting,\r\nCLMPS Book of Abstracts\r\n(Ilona Nevalainen, Mikko Virtanen and Paivi Seppala, editors),\r\nprinted at University of Helsinki, 2015, pp.&nbsp;409\u2013410.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c191');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Alexander Roberts, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c193')\"><i>Modal Expansionism<\/i><\/a>\r\n<div id=\"absbox-c193\" class=\"abstract_box\">\r\nAlexander Roberts, <i>Modal Expansionism<\/i>\r\n<br><br>\r\nIn arguing that a certain situation is metaphysically possible, philosophers frequently employ recombinatorial principles. Such principles articulate the idea that some possible individuals satisfying certain properties can be recombined into a single possibility at which they all satisfy those properties. Indeed, the theoretical plausibility of many recombinatorial principles is often taken for granted, since rejecting them would equate to imposing unwelcome arbitrary limits on the extent of modal space. However, an apparent paradox due to Kit Fine (2002, pp.223-224), later reformulated by Peter Fritz (manuscript), is seen to allegedly demonstrate that two plausible recombinatorial principles are inconsistent with one another and cannot therefore both feature in modal theory. The more widely known `Kaplan's paradox' due to David Kaplan (1995) also might be taken to undermine another similar recombinatorial principle. Nevertheless, the paper offers a solution to all aforementioned paradoxes in the form of a novel conception of metaphysical modality, according to which for any metaphysical modality there will always be some further, more-inclusive metaphysical modality; or, in other words, the expression `metaphysically necessary' and its interdefinable notions are indefinitely extensible\u2013call this view <i>modal expansionism<\/i>. In accordance with recent modal treatments of the indefinite extensibility of set-theoretic expressions (see Fine (2006); Linnebo (2010); Studd (2013), and Uzquiano (2015)), where the modality used to explicate indefinite extensibility is interpretational, the paper thus provides a semantics for a quantified bimodal logic containing metaphysical and interpretational modal operators. This semantics is used to more precisely articulate the modal expansionist's diagnoses of the apparent paradoxes.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nK. Fine,\r\nProblem of Possibilia,\r\nModality and Tense: Philosophical papers\r\n(K. Fine),\r\nOxford Clarendon Press,\r\n2005.\r\n<br>\r\n[2]\r\nK. Fine,\r\nRelatively Unrestricted Quantification,\r\nAbsolute Generality\r\n(A. Rayo, G. Uzquiano, editors),\r\nClarendon Press,\r\nOxford,\r\n2006.\r\n<br>\r\n[3]\r\nP. Fritz,\r\nA Purely Recombinatorial Puzzle,\r\nUnpublished manuscript. Hosted at <tt>https:\/\/dl.dropboxusercontent.com\/u\/15106063\/drafts\/A%20Purely%20Recombinatorial%20Puzzle.pdf<\/tt>\r\n<br>\r\n[4]\r\nD. Kaplan,\r\nA problem in Possible-World Semantics,\r\nModality, Morality, and Belief: Essays in Honour of Ruth Barcan Marcus.\r\n(W. Sinnot-Armstrong, editor),\r\nCambridge University Press,\r\nCambridge,\r\n1995.\r\n<br>\r\n[5]\r\n\u00d8. Linnebo,\r\nPluralities and Sets,\r\nJournal of Philosophy,\r\nvol.&nbsp;107 (2010), pp.&nbsp;144\u2013164.\r\n<br>\r\n[6]\r\nJ. Studd,\r\nThe Iterative Conception of Set: A (Bi-)Modal Axiomatisation,\r\nThe Journal of Philosophical Logic,\r\nvol.&nbsp;42 (2013), pp.&nbsp;697\u2013725.\r\n<br>\r\n[7]\r\nG. Uzquiano,\r\nVarieties of Indefinite Extensibility,\r\nNotre Dame Journal of Formal Logic,\r\nvol.&nbsp;56 (2015), pp.&nbsp;147\u2013166.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c193');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">William Stafford, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c195')\"><i>On the Modal Analogue of Frege's Theorem<\/i><\/a>\r\n<div id=\"absbox-c195\" class=\"abstract_box\">\r\nWilliam Stafford, <i>On the modal analogue of Frege\u2019s theorem<\/i>\r\n<br><br>\r\nFrege\u2019s Theorem says that a second-order system (Hume\u2019s Principle), equipped with a type-lowering \u201cnumber of\u201d operator, interprets full second-order Peano arithmetic (cf. [1] Chapter 1, [4] Chapter 4). This implies, among other things, that all models of this system must have infinite domains. Some have thought that this is inconsistent with the logicism which Frege, Wright, and Hale hoped to secure by means of this theorem. We examine the obstacles to, and resources required, for establishing a version of Frege\u2019s Theorem in a modal setting, where it is consistent that each world contains only finitely many objects. The particular resources we consider are higher-order logic and resources for cross-world predication (cf. [3]). Further, we consider the relation of all this to (i) Hodes\u2019 suggestion that \u201cmathematics is higher-order modal logic\u201d ([2]) and (ii) recent work by Linnebo [4] and Studd [5], which similarly embeds set theories and abstraction principles within the framework of higher-order modal logic.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nJohn P. Burgess,\r\nFixing Frege,\r\nPrinceton Monographs in Philosophy,\r\nPrinceton University Press, Princeton,\r\n2005.\r\n<br>\r\n[2]\r\nHarold Hodes,\r\nLogicism and the Ontological Commitments of Arithmetic,\r\nThe Journal of Philosophy,\r\nvol.&nbsp;81 (1984), no.&nbsp;3, pp.&nbsp;123\u2013149.\r\n<br>\r\n[3]\r\nAlex Kocurek,\r\nThe problem of cross-world predication,\r\nJournal of Philosophical Logic,\r\n2016.\r\n<br>\r\n[4]\r\n\\O{}ystein Linnebo,\r\nThe Potential Hierarchy of Sets,\r\nThe Review of Symbolic Logic,\r\nvol.&nbsp;6 (2013), no.&nbsp;2, pp.&nbsp;205\u2013228.\r\n<br>\r\n[5]\r\nJames P Studd,\r\nAbstraction reconceived,\r\nThe British Journal for the Philosophy of Science,\r\n2015.\r\n<br>\r\n[6]\r\nCrispin Wright,\r\nFrege\u2019s Conception of Numbers as Objects,\r\nScots Philosophical Monographs,\r\nAberdeen University Press, Aberdeen,\r\n1983.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c195');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">S\u0142awomir Kost, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c197')\"><i>Connected frames for fusions of multimodal logics<\/i><\/a>\r\n<div id=\"absbox-c197\" class=\"abstract_box\">\r\nS\u0142awomir Kost, <i>Connected frames for fusions of multimodal logics<\/i>\r\n\r\n<br><br>\r\nMost of monomodal logics are characterized by classes of frames. It is even possible to use single connected frames for some logics. The additional modalities make the problem of seeking one connected frame more demanding.\r\nLet us consider multimodal logics $L_1$ and $L_2$. We assume that $L_1$ is characterized by a class $\\mathcal{C}_1$ of connected $n$-frames and $L_2$ is characterized by a class $\\mathcal{C}_2$ of connected $m$-frames. Classes $\\mathcal{C}_1'$ and $\\mathcal{C}_2'$ are closures of $\\mathcal{C}_1$ and $\\mathcal{C}_2$, respectively, under the formation of disjoint unions and isomorphic copies. It is already known that fusion $L_1\\oplus L_2$ is characterized by the class $\\mathcal{C}_1'\\oplus \\mathcal{C}_2'$ (see e.g. [2]).\r\nLet $\\mathcal{C}=\\{\\mathfrak{F}_i;i\\in I\\}$ be the family of connected frames and $\\mathfrak{F}$ be a connected frame. A point $x_0$ from $\\mathfrak{F}$ is a $\\mathcal{C}$-<i>starting point<\/i> if every mapping $f:\\{x_0\\}\\to \\mathfrak{F}_i$ can be extend to a $p$-morphism $f:\\mathfrak{F}\\to \\mathfrak{F}_i$, for $i\\in I$.\r\nWe assume that there exists a $L_1$-frame $\\mathfrak{F}^1$ with $\\mathcal{C}_1$-starting point and there exists a $L_2$-frame $\\mathfrak{F}^2$ with $\\mathcal{C}_2$-starting point. Using isomorphic copies of the frames $\\mathfrak{F}^1$ and $\\mathfrak{F}^2$ we construct a connected frame $\\mathfrak{F}^s$ characterizing the fusion $L_1\\oplus L_2$. The obtained frame has some useful properties. Among others, $\\mathfrak{F}^s$ is countable if both $\\mathfrak{F}^1$ and $\\mathfrak{F}^2$ are countable. Moreover, the frame $\\mathfrak{F}^s$ has a $CS(\\mathcal{C}_1'\\oplus \\mathcal{C}_2')$-starting point, where $CS(\\mathcal{C}_1'\\oplus \\mathcal{C}_2')$ is a subclass of $\\mathcal{C}_1'\\oplus \\mathcal{C}_2'$ such that $$L_1\\oplus L_2=\\mbox{Log}\\{ \\mathcal{C}_1'\\oplus \\mathcal{C}_2' \\}=\\mbox{Log}\\{CS(\\mathcal{C}_1'\\oplus \\mathcal{C}_2')\\}.$$\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nPatrick Blackburn, Maarten De Rijke, Yde Venema,\r\nModal Logic,\r\nCambridge Tracts in Theoretical Computer Science (No.53),\r\nCambridge University Press,\r\n2001.\r\n<br>\r\n[2]\r\nEdited by D.M. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev.\r\nMany-Dimensional Modal Logics: Theory and Applications,\r\nElsevier,\r\n2003.\r\n<br>\r\n[3]\r\nKit Fine, Gerhard Schurz,\r\nTransfer Theorems for Multimodal Logics,\r\nLogic and Reality,\r\nCambridge University Press,\r\n1996,\r\npp.&nbsp;169\u2013213.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c197');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">Zofia Kostrzycka, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c199')\"><i>On interpolation in Brouwer modal logics<\/i><\/a>\r\n<div id=\"absbox-c199\" class=\"abstract_box\">\r\nZofia Kostrzycka, <i>On interpolation in Brouwer modal logics<\/i>\r\n<br><br>\r\nWe study the Brouwer modal logic $\\mathbf{KTB}$ and its normal extensions, which are determined by a class of reflexive and symmetric Kripke frames with a given degree of branching.  Hence we consider logics  ${\\bf KTBAlt(n)}:=\\mathbf{KTB}\\oplus alt_n$, where\r\n\\begin{eqnarray*}\r\n&&alt_n:=\\Box p_1 \\vee \\Box( p_1 \\to  p_2)\\vee ...\\vee \\Box(( p_1 \\wedge ... \\wedge  p_n) \\to  p_{n+1}),\\;\\;\\;\\;n\\ge 3.\r\n\\end{eqnarray*}\r\nFor $n=3$ we get a linear Brouwer logic ${\\bf KTBAlt(3)}$. It is proved in\r\n\r\n\r\n[1],[2] that all logics from $NEXT(\\mathbf{KTBAlt(3)})$ have finite model property and\r\nare finitely axiomatizable. It is easily seen by the above theorem that the cardinality of the\r\nclass  $NEXT({\\bf KTBAlt(3)})$ is only countably infinite.\r\nOn the other side it is known that the cardinality of the\r\nclass $NEXT({\\bf KTBAlt(4)})$ is uncountably infinite [3].\r\nWe shall look for locally finite logics from $NEXT(\\mathbf{KTB.Alt(n)})$ that have Craig interpolation property or at least interpolation property for deducibility. To do this, we shall characterize Halld\u00e9n complete logics since there is an important connection between the Craig interpolation\r\nproperty and Halld\u00e9n completeness of modal logics, see  G. F. Schumm [6].\r\nAmong Halld\u00e9n complete and locally finite logics from $NEXT(\\mathbf{KTB.Alt(n)})$ we shall characterize the logics which have (CIP) or (IPD).\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]  Byrd, M., Ullrich, D. The extensions of $BAlt_3$,\r\nJournal of Philosophical Logic, vol.&nbsp;6 (1977), pp.&nbsp;109\u2013117.\r\n<br>\r\n[2]  Byrd, M., The extensions of $BAlt_3$ - revisited,\r\nJournal of Philosophical Logic, vol.&nbsp;7 (1978), pp.&nbsp;407\u2013413.\r\n<br>\r\n[3]  Kostrzycka, Z., {Miyazaki, Y.}, Normal modal logics determined by aligned clusters, submitted.\r\n<br>\r\n[4]   Kostrzycka, Z., On Halld\u00e9n completeness of modal logics determined by homogeneous Kripke frames,  Bulletin of the Section of Logic, vol.&nbsp;44 (2015),  no.&nbsp;3\/4, pp.&nbsp;1\u201320.\r\n<br>\r\n[5]  Maksimowa, L., Amalgamation and Interpolation in Normal Modal Logics, Studia Logica, vol.&nbsp;50 (1991), no.&nbsp;3\/4,  pp.&nbsp;457\u2013471.\r\n<br>\r\n[6]  Schumm, G. F., Some failures of interpolatin in modal logic, Notre Dame Journal of Formal Logic, vol.&nbsp;27 (1986), no.&nbsp;(1), pp.&nbsp;108\u2013110.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c199');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Model Theory: Fields and Algebra <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Martin Bays and Jonathan Kirby, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c201')\"><i>Exponential-algebraic closedness and quasiminimality<\/i><\/a>\r\n<div id=\"absbox-c201\" class=\"abstract_box\">\r\nMartin Bays and Jonathan Kirby, <i>Exponential-algebraic closedness and quasiminimality<\/i>\r\n<br><br>\r\nIt is well-known that the complex field $\\mathbb C$, considered as a structure in the ring language, is strongly minimal: every definable subset of $\\mathbb C$ itself is finite or co-finite. Zilber conjectured that the complex exponential field $\\mathbb{C}_\\mathrm{exp}$ is quasiminimal, that is, every subset of $\\mathbb C$ definable in this structure is countable or co-countable.\r\nHe later showed that if Schanuel's conjecture of transcendental number theory is true and $\\mathbb{C}_\\mathrm{exp}$ is <i>strongly exponentially-algebraically closed<\/i> then his conjecture holds [1]. Schanuel's conjecture is considered out of reach, and proving strong-exponential algebraic closedness involves finding solutions of certain systems of equations and then showing they are generic, the latter step usually done using Schanuel's conjecture.\r\nWe show that if $\\mathbb{C}_\\mathrm{exp}$ is <i>exponentially-algebraically closed<\/i> then it is quasiminimal. Thus Schanuel's conjecture can be dropped as an assumption, and strong exponential-algebraic closedness can be weakened to exponential-algebraic closedness which requires certain systems of equations to have solutions, but says nothing about their genericity.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nBoris Zilber,\r\nPseudo-exponentiation on algebraically closed fields of\r\ncharacteristic zero,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;132 (2005), no.&nbsp;1, pp.&nbsp;67\u201395.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c201');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Dimitra Hobitaki and Thanases Pheidas, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c203')\"><i>Undecidability of the existential theory of the ring of exponential sums<\/i><\/a>\r\n<div id=\"absbox-c203\" class=\"abstract_box\">\r\nDimitra Hobitaki and Thanases Pheidas, <i>Undecidability of the existential theory of the ring of exponential sums<\/i>\r\n<br><br>\r\nDefine the set of <i>exponential sums<\/i>, EXP($\\mathbb{C}$), to  be the the set of expressions\r\n\\begin{equation}\r\na=\\alpha _0+\\alpha _1e^{\\mu_1z}+\\dots +\\alpha _Ne^{\\mu_Nz}\r\n\\end{equation}\r\nwhere $\\alpha_i, \\mu _j \\in \\mathbb{C}$.  We ask whether the positive existential first order theory of EXP($\\mathbb{C}$), as a  structure of the language\r\n$$\r\n{\\bf L}=\\{ +,\\mathbb{C}dot, 0,1,e^z\\}\r\n$$ is decidable or undecidable. In a recent unpublished paper P. D Aquino, Th. Pheidas and G. Terzo have proven a negative answer (actually they prove a stronger result), even for the positive theory, pending on a number theoretic hypothesis that is still being checked, We provide a new proof, based on theirs, but using different tools (`Pell Equations' instead of Elliptic Curves). Our approach  has been  suggested by A. Macintyre. Our result may be considered as an analogue of Hilbert's Tenth Problem for this structure and as a  step to answering the similar problem for the ring of `exponential polynomials', which is still open.\r\nWe prove:\r\n{\\bf Theorem 1:}\r\nThe solutions of the equation\r\n\\begin{equation}\\label{MD}\r\n(e^{2z}-1)y^2=x^2-1\r\n\\end{equation}\r\nwhere the unknowns $x$ and $y$ range over  EXP($\\mathbb{C}$)\r\nare given by\r\n\\begin{equation}\r\n(x,y)=m\\mathbb{C}dot (\\pm e^z,1)\\oplus n\\mathbb{C}dot (\\pm e^{-z},\\i e^{-z})\r\n\\end{equation}\r\nwhere, for any solutions $(a_1,b_1)$, $(a_2,b_2)$ of (2) the law $\\oplus$ is defined by\r\n$(a_1,b_1)\\oplus (a_2,b_2)=(a_1a_2+(e^{2z}-1)b_1b_2, a_1b_2+a_2b_1)$.\r\nThe proof uses techniques of [4], [1] and [3]. From this, by adapting techniques of  [2] we are able to prove\r\n{\\bf Theorem 2:} The ring of rational integers $\\mathbb{Z}$ is positive existentially definable over EXP($\\mathbb{C}$), as an ${\\bf L}$-structure. Hence the positive existential theory of this structure is undecidable.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nT. Pheidas, P. D'Aquino, G. Terzo,\r\nUndecidability of the diophantine theory of exponential sums,\r\nmanuscript\r\n<br>\r\n[2]\r\nJ. Denef,\r\nThe diophantine problem for polynomial\r\nrings and fields of rational functions,\r\nTransactions of the\r\nAmerican Mathematical Society,\r\nvol.&nbsp;242 (1978), pp.&nbsp;391\u2013399.\r\n<br>\r\n[3]\r\nTh. Pheidas and K. Zahidi ,\r\nUndecidable existential theories of polynomial rings and function fields,\r\nCommunications in Algebra,\r\nvol.&nbsp;27 (1999), no.&nbsp;10, pp.&nbsp;4993\u20135010.\r\n<br>\r\n[4]\r\nL. van den Dries,\r\nExponential rings, exponential\r\npolynomials and exponential functions,\r\nPacific Journal of\r\nMathematics,\r\nvol.&nbsp;113 (1984), no.&nbsp;1, pp.&nbsp;51\u201366.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c203');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Erick Garcia Ramirez, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c205')\"><i>Tangent cones and stratifications in real closed valued fields<\/i><\/a>\r\n<div id=\"absbox-c205\" class=\"abstract_box\">\r\nErick Garcia Ramirez, <i>Tangent cones and stratifications in real closed valued fields<\/i>\r\n<br><br>\r\nWe show that for a definable set $X$ in a real closed valued field, the stratifications defined in&nbsp;[1] induce stratifications of the same nature on the tangent cones of $X$. This shows in particular that those stratifications are stronger than classical Whitney stratifications (introduced in&nbsp;[2]), as the latter do not induce (Whitney) stratifications on tangent cones.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nI. Halupczok,\r\nNon-archimedean Whitney stratifications,\r\nProceedings of the London Mathematical Society,\r\n(2013), no. 109, pp. 1304\u20131362.\r\n<br>\r\n[2]\r\nH. Whitney\r\nTangents to an analytic variety,\r\nAnnals of Mathematics,  Second Series,\r\nvol. 81 (1965), no. 3, pp.&nbsp;496\u2013549.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c205');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Aibat Yeshkeyev and Olga Ulbrikht, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c207')\"><i>Cosemanticness and JSB-property for Abelian groups<\/i><\/a>\r\n<div id=\"absbox-c207\" class=\"abstract_box\">\r\nAibat Yeshkeyev and Olga Ulbrikht, <i>Cosemanticness and JSB-property for Abelian Groups<\/i>\r\n<br><br>\r\nWe will give some model-theoretic results of Abelian groups in the frame of Jonsson theories. In the [1] John Goodrick gave necessary and sufficient conditions of the Schroder-Bernstein (SB) property for complete theories of Abelian groups. We consider Jonsson analogue of Theorem 1 from [1].\r\n<br>\r\nJonsson theory $T$ has the Schroder-Bernstein (JSB) property  if for any two models $A, B\\in E_T$ from the fact that they are mutually isomorphically embeddable each other follows that they are isomorphic.\r\n<br>\r\nWe know that theory of Abelian groups will be a Jonsson theory and also perfect.\r\n<br>\r\nIn connection with this concept, we got the result that the following theorem is true:\r\n<br>\r\n<br>\r\n<b>Theorem<\/b>.\r\nLet $T$ be a Jonsson theory of Abelian groups, then the following conditions are equivalent:\r\n<br>\r\n(1) $T$ is $J-\\omega-$stable;\r\n<br>\r\n(2) $T^*$ is $\\omega-$stable;\r\n<br>\r\n(3) $T$ has JSB property.\r\n<br><br>\r\nLet $A\\in Mod$ $\\sigma_{AG}$, where $\\sigma_{AG}=\\langle +,-, 0\\rangle$, i.e. our considered theories are universal. Denote through $JSp(A)$ Jonsson spectrum of Abelian group $A$, where $JSp(A)=\\{T | T$ is a Jonsson theory in language $\\sigma_{AG}$ and $A\\in Mod T\\}$.\r\n<br>\r\nWe say that $T_1$ is cosemantic to $T_2$ ($T_1\\bowtie T_2$) if $C_{T_1}=C_{T_2}$, where $C_{T_i}$ is semantic model of $T_i$, $i=1, 2$. Then it is easy to notice that $JSp(A)\/_{\\bowtie}$ is a factor set by relation $\\bowtie$ and let its power equals $\\mu$, i.e. $|JSp(A)\/_{\\bowtie}|=\\mu$.\r\n<br>\r\n<br>\r\n<b>Theorem<\/b>.\r\nLet $T$ is Jonsson theory of  Abelian groups then $C_T\\in E_T$ and $C_T$ is divisible group and its a Shmelev's standart group is $\\underset{\\kappa}\\oplus\\mathbb{Z}_{p^\\infty}\\underset{\\kappa}\\oplus\\mathbb{Q}$, where $\\kappa=|C|$.\r\n<br>\r\n<br>\r\nLet's call a pair $(\\alpha,\\beta)^A_C$ as Jonsson invariant of Abelian group $A$ if a Shmelev's standart group of a group $A$ is a group of the following form $\\underset{\\alpha}\\oplus\\mathbb{Z}_{p^\\infty}\\underset{\\beta}\\oplus\\mathbb{Q},$ where $C$ is semantic model of $[T]\\in JSp(A)\/_{\\bowtie}$.\r\n<br>\r\nThe following result is a Jonsson analogue of the well-known  Shmelev's theorem about the elementary classification of Abelian groups.\r\n<br>\r\n<br>\r\n<b>Theorem<\/b>.\r\nLet $A,B\\in Mod$ $\\sigma_{AG}$ then $A \\bowtie B \\Leftrightarrow (\\alpha,\\beta)^A_{C_i}=(\\alpha,\\beta)^B_{C_i}$, $i\\in I$, $|I|=\\mu$.\r\n<br>\r\n<br>\r\nAll additional information regarding Jonsson theories can be found in [2].\r\n<br><br><b>References<\/b>\r\n<br>[1]\r\n Goodrick J.,\r\n The Schroder-Bernstein property for theories of abelian groups,\r\narXiv.org $>$ math $>$ arXiv:0705.1850v1, 2007.\r\n<br>[2]\r\n Yeshkeyev A.R.,\r\n Jonsson Theories,\r\nPublisher of the Karaganda state university,\r\n2009.\r\n\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c207');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Computability Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Dino Rossegger and Ekaterina Fokina, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c209')\"><i>Enumerable functors<\/i><\/a>\r\n<div id=\"absbox-c209\" class=\"abstract_box\">\r\nDino Rossegger and Ekaterina Fokina, <i>Enumerable functors<\/i>\r\n<br><br>\r\nWe propose a new notion of reducibility between structures, <i>enumerable functors<\/i>, inspired by the recently investigated notion of computable functors&nbsp;[1],&nbsp;[2]. An enumerable functor from a structure $\\mathcal{A}$ to a structure $\\mathcal{B}$  is a pair $(\\Psi, \\Phi)$ where $\\Psi$ is an enumeration operator transforming every presentation of $\\mathcal{A}$ to a presentation of $\\mathcal{B}$ and $\\Phi$ is a Turing functional transforming every isomorphism between two presentations of $\\mathcal{A}$ to an isomorphism of their image. Our main results are that enumerable functors preserve $\\Sigma_n$\u2013spectra&nbsp;[3] and that they are equivalent to a restricted version of effective interpretability. We also extend this equivalence to effective bi-interpretability and reducibility between classes of structures by effective bi-interpretability.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nMatthew Harrison-Trainor, Alexander Melnikov, Russel Miller, and Antonio Montalb\u00e1n,\r\nComputable functors and effective interpretability,\r\nsubmitted for publication\r\n<br>\r\n[2]\r\nRussel Miller, Bjorn Poonen, Hans Schoutens, and Alexandra Shlapentokh,\r\nA computable functor from graphs to fields,\r\nsubmitted for publication\r\n<br>\r\n[3]\r\nEkaterina Fokina, Pavel Semukhin, and Daniel Turetsky,\r\nDegree spectra of sructures under equivalence relations,\r\nin preparation\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c209');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Sergey Ospichev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c211')\"><i>Minimal numberings in partial computable functionals<\/i><\/a>\r\n<div id=\"absbox-c211\" class=\"abstract_box\">\r\nSergey Ospichev, <i>Minimal numberings of partial computable functionals<\/i>\r\n<br><br>\r\nOne of the main questions in numbering theory is studying extremal elements of Rogers semilattices of different families. Here we concentrate our interest on partial computable functionals of finite types.\r\nLet's define \\textsl{functional type}. Let $T$ will be the set of\r\nall types.\r\n1. $0\\in{T}$;\r\n2. if $\\sigma$,$\\tau$ are types, then $(\\sigma\\times\\tau)$ and\r\n$(\\sigma|\\tau)$ are also types;\r\n3. $T$ - minimal set, satisfying  1 and 2.\r\nNow we define \\textsl{partial computable functionals}. Let\r\n$C_\\sigma$ be family of all partial computable functionals of type\r\n$\\sigma$. Let $C_0$ be the family of\r\nall partial computable functions or the family of all computable enumerable sets. If $C_\\sigma$ and $C_\\tau$ are\r\nalready defined, then $C_{(\\sigma\\times\\tau)}\\rightleftharpoons\r\nC_\\sigma\\times C_\\tau$ and\r\n$C_{(\\sigma|\\tau)}\\rightleftharpoons\\mathfrak{Mor}(C_\\sigma,C_\\tau)$.\r\nAny $C_\\sigma$ has natural universal numbering $\\nu_\\sigma$ by definition. So we call numbering $\\mu$ $\\sigma$-computable if $\\mu$ is reducible to $\\nu_\\sigma$.\r\nIn work are proven\r\n<br><br>\r\n<b>Theorem.<\/b> For any $\\sigma\\in{T}$ there are infinitely many nonequivalent $\\sigma$-computable friedberg\r\nnumberings of family $C_\\sigma$.\r\n<br><br>\r\n<b>Theorem.<\/b> For any $\\sigma\\in{T}$ there are infinitely many nonequivalent $\\sigma$-computable positive\r\nundecidable numberings of family $C_\\sigma$.\r\n<br><br>\r\n<b>Theorem.<\/b> For any $\\sigma\\in{T}$ there is infinite $\\sigma$-computable family $\\mathcal{S}\\subset C_\\sigma$ without $\\sigma$-computable friedberg\r\nnumberings.\r\n<br><br>\r\nThe reported study was partially supported by RFBR, research project No. 14-01-00376 and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1).\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c211');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Arno Pauly, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c213')\"><i>Computability on the space of countable ordinals<\/i><\/a>\r\n<div id=\"absbox-c213\" class=\"abstract_box\">\r\nArno Pauly, <i>Computability on the space of countable ordinals<\/i>\r\n<br><br>\r\nWhile there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far (but cf.&nbsp;[1]). In order to remedy this, we explore various potential representations (in the sense of computable analysis [6, 3]) of the set of countable ordinals. An equivalence class of representations is then suggested as a standard, as it offers the desired closure properties. This class is characterized exactly by the computability of four specific operations.\r\nWe show that the supremum of a continuous function from Baire space into the countable ordinals can be computed from a name of such function. The binary infimum is also a computable operation, and with some caveat regarding finite values, even countable infima are computable.\r\nWith a decent notion of computability on the space of countable ordinals in place, we can then state and prove a computable uniform version of the Hausdorff-Kuratowski theorem. We can also show yet another proof of the computable Lusin separation theorem (cf.&nbsp;[2]).\r\nA preprint is available on the arXiv [5], and a preliminary version appeared as [4].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nZhenhao Li and Joel D. Hamkins,\r\nOn effectiveness of operations on countable ordinals,\r\nunpublished notes.\r\n<br>\r\n[2]\r\nAuthor's Name,\r\nClassical descriptive set theory as a refinement of effective descriptive set theory,\r\nAnnals of Pure and Applied Logic,\r\nvol.&nbsp;162 (2010), pp.&nbsp;243\u2013255.\r\n<br>\r\n[3]\r\nArno Pauly,\r\nOn the topological aspects of the theory of represented spaces,\r\nComputability,\r\nto appear (2016), available at http:\/\/arxiv.org\/abs\/1204.3763.\r\n<br>\r\n[4]\r\n\u2015\r\nComputability on the Countable Ordinals and the Hausdorff-Kuratowski Theorem (Extended Abstract),\r\nMathematical Foundations of Computer Science 2015\r\n(G.&nbsp;Italiano and G.&nbsp;Pighizzini and D.&nbsp;Sannella)\r\nLecture Notes in Computer Science, vol.&nbsp;9234,\r\nSpringer,\r\n2015.\r\n<br>\r\n[5]\r\n\u2015,\r\nComputability on the countable ordinals and the {H}ausdorff-{K}uratowski theorem,\r\narXiv 1501.00386, 2015.\r\n<br>\r\n[6]\r\nKlaus Weihrauch,\r\nComputable Analysis,\r\nSpringer,\r\n2000.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c213');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Siddharth Bhaskar, <i>Revisiting Tiuryn's separation: an example of recursion-theoretic tameness<\/i><\/td><\/tr>\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Categorical Logic and Type Theory <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Christian Espindola, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c216')\"><i>Completeness of Infinitary Intuitionistic Logics<\/i><\/a>\r\n<div id=\"absbox-c216\" class=\"abstract_box\">\r\nChristian Espindola, <i>Completeness of Infinitary Intuitionistic Logics<\/i>\r\n<br><br>\r\nCompleteness theorems for infinitary classical logics $\\mathcal{L}_{\\kappa, \\kappa}$ (for, say, an inaccessible $\\kappa$) have been known for decades. When removing excluded middle, however, the situation is more difficult to analyze even in the propositional case, as the main difficulty in studying infinitary intuitionistic logics is the huge variety of non-equivalent formulas that one can obtain. Completeness results for the propositional fragment $\\mathcal{L}_{\\omega_1, 0}$ have been obtained, but the general case has not been addressed. The purpose of this talk is to outline set-theoretical and category-theoretical techniques that allow the study of completeness theorems for infinitary intuitionistic logics in the general case, both for propositional and first-order logics, in terms of an infinitary Kripke semantics. We will also analyze to what extent the use of large cardinal axioms (more precisely, the condition that $\\kappa$ be weakly compact) is necessary, and some applications of the completeness results will be presented.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1] P. Johnstone,Sketches of an Elephant - A Topos Theory Compendium - Vol I and II, Oxford University Press, 2002.\r\n<br>\r\n[2] A. Kanamori,The higher infinite, Springer Verlag, 1994.\r\n<br>\r\n[3] C. Karp,Languages with expressions of infinite length, North-Holland Publishing Co, 1964.\r\n<br>\r\n[4] S. Maclane, I. Moerdijk,Sheaves in geometry and logic, Springer Verlag New York, 1994.\r\n<br>\r\n[5] M. Makkai,A theorem on Barr-exact categories, with an infinite generalization,Annals of Pure and Applied Logic,vol.&nbsp;47 (1990), pp.&nbsp;225\u2013268.\r\n<br>\r\n[6] M. Nadel,Infinitary intuitionistic logic from a classical point of view,Annals of Mathematical Logic, vol.&nbsp;14 (1978), no.&nbsp;2, pp.&nbsp;159\u2013191.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c216');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Henrik Forssell, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c218')\"><i>Constructive reflection principles for regular theories<\/i><\/a>\r\n<div id=\"absbox-c218\" class=\"abstract_box\">\r\nHenrik Forssell, <i>Constructive reflection principles for regular theories<\/i>\r\n<br><br>\r\nClassically, any structure for a signature $\\Sigma$ may be completed to a model of a desired regular theory $\\mathbb{T}$ by means of the <i>chase construction<\/i> or <i>small object argument<\/i>.\r\nMoreover, this exhibits $\\operatorname{Mod}({\\mathbb{T}})$ as weakly reflective in $\\operatorname{Str}({\\Sigma})$.\r\nWe investigate this in the constructive setting.\r\nThe basic construction is unproblematic; however, it is no longer a weak reflection.\r\nIndeed, we show that various reflection principles for models of regular theories are equivalent to choice principles in the ambient set theory.\r\nHowever, the embedding of a structure into its chase-completion still satisfies a <i>conservativity<\/i> property, which suffices for applications such as the completeness of regular logic with respect to Tarski (i.e.&nbsp;set) models.\r\nWe also consider the extent to which this analysis can be carried over to stronger fragments of first-order logic, notably involving disjunctions.\r\nUnlike most constructive developments of predicate logic, we do not assume that equality between symbols in the signature is decidable.\r\nThis is joint work with Peter LeFanu Lumsdaine.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c218');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Henning Urbat, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c220')\"><i>Eilenberg-Reiterman Theory for a Monad<\/i><\/a>\r\n<div id=\"absbox-c220\" class=\"abstract_box\">\r\nHenning Urbat, <i>Eilenberg-Reiterman Theory for a Monad<\/i>\r\n<br><br>\r\nAlgebraic language theory investigates machine behaviours by relating them to\r\nalgebraic structures. Its core\r\nresult is Eilenberg's variety theorem [2]: <i>varieties of languages<\/i> (classes of regular languages closed\r\nunder boolean operations, derivatives, and preimages of monoid morphisms) correspond  to <i>pseudovarieties of monoids<\/i> (classes of finite monoids\r\nclosed under quotients, submonoids, and finite products). This together with Reiterman's theorem [3], stating that pseudovarieties can be specified by <i>profinite equations<\/i>, establishes a firm connection between automata, regular languages, and algebra.\r\nIn the meantime, numerous further Eilenberg-type correspondences have been discovered, either dealing with classes of regular languages with weaker closure properties, or considering other types of languages such as languages of infinite words,\r\ntree\r\nlanguages, or\r\ncost functions. This plethora of similar results has spurred interest in\r\ncategorical approaches to algebraic language theory, putting a common roof over the various developments. In this talk, we present a general Eilenberg-Reiterman theorem  [1, 4] that achieves the desired unification. The idea is to model languages, and the algebras recognizing them, by monads on some algebraic category. Then Stone-type dualities are used to relate pseudovarieties of finite algebras to profinite equational theories and classes of recognizable languages. Our framework covers the bulk of Eilenberg-Reiterman theorems in the literature, and produces new correspondences for free.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nL.-T. Chen, J. Ad\u00e1mek, S. Milius, and H. Urbat.\r\nProfinite monads, profinite equations and Reiterman\u2019s theorem.\r\nProc. FoSSaCS\u201916.\r\nLNCS 9634, Springer, 2016.\r\n<br>\r\n[2]\r\nS.&nbsp;Eilenberg.\r\nAutomata, Languages, and Machines, Vol.&nbsp;B,\r\nAcademic Press, New York, 1976.\r\n<br>\r\n[3]\r\nJ.&nbsp;Reiterman.\r\nThe Birkhoff theorem for finite algebras.\r\nAlgebra Universalis, 14(1):1\u201310, 1982.\r\n<br>\r\n[4]\r\nH. Urbat, J. Ad\u00e1mek, L.-T. Chen, and S. Milius.\r\nOne\r\nEilenberg theorem to rule them all.\r\nPreprint: <tt>http:\/\/arxiv.org\/abs\/1602.05831<\/tt>, 2016.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c220');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Philosophical Logic <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Richard Lawrence, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c222')\"><i>Concepts and objects in game-theoretical semantics<\/i><\/a>\r\n<div id=\"absbox-c222\" class=\"abstract_box\">\r\nRichard Lawrence, <i>Concepts and objects in game-theoretical semantics<\/i>\r\n<br><br>\r\nFrege's distinction between objects and concepts represents an\r\nimportant break he makes from the earlier logical tradition, and it\r\nhas had an outsized influence ever since: it is the basis of the\r\ndistinction between first- and second-order logic, as well as the type\r\nsystem used in studying natural language semantics.  But the\r\ndistinction is also the source of some of Frege's most puzzling\r\nremarks.  I will re-examine the distinction, offering a new\r\ninterpretation of it based on game-theoretical semantics.\r\nA standard reading construes Frege's distinction as a distinction\r\nbetween two ontological categories: objects are `saturated' entities,\r\nwhile concepts are `unsaturated' entities.  This reading faces serious\r\ninterpretive challenges, however.  A better interpretation conceives\r\nobjects and concepts as epistemological roles: something belongs to\r\none category or the other depending on how it is apprehended in\r\nscientific thought.  Unfortunately, Frege's metaphors are little help\r\nin deciding when something plays one role or the other.\r\nClearer criteria can be derived from Hintikka's game-theoretical\r\nsemantics for first-order logic.  We can understand the distinction\r\nbetween objects and concepts in terms of their different roles in a\r\nsemantic game that defines truth in a model.  An object is what\r\nplayers seek, find, and specify when executing quantifier moves in\r\nthis game.  A concept is what guides a player as she makes such moves:\r\nit distinguishes a strategic choice from a non-strategic one.  This\r\ncharacterization of objects and concepts has the virtue of ontological\r\nneutrality, offering a clear sense in which numbers, colors, and\r\nplanets all count as objects, without committing us to finding\r\nanything common in their ontologies.  It also suggests a new\r\nunderstanding of second-order quantification.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c222');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Jonathan Dittrich, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c224')\"><i>The inadequacy of nontransitive solutions to paradox<\/i><\/a>\r\n<div id=\"absbox-c224\" class=\"abstract_box\">\r\nJonathan Dittrich, <i>The inadequacy of nontransitive solutions to paradox<\/i>\r\n<br><br>\r\n[1] has argued that a nontransitive substructural logic (NT) provides both a solution to semantic paradoxes and preserves full classical logic. Here we argue that NT fulfils these goals only inadequately.\r\nWe distinguish between weak and strong inconsistency: A language is weakly inconsistent for some formula $\\phi$ iff it proves both &nbsp;&nbsp; $\\vdash \\phi$ and $\\phi \\vdash$ &nbsp;&nbsp;. It is strongly inconsistent iff it proves the empty sequent. Although NT is strongly consistent, it remains weakly inconsistent. For even without Cut, NT derives both &nbsp;&nbsp; $\\vdash$ T($\\ulcorner \\lambda \\urcorner$) and T($\\ulcorner \\lambda \\urcorner$) $\\vdash$ &nbsp;&nbsp; for the Liar sentence $\\lambda$.\r\nAccording to Ripley, the characteristic of any paradoxical sentence $\\psi$ is that both &nbsp;&nbsp; $\\vdash \\psi$ and $\\psi \\vdash &nbsp;&nbsp;$ are provable. However, one can show that NT is weakly inconsistent for many central theorems of classical logic as well; including the law of non-contradiction, excluded middle and identity. This is straightforward in first-order logic when these principles are understood in terms of the Truth-predicate. With second-order logic, this can be extended to hold of these theorems understood with arbitrary predicates.\r\nThere are two problems. First, NT fails to provide an adequate distinction between paradoxical and non-paradoxical sentences. For classical theorems bear the characteristic of paradoxes. Second, it casts doubt on the classicality of NT. Intuitively, it is not sufficient to merely prove all theorems of classical logic. For even an inconsistent system with explosion will do this. One would further have to ensure that the system does not prove anything weakly inconsistent with these theorems - again, NT fails to do so.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nDavid Ripley,\r\nConservatively Extending Classical Logic with Transparent Truth,\r\nReview of Symbolic Logic,\r\nvol.&nbsp;5 (2012), no.&nbsp;2, pp.&nbsp;354\u2013378.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c224');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Malte Klie\u00df, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c226')\"><i>Strong Predicate Exchangeability in inductive logic<\/i><\/a>\r\n<div id=\"absbox-c226\" class=\"abstract_box\">\r\nMalte Klie\u00df, <i>Strong Predicate Exchangeability in Inductive Logic<\/i>\r\n<br><br>\r\nIn Pure Inductive Logic we study the consequences for an agent's rational belief function\r\ngiven that she assumes a list of rational principles [4], in the\r\nframework following Carnap's Inductive Logic [1].\r\nThe Principle of Strong Predicate Exchangeability, or SPx, is a rational principle\r\nstating that an agent's belief in a sentence $\\phi$ should be the same as that in\r\nthe statement $\\psi$, where $\\psi$ is obtained from $\\phi$ by permuting the atoms\r\noccurring in $\\phi$ freely as long as the number of negations remains fixed.\r\nThe principle was discovered during work on Predicate Exchangeability [2],[3].\r\nThe principle is in strength between Predicate Exchangeability and Atom Exchangeability,\r\nin the sense that each probability function satisfying a stronger principle also\r\nsatisfies the weaker one, but not vice versa. Representation Theorems for SPx\r\nshow a remarkable similarity to those for Atom Exchangeability.\r\nIn terms of justification, we can obtain SPx both through a generalization of\r\nAtom Exchangeability and a weakening of Johnson's Sufficientness Postulate.\r\nWe will give an outline of the representation theorems and show the relation between\r\nthe rational principles.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nRudolf Carnap,\r\nA Basic System of Inductive Logic,\r\nStudies in Inductive\r\nLogic and Probability\r\n(R. Carnap and R.C. Jeffrey, editors),\r\nUniversity of California Press,\r\n1971,\r\npp.&nbsp;33\u2013165.\r\n<br>\r\n[2]\r\nMalte S. Klie\u00df,\r\nThe Principle of Predicate Exchangeability in Pure Inductive Logic,\r\nPhD Thesis, University of Manchester, 2013.\r\n<br>\r\n[3]\r\nMalte Klie\u00df and Jeff Paris,\r\nPredicate Exchangeability and Language Invariance in Pure Inductive Logic,\r\nLogique &amp; Analyse,\r\nvol.&nbsp;57 (2014), no.&nbsp;228, pp.&nbsp;513\u2013540.\r\n<br>\r\n[4]\r\nJeffrey Paris and Alena Vencovsk\u00e1,\r\nPure Inductive Logic,\r\nPerspectives in Logic,\r\nCambridge University Press,\r\n2015.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c226');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Eric Epstein, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c228')\"><i>Component Contexts, Reference Determination, and the Liar Paradox<\/i><\/a>\r\n<div id=\"absbox-c228\" class=\"abstract_box\">\r\nEric Epstein, <i>Component contexts, reference determination, and the Liar Paradox<\/i>\r\n<br><br>\r\nConsider the following:\r\n<br>\r\n(A)\tSentence A is not true.\r\n<br>\r\nReflection on this sentence leads quickly to contradictions. This is the Liar paradox.\r\nI develop a diagnosis and solution for this paradox. There is no contradiction, because sentences like A fail to say what they appear to say. This is because the key occurrence of the word 'true' in any such sentence fails to refer to truth. Instead of referring to truth, such occurrences are indeterminate in reference as between two other truth-like properties, a-truth and d-truth, discussed by (Scharp 2013). It is therefore indeterminate whether a Liar sentence like A says of itself that it is not a-true or that it is not d-true. Still, one needn't hold that Liar sentences are meaningless. Rather, they come quite close to saying of themselves that they are not true, since a-truth and d-truth are very similar to truth.\r\nContextualist and indexicalist approaches to the Liar paradox also maintain that certain problematic sentences containing the truth-predicate fail to express propositions. Unlike those approaches, mine does not take the truth-predicate to refer to different properties across different contexts of use. Instead, I take it to be sensitive to its linguistic context of occurrence. My view better navigates the difficulties raised for contextualist and indexicalist approaches. E.g., my solution allows for unrestricted quantification over contexts, and better respects the linguistic data about how `true' behaves in non-paradoxical sentences.\r\nOn my view, the word `true' refers to truth, but its occurrence in any Liar sentence does not. Therefore, sometimes an occurrence of an expression differs in reference from the expression of which it is an occurrence. This may seem surprising, but compared to the demands of alternative approaches it is a small price to pay.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nKevin Scharp,\r\nReplacing Truth,\r\nOxford University Press\r\n2013.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c228');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Non-Classical Logics, Computability <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Diana Costa, Manuel A. Martins and Jo\u00e3o Marcos, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c230')\"><i>Inconsistent Accessibility Relation in Hybrid Logic<\/i><\/a>\r\n<div id=\"absbox-c230\" class=\"abstract_box\">\r\nDiana Costa, Manuel A. Martins and Jo\u00e3o Marcos, <i>Inconsistent accessibility relation in Hybrid logic<\/i>\r\n<br><br>\r\n.\r\n<br>\r\n[1]\r\nBesnard, P. and Hunter, A.,\r\nQuasi-classical logic: Non-trivializable classical reasoning from inconsistent information,\r\nSymbolic and Quantitative Approaches to Reasoning and Uncertainty \u2013 Lecture Notes in Computer Science\r\nvol.946,\r\n(Froidevaux, C. and Kohlas, J., editors),\r\nSpringer Berlin Heidelberg,\r\n1995,\r\npp.44-51.\r\n<br>\r\n[2]\r\nBlackburn, P.,\r\nRepresentation, reasoning, and relational structures: A hybrid logic manifesto,\r\nLogic Journal of the IGPL,\r\nvol.8 (2000), no.3, pp.339\u2013365.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c230');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Yuna Won, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c232')\"><i>Chisholm's Paradox Revisited: A Dynamic Implementation of Kratzer Semantics for Deontic `Ought's<\/i><\/a>\r\n<div id=\"absbox-c232\" class=\"abstract_box\">\r\nYuna Won, <i>Chisholm's Paradox Revisited: A Dynamic Implementation of Kratzer Semantics for Deontic `Ought's<\/i>\r\n<br><br>\r\nChisholm's Paradox is one of the most famous puzzles in the deontic logic literature. It is commonly believed that the Chisholm's Paradox arises only in Standard Deontic Logic (SDL), but not in ordering semantics, which is now the orthodox semantics for conditionals and modals. Although ordering semantics is free from this classic puzzle, I argue that ordering semantics fails to meet the real challenge raised by Chisholm's example, which is about the nature of contrary-to-duty (CTD) obligations\\textemdash a type of obligations that takes effect when a corresponding primary obligation is violated. To show that ordering semantics cannot provide an adequate formal representation of CTD obligations, I put forward a new puzzle, the CTD trilemma, extending the familiar example from Chisholm's Paradox. The source of their limitation will be identified, and possible resistance to the CTD trilemma will be carefully examined. To solve this new puzzle and adequately capture the notion of CTD obligations in a formal system, I maintain that we need to acknowledge two different normative uses of deontic modal sentences corresponding to the world-to-word direction of fit and the word-to-world direction of fit; I call the former type the axiological use of `ought'-statements and the latter the deontological use of `ought'-statements. Finally, I propose a dynamic approach which accounts for these two meanings of `ought's based on Kratzer semantics and show how it solves the CTD trilemma.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c232');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Anahit Chubaryan, Artur Khamisyan and Arman Tshitoyan, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c234')\"><i>Some new proof systems for a version of many-valued logics and proof complexities in it<\/i><\/a>\r\n<div id=\"absbox-c234\" class=\"abstract_box\">\r\nAnahit Chubaryan, Artur Khamisyan and Arman Tshitoyan, <i>Some new proof systems for a version of many-valued logics and proof complexities in it<\/i>\r\n<br><br>\r\nSome method for construction a deductive full propositional calculi for some version of $k$-valued ($k\\geq 3$.) logic is described in the paper. The propositional connectives are defined as follows: conjunction is min, disjunction is max, negation  is  defined by permuting the truthvalues cyclically. We use as literals the propositional variables, variables with negation, with double  negations, with triple negations etc. We generalize the notions of determinative conjunct and determinative disjunctive normal form (dDNF), introduced by first coauthor for two-valued Boolean functions in [1], and  on the base of it construct the systems $E_k$, axioms of which are not fixed. Each conjunct from some dDNF of given formula can be considered as an axiom . The elimination rule ($e$-rule) infers conjunct $K'\\cup K\"\\cup K\"'\\cup\\cdots$ from  conjuncts $K'\\cup \\{p\\}$, $K\"\\cup \\{^{\\sim}p\\}$, $K\"'\\cup \\{^{\\sim\\sim}p\\}$ etc. for a propositional variable $p$. $E_k$-proof is defined as usually. It is obvious that some DNF $D=\\{K_1, K_2, \\ldots, K_i\\}$ is $k$-valued tautology iff using $e$-rule  we can derive the empty conjnct from axioms $\\{K_1, K_2, \\ldots, K_i\\}$. We prove also that for every $k$ there is some sequence of $k$-valued tautologies, which have in described systems the same by order upper and lower bounds for the main proof complexity characteristics: exponential for lines and size, polynomial for space and width.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nAn. Chubaryan, Arm. Chubaryan,\r\nA new conception of Equality of Tautologies,\r\nL{&amp;}PS, Triest, Italy,\r\nVol.&nbsp;V, No&nbsp;1, 2007, pp.&nbsp;3-8.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c234');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Hubert Bo\u017cek, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c236')\"><i>Analysis of Functional Expressions in Leon Chwistek's Elementary Semantics and $\\lambda$-calculus by Alonzo Church<\/i><\/a>\r\n<div id=\"absbox-c236\" class=\"abstract_box\">\r\nHubert Bo\u017cek, <i>Analysis of functional expressions in Leon Chwistek's elementary semantics and $\\lambda$-calculus by Alonzo Church<\/i>\r\n<br><br>\r\nIn my presentation I wish to discuss some similarities between Leon Chwistek's elementary semantics (ES) and $\\lambda$-calculus introduced by Alonzo Church, both systems dating back to early 1930's (1, 2, 3, 4, 5) Chwistek: 1929, 1932, 1933 ; Church: 1932, 1936). Instead of comprehensive comparison of the two formal systems, I will focus on the representations of functional expressions they offer. My goal is to demonstrate that both construction and transformation rules of the systems in question are in fact mutually reducible or interchangeable, at least, when applied to functions of the same assumed logical type.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nLeon Chwistek,\r\nNeue Grundlagen der Logik und Mathematik,\r\nMathematische Zeitschrift,\r\nvol.30 (1929), pp.704-724.\r\n<br>\r\n[2]\r\n\u2015,\r\nNeue Grundlagen der Logik und Mathematik. Zweite Mitteilung,\r\nMathematische Zeitschrift,\r\nvol.34 (1932), pp.527-534.\r\n<br>\r\n[3]\r\n\u2015,\r\nDie nominalistische Grundlegung der Mathematik,\r\nErkenntnis,\r\nvol.3(1933), no. 4-6, pp. 367-388.\r\n<br>\r\n[4]\r\nAlonzo Church,\r\nA Set of Postulates for the Foundation of Logic,\r\nAnnals of Mathematics,\r\nvol.33 (1932), no.2, pp.346\u2013366.\r\n<br>\r\n[5]\r\n\u2015,\r\nAn Unsolvable Problem of Elementary Number Theory,\r\nAmerican Journal of Mathematics,\r\nvol.58 (1936), pp. 354-363.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c236');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.40<\/td>\r\n<td style=\"vertical-align: top;\">\u0130brahim \u015eent\u00fcrk, Tahsin Oner and Urfat Nuriyev, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c238')\"><i>Completeness of Categorical Syllogisms By Means of Diagrammatic Method<\/i><\/a>\r\n<div id=\"absbox-c238\" class=\"abstract_box\">\r\n\u0130brahim \u015eent\u00fcrk, Tahsin Oner and Urfat Nuriyev, <i>Completeness of Categorical Syllogisms By Means of Diagrammatic Method<\/i>\r\n<br><br>\r\nIn this study, our goal is to show the completeness of categorical syllogisms by means of diagrammatic method. For this, we firstly construct a formal system SLCD (Syllogistic Logic with Caroll Diagrams), which gives us a formal approach to logical reasoning with diagrams, for representations of the fundamental Aristotelian categorical propositions and show that they are closed under the syllogistic criterion of inference which is the deletion of middle term. Therefore, it is implemented to let the formalism comprise synchronically bilateral and trilateral diagrammatical appearance and a naive algorithmic nature. And also, there is no need specific knowledge or exclusive ability to understand as well as to use it.\r\nIn other respects, we scrutinize algebraic properties of categorical syllogisms together with a representation of syllogistic arguments by using sets in SLCD. To this end, we explain quantitative relation between two terms by means of bilateral diagrams. Thereupon, we enter the data, which are taken from bilateral diagrams, on the trilateral diagram. With the help of elemination method, we obtain a conclusion which is transformed from trilateral to bilateral diagram. A categorical syllogistic system consists of 256 syllogistic moods, 15 of which are unconditionally and 9 are conditionally; in total 24 of them are valid. Those syllogisms in the conditional group are also said to be <i>strengthened<\/i>, or valid under <i>existential import<\/i>, which is an explicit assumption of existence of some <i>S<\/i>, <i>M<\/i> or <i>P<\/i>. So, we add a rule, which is <i>Some X is X<\/i> when $X$ exists, to SLCD. Therefore, we obtain the formal system SLCD$^\\dagger$ from SLCD.\r\nFinally, we show that syllogism is valid if and only if it is provable in SLCD and strengthened syllogism is valid if and only if it is provable in SLCD$^\\dagger$. This means that SLCD is sound and complete.\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nLewis Caroll.\r\nSymbolic Logic,\r\nClarkson N. Potter,\r\n1896.\r\n<br>\r\n[2]\r\nJan \\L{}ukasiewicz.\r\nAristotle's Syllogistic From the Standpoint of Modern Logic,\r\nClarendon Press,\r\nOxford,\r\n1951.\r\n<br>\r\n[3]\r\nI. Pratt-Hartmann and L. S. Moss,\r\nOn the Computational Complexity of the Numerically Definite Syllogistic and Related Logics,\r\nReview of Symbolic Logic,\r\nvol.2, no.4, pp.647\u2013683, 2009.\r\n<br>\r\n[4]\r\nRuggero Pagnan,\r\nA Diagrammatic Calculus of Syllogisms,\r\njournal of logic language and information,\r\nno.21, pp.347\u2013364, 2012.\r\n<br>\r\n[5]\r\nA. E. Kulinkovich,\r\nAlgorithmization of Resoning in Solving Geological Problems,\r\nProceedings of the Methodology of Geographical Sciences\r\nNaukova Dumka,\r\n1979,\r\npp.&nbsp;145\u2013161.\r\n<br>\r\n[6]\r\nIbrahim Senturk, Tahsin Oner, Urfat Nuriyev,\r\nAn Algebraic Approach to Categorical Syllogisms By Using Bilateral Diagrams,\r\nTheoretical and Applied Aspects of Cybernetics. Proceedings of the 5th International Scientific Conference of Students and Young Scientists\r\n( Kyiv-Ukraine),\r\n2015,\r\npp. 14\u201321.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c238');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td><br><\/td><td><br><\/td><\/tr>\r\n\r\n<tr><td><\/td><td style=\"vertical-align: top;\"><b>Computer Science <\/b><\/td><\/tr>\r\n<tr><td style=\"vertical-align: top;\">16.00<\/td>\r\n<td style=\"vertical-align: top;\">Pavel Semukhin and Igor Potapov, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c240')\"><i>Decidability of reachability problems in SL(2,Z)<\/i><\/a>\r\n<div id=\"absbox-c240\" class=\"abstract_box\">\r\nPavel Semukhin and Igor Potapov, <i>Decidability of reachability problems in $\\mathrm{SL}(2,\\mathbb{Z})$<\/i>\r\n<br><br>\r\nThe <i>vector reachability problem<\/i> for semigroups in $\\mathrm{SL}(2,\\mathbb{Z})$ is defined as follows: Given two vectors $\\mathbf{x}$ and $\\mathbf{y}$ with integer coefficients and a finite collection of matrices $M_1,\\ldots,M_n$ from $\\mathrm{SL}(2,\\mathbb{Z})$, decide whether there exists a matrix $M$ from the semigroup $\\langle M_1,\\ldots,M_n\\rangle$ generated by the matrices $M_1,\\ldots,M_n$ such that $M\\mathbf{x}=\\mathbf{y}$.\r\nSimilarly, we define <i>reachability problem by fractional linear transformations<\/i> in $\\mathrm{SL}(2,\\mathbb{Z})$: Given two rational numbers $x$ and $y$ and a finite collection of matrices $M_1,\\ldots,M_n$ from $\\mathrm{SL}(2,\\mathbb{Z})$, decide whether there exists a matrix $\\begin{bmatrix} a & b\\\\\r\nc & d \\end{bmatrix}\\! \\in\\! \\langle M_1,\\ldots,M_n\\rangle$ such that $\\dfrac{ax+b}{cx+d}=y$.\r\nIn this talk we present a proof that the above-mentioned reachability problems in $\\mathrm{SL}(2,\\mathbb{Z})$ are decidable. Our approach to solving these problems relies on the translation of numerical questions on matrices into computational\r\nproblems on words and regular languages. We will also consider a geometric interpretation of reachability paths and use this technique to prove decidability of a special case of the {\\sl scalar reachability problem} in $\\mathrm{SL}(2,\\mathbb{Z})$, which can be viewed as a generalization of the vector reachability problem.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c240');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.25<\/td>\r\n<td style=\"vertical-align: top;\">Christoph-Simon Senjak, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c242')\"><i>A theory of parsers<\/i><\/a>\r\n<div id=\"absbox-c242\" class=\"abstract_box\">\r\nChristoph-Simon Senjak, <i>A Theory of Parsers<\/i>\r\n<br><br>\r\nParsing is an essential problem in Computer Science, and especially since streaming of data has become popular, parsing data \"online\" and getting partial results as fast as possible with a reasonably small amount of memory is a main objective. There are several formal approaches to this problem (e.g. the pi calculus). We present an approach that uses relations between input and output of a parser, and properties that, when proved about those relations, make them well-suited for parsing, and usually lead to efficient parsers by program extraction. We present some monad-style combinators that we use to build up more complex relations from simpler ones. We used our approach to write an implementation of Deflate (compression standard) in Coq, which shows that it is not only theoretically beautiful but also practically usable. While our own research only uses Coq, it should be easily adaptable to any dependently typed programming language.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c242');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">16.50<\/td>\r\n<td style=\"vertical-align: top;\">Svetlana Aleksandrova and Nikolay Bazhenov, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c244')\"><i>Automatic and tree-automatic list structures<\/i><\/a>\r\n<div id=\"absbox-c244\" class=\"abstract_box\">\r\nSvetlana Aleksandrova and Nikolay Bazhenov, <i>Automatic and tree-automatic list structures<\/i>\r\n<br><br>\r\nMoore and Russell introduced their formal theory of linear lists in\r\n[1]. Generalizing this theory, Goncharov\r\n\r\n[2] constructed the axiomatic theory of linear\r\nlists over the elements of a given data type.\r\n<br>\r\nIn this talk we will\r\ndiscuss algorithmic complexity of models of this theory of lists. We\r\nexhibit automatic properties of different classes of list structures\r\nusing the framework of automatic and tree-automatic structures. In\r\nparticular, we show that list structures over certain sets do not\r\nhave automatic copies but have tree-automatic presentations, while\r\nstronger hereditarily finite list superstructure is not\r\ntree-automatically presentable.\r\n<br>\r\nWe will also explore properties of list structures with respect to ordinal-automatic structures\r\nintroduced by Schlicht and Stephan [3].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nD.J. Moore and B. Russell, Axiomatic\r\nData Type Specifications: a First Order Theory of Linear Lists,\r\nActa Informatica, 15(3):193-207, 1981.\r\n<br>\r\n[2]\r\nS.S. Goncharov, A Theory of Lists and\r\nIts Models, Vychisl. Sistemy, 114:84-95,  1986 (In Russian).\r\n<br>\r\n[3]\r\nPh. Schlicht and F. Stephan, Automata\r\non ordinals and automaticity of linear orders, Annals of Pure and\r\nApplied Logic, 164(5):523&nbsp;527, 2013.\r\n<br>\r\nSupported by the Grants Council (under RF President) for State\r\nAid of Leading Scientific Schools (grant NSh-6848.2016.1)\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c244');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<tr><td style=\"vertical-align: top;\">17.15<\/td>\r\n<td style=\"vertical-align: top;\">Joshua Blinkhorn, <a href=\"javascript:void(0)\" onclick=\"openAbstract('absbox-c246')\"><i>Dependency schemes and soundness in QBF calculi<\/i><\/a>\r\n<div id=\"absbox-c246\" class=\"abstract_box\">\r\nJoshua Blinkhorn, <i>Dependency schemes and soundness in QBF calculi<\/i>\r\n<br><br>\r\nThe tremendous success of SAT solvers in recent years is motivating advances in quantified Boolean formula (QBF) solving.\r\nAfforded by its PSPACE-completeness, QBF allows natural and compact encodings of real-world problems [4], due to the addition of a quantifier prefix to the propositional formula.\r\nThe quantifier prefix of a QBF imposes a linear order on the variables, in which a given variable may depend upon any preceding variable.\r\nIt need not, however, necessarily depend on <i>all<\/i> the preceding variables.\r\nA dependency scheme is an algorithm that attempts to identify cases of independence by appeal to the syntactic form of an instance, producing a partial order on the variables that describes the dependency structure more accurately.\r\nHarnessing independence in this way, a solver enjoys greater freedom to navigate the search space, and frequently solves the instance faster despite the computational overhead incurred in computing the dependency scheme [2].\r\nWhereas there is potential to implement independence further in QBF solving, a major concern is whether the use of a given scheme preserves the correctness of the solving method [3];\r\nthat is, does the underlying proof system remain sound when parametrised by the dependency scheme?\r\nTo that end, we show how to implement dependency schemes in stronger `long-distance' QBF calculi, and demonstrate that the notion of `full exhibition', which is a property of dependency schemes, is sufficient for soundness in all the resulting systems.\r\nFurther, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky [1].\r\n<br><br>\r\n<b>References<\/b>\r\n<br>\r\n[1]\r\nSlivovsky, F.,\r\nStucture in \\#SAT and QBF,\r\nPhD thesis (2015), Vienna University of Technology.\r\n<br>\r\n[2]\r\nLonsing, F.,\r\nDependency shemes and search-based QBF solving: theory and practice,\r\nPhD thesis (2012), Johannes Kepler University.\r\n<br>\r\n[3]Room\r\nSlivovsky, F., Szeider, S.,\r\nSoundness of Q-resolution with dependency schemes,\r\nTheoretical Computer Science,\r\nvol.&nbsp;612 (2016), pp.&nbsp;83\u2013101.\r\n<br>\r\n[4]\r\nBenedetti, M., Mangassarian, H.,\r\nQBF-based formal verification: experience and perspectives,\r\nSatisfiability, Boolean Modeling and Computation,\r\nvol.&nbsp;5 (2008), nos.&nbsp;1\u20134, pp.&nbsp;133\u2013191.\r\n\r\n<br><a href=\"javascript:void(0)\" onclick=\"closeAbstract('absbox-c246');\">[Close]<\/a><\/div>\r\n<\/td><\/tr>\r\n\r\n<\/table>\r\n\r\n\r\n","protected":false},"excerpt":{"rendered":"Monday 1st August 9.00 - 9.15 Opening 9.15 - 10.15 Plenary (BLC Lecture): Laurent Bienvenu, Randomized algorithms in computability theory [Slides] Laurent Bienvenu, Randomized algorithms in computability theory Are randomized algorithms more powerful than deterministic ones? This is perhaps one of the most important general questions in computational complexity, the problem P ?= BPP perhaps...","protected":false},"author":81,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-114","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/pages\/114","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/users\/81"}],"replies":[{"embeddable":true,"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/comments?post=114"}],"version-history":[{"count":0,"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/pages\/114\/revisions"}],"wp:attachment":[{"href":"https:\/\/conferences.leeds.ac.uk\/lc2016\/wp-json\/wp\/v2\/media?parent=114"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}