
Plenary Talks
Model Theory Session
Set Theory Session
Recursion and Computer Science Logic Session
Algebraic and NonClassical Logics SessionJosep Maria Font (University of Barcelona, Spain)
Logics preserving degrees of truth and the hierarchies of abstract algebraic logic (slides)In this talk I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of the idea of consequence in a manyvalued context than the usual truthpreserving scheme, which uses a single designated element or subset. I will review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic, in the framework of Lukasiewicz logics and in the larger one of substructural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices.Deirdre Haskell (McMaster University, Canada)
Model theory and motivic integration (slides)The model theory of valued fields has recently drawn the attention of mathematicians working in subjects from differential geometry to algebraic number theory because of its essential role in the theory of motivic integration. Motivic integration is a measure theory on varieties defined over a valued field. It differs from a standard integration theory in two fundamental ways, both of which use logic. The first is that the measure is a function into a ring, instead of into the real numbers. The second is that the measure is a function defined on a formula, instead of on the set defined by the formula. These two abstractions make the calculations uniform across different fields, as well as allowing for a transfer principle; that is, if a firstorder property holds for all but finitely many primes then it holds in characteristic zero. This level of generality is made possible by working with a valued field as a structure in a firstorder language.Modeltheoretic results about valued fields are used to develop the integration theory. My goal in this talk is to explain both the algebraic context and the relevant modeltheoretic results which go into constructing the integration theory. No previous knowledge about valued fields will be assumed.
Denis Hirschfeldt (University of Chicago, USA)
The reverse mathematics of realizing and omitting typesI will discuss recent work on the computability theoretic and proof theoretic strength of constructions of special models (such as atomic and homogeneous models) that involve realizing and omitting types. The combinatorial character of these constructions place many of them in the rich reverse mathematical world of principles provable from Ramsey’s Theorem for Pairs.Ulrich Kohlenbach (Technische Universität Darmstadt, Germany)
Recent Developments in Proof Mining: Bounds from Proofs in Nonlinear Ergodic TheoryIn this talk we give a survey on recent applications of the ‘proof mining’ program. This program is concerned with the extraction of effective uniform bounds from ineffective proofs in analysis using techniques from proof theory.We will focus on problems in ergodic theory.
More specifically, we discuss new explicit uniform rates of metastability (in the sense of Tao) for the von Neumann Mean Ergodic Theorem (for uniformly convex Banach spaces) as well as Baillon’s weak nonlinear ergodic theorem. We then discuss rates of asymptotic regularity as well as metastability of a strong nonlinear ergodic theorem for Halpern iterations due to Wittmann in the Hilbert space case, Shioji/Takahashi for Banach spaces with uniformly Gateaux differentiable norm and Saejung for CAT(0)spaces.
The bound on the latter two results have been established in joint work with L. Leucstean using a novel method of eliminating Banach limits as well as a new logical metatheorem for uniformly smooth Banach spaces. We also present a quantitative version due to Safarik of another strong nonlinear ergodic theorem by Baillon (for odd operators) and Wittmann for much more general operators. Finally, we discuss a quadratic rate of asymptotic regularity recently extracted (jointly with Körnlein) for the important class of accretive operators in Banach space.
Benedikt Loewe (University of Amsterdam, Netherlands)
Looking upwards and downwards in the settheoretic multiverseIn 2008, we determined the “modal logic of forcing”, the set of those modal statements that are valid if you interpret the modal necessity operator as “in all forcing extensions” and replace the propositional variables by arbitrary sentences of the language of set theory. In terms of the settheoretic multiverse, this modal fragment corresponds to looking upwards from a particular model of set theory. In this talk reporting on joint work with Joel Hamkins, we shall consider the natural dual operation of looking downwards, i.e., finding a ground model of which the current model of set theory is a forcing extension. We shall explore some properties of the modal logic of this operation and determine which combinations of modal logics for the upwards and downwards part of the multiverse are possible.Antonio Montalbán (University of Chicago, USA)
The boundary of determinacy within second order arithmeticWe find the exact level at which determinacy stops being provable in second order arithmetic, or equivalently for this type of sentences, in ZFC without the power set axiom. This is joint work with Richard A. Shore.Justin Moore (Cornell University, USA)
Iterated forcing and the Continuum Hypothesis (slides)Building on Jensen’s proof that the Souslin Hypothesis is consistent with the Continuum Hypothesis, Shelah developed a criteria for proving that an iteration of forcings does not introduce new real numbers. By work of Devlin and Shelah, this criteria necessarily involves more than just the requirement that the individual iterands do not add reals and that they preserve stationary subsets of $\omega_1$.Parts of Shelah’s criteria had an ad hoc nature to them. Additionally, new problems arose in the 1990s which required weakening Shelah’s criteria. In some cases this was possible, while in others the problems resisted all attempts to solve them.
After developing this context, the talk will discuss two recent developments which show that the incremental progress made in the 1990s can not be subsumed into a more general theory of iterated forcing. First, there are two forcing axioms which are each consistent with the continuum hypothesis but which jointly negate it. Second, there is a new example of a combinatorial consequence of CH which “stands behind” the previously ad hoc part of Shelah’s criteria, making it more essential than it had previously been believed.
Daniele Mundici (University of Florence, Italy)
Geometry of the DubucPoveda strong semisimplicity propertyAn algebra A is strongly semisimple (in the sense of DubucPoveda) if every principal congruence of A is the intersection of maximal congruences of A. While this notion is stated in very general and purely algebraic terms, for latticeordered abelian groups (with or without unit) and for MValgebras, strong semisimplicity turns out to be deeply related to the tangent space of their maximal spectra. We will discuss the geometry of strong semisimplicity, and its role in the visualization of syntactic and semantic consequence in infinitevalued Lukasiewicz logic, the underlying logic of MValgebras.Alf Onshuus (Universidad de los Andes, Colombia)
VCdensity and dprank in model theory (slides)VCdimension and VCdensity were introduced by Vapnik and Chervonenkis to measure, in some way, the complexity of a family of subsets of a given universe. This notion has had important applications in statistics an learning theory, usually in the framework where one looks at a definable family of definable sets (usually in a cartesian power of the real or the real exponential field). In this context model theory becomes relevant; in fact, any model where all uniform definable families of definable sets have finite VCdimension is precisely what Shelah independently defined as a dependent theory (also known as a theory with NIP). This connection of “VCtheory” with model theory was discovered by Laskowski and is the setting of the talk.We will study another connection discovered later. Turns out that another important notion of VCtheory, VCdensity, is quite related to type counting and to dprank (a rank introduced by Shelah in the context of dependent theories). We will show some of the connections of model theory with VCdensity, and show some evidence of a strong relation between VCdensity and dprank.
Valeria de Paiva (Rearden Commerce, USA)
Dialectica Categories Surprising Application — mapping cardinal invariants (slides)Goethe famously said that “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.” True. Even more true of category theorists. Following this great tradition of appropriating other people’s work, I want to tell you how I learned about “cardinalities of the continuum” from Blass and Morgan and da Silva and how I want to rock their boat, just a little, in the direction of my kind of mathematics. To this end I will introduce you to Dialectica Categories, an internal model of Goedel’s Dialectica Interpretation that happens to be a very clever model of Linear Logic too. Time permitting I will mention some of its other applications.Thomas Scanlon (University of California, Berkeley, USA)
Arithmetic dynamics from logicDynamics, as the study of iterations of operations, is central to many parts of mathematics. From the logical point of view, one would expect that questions about discrete dynamical systems, even when restricted to reasonably tame categories, would suffer from Gödel incompleteness phenomena, and this, of course, is true. However, for many problems in algebraic dynamics, that is, those dynamical systems defined by rational functions, ideas from the model theory of difference fields, valued fields, and ominimality can bring some order to the chaos.Simon Thomas (Rutgers University, USA)
On the Bergman Property for infinite products of finite groups (slides)For each prime $p$, let $SL(2,p)$ be the corresponding special linear group over the field with $p$ elements. Then, by considering the ultraproduct with respect to any nonprincipal ultrafilter, it is easily seen that the full direct product $\prod SL(2,p)$ has countable cofinality. In recent joint work, Jindra Zapletal and I considered the question of whether the existence of a nonprincipal ultrafilter is either necessary or sufficient in order to prove this result. Perhaps surprisingly, the answer turned out to involve large cardinals, a suitable Ramsey theorem and the asymptotic representation theory of finite groups.Jouko Väänänen (University of Helsinki and University of Amsterdam)
Dependence and independence in logic (slides)Juan Diego Caycedo (Universität Freiburg, Germany)
Tameness of expansions of the real and complex fieldsThe complex field and the real ordered field are prototypical examples of wellbehaved structures from the modeltheoretic viewpoint, although in rather different ways. Indeed, in both cases there is a certain geometric understanding of definable sets. Natural questions are what expansions of these structures introducing new definable sets admit similar results and where is there a dividing line between “tame” and “wild” expansions.In the case of the real field, a rich theory has been developed for ominimal expansions. Also, a meaningful division is obtained when tame expansions are defined to be those which do not define the set of integer numbers.
For expansions of the complex field, it is less clear what the notion of tame expansions should be. In this context, I will discuss some examples of structures related to complex exponentiation and some connections with the real case.
Artem Chernikov (Université Claude Bernard Lyon 1, France)
NTP2 theoriesTheories without the tree property of the second kind, or NTP2 theories, is a large class of firstorder theories introduced by Shelah. This class is interesting from the purely model theoretic point of view (as it is a natural common generalization of both simple and NIP theories) and also due to some new important examples that it contains (e.g. any ultraproduct of padics and certain valued difference fields). I will give an overview of recent progress on developing the theory of forking calculus and weight for this class.Vinicius Cifú Lopes (Universidade Federal do ABC, Brazil)
Pseudofinite and Pseudocompact Metric StructuresWe initiate the study of pseudofiniteness in continuous logic. We introduce a related concept, namely that of pseudocompactness, and investigate the relationship between the two concepts. We establish some basic properties of pseudoniteness and pseudocompactness and provide many examples. We also investigate the injectivesurjective phenomenon for definable endofunctions in pseudofinite structures. This is joint work with Isaac Goldbring, then UCLA, now at UIChicago.Alf Dolich (City University of New York, USA)
Expansions of ominimal theories by a dense independent predicateLet $T$ be an ominimal theory in a language $L$. Let $L_H$ be $L$ augmented with a new unary predicate $H$ and let $T_H$ be the $L_H$theory containing $T$ and stating that $H$ is a dense independent (in the sense of the definable closure operator for $T$) subset. $T_H$ is complete and any $L_H$formula is $T_H$ equivalent to a Boolean combination of certain special existential formulas. We study models, $M$, of $T_H$. In particular we will discuss a decomposition theorem for definable subsets of arbitrary Cartesian powers of $M$ and a result indicating that definable functions of arbitrary arity are “close” to being definable in the underlying ominimal structure.John Goodrick (Universidad de los Andes, Colombia)
The SchröderBernstein property and asaturated models (slides)We say that a class $(K, Mor)$ of structures $K$ with a distinguished class $Mor$ of injections between them is SchröderBernstein (or SB) if any two Morbiembeddable structures in $K$ are isomorphic. We focus on the case where $K$ is a subclass of $Mod(T)$ for some firstorder theory $T$ and $Mor = Elem$ are the elementary embeddings between them.While characterizing the theories $T$ such that $(Mod(T), Elem)$ is SB seems to be complicated, it turns out to be much easier to consider the case where $K$ consists of all the sufficiently saturated models of $T$ (we use asaturation, which is slightly weaker than $T^{+}$saturation). We can completely characterize the theories $T$ for which the class of asaturated models is SB when $T$ is unstable (in which case this will never be SB) and when $T$ is superstable. Extending these results to all theories $T$ would seem to require answering difficult longopen questions on the existence of regular types in strictly stable theories.
Janak Daniel Ramakrishnan (CMAF, Portugal)
Partial orders in ominimal structuresWe show that partial orders definable in several “tame” ordered structures can be definably extended to linear orders. Our technique extends a straightforward one on wellordered structures, and holds for many variants of ominimal structures, as well as for ordered structures not usually considered tame, such as nonstandard models of PA. Joint work with C. Steinhorn.Margaret Thomas (Universität Konstanz, Germany)
Rational points and integervalued definable functionsPila and Wilkie’s theorem concerning the density of rational and algebraic points lying on sets definable in ominimal expansions of the real field has already had farreaching consequences for Diophantine geometry. Wilkie has conjectured an improvement to their result for sets definable in the real exponential field. We shall illustrate how the already proven onedimensional case of this conjecture has consequences for the growth behaviour of the analytic functions definable in this structure which take integer values at natural number arguments. These results are related to a classical theorem of P\’olya; we show that if a definable, analytic function $f : [0, \infty)^{k} \rightarrow \mathbb{R}$ is such that $f(\mathbb{N}^{k}) \subseteq \mathbb{Z}$, and, for every $\epsilon > 0$, $f$ grows slower than $\exp(r^{\epsilon})$ (where $r$ is the distance from the origin), then $f$ is a polynomial over $\mathbb{Q}$. This result is joint work with Gareth O. Jones and Alex J. Wilkie.Xavier Vidaux (Universidad de Concepción, Chile)
Uniform Definability and Undecidability in Classes of StructuresWe present a concept of uniform encodability of theories and develop tools related to this concept. As an application we obtain general undecidability results which are uniform for large families of structures. For example we show that there is no algorithm to answer whether a system of polynomial equations over $Z[z]$ has or does not have solutions in all but finitely many polynomial rings $F_p[z]$ with $F_p$ a finite field.In the way, we define uniformly among odd characteristics the equivalence relation “$x\sim y$ if and only if there is some integer $s$ for which $y=x^{p^s}$” in large classes of function fields and in polynomial rings.
This is joint work with Hector Pasten (Queen’s University, Canada) and Thanases Pheidas (University of Crete, Greece).
Contributed Talks (Model Theory)
María Victoria Cifuentes (Universidad Nacional de Colombia)
Grupos extremadamente llevaderos y Construcciones de HrushovskiRetomando el resultado de Kechris, Pestov y Todorcevic sobre dinámicas topológicas exploradas en el grupo de automorfismos del límite de una clase de Fraïssé, revisamos el comportamiento del grupo de automorfismos de la estructura genérica asociada a una construcción de Hrushovski. Bajo ciertas condiciones adicionales sobre la construcción se obtiene un resultado análogo al de KPT.Vincent Guingona (University of Notre Dame, USA)
On VCMinimal Theories (slides)I discuss recent developments in VCminimal theories. This includes examining VCminimality and variants in abelian groups, ordered abelian groups, ordered fields, and valued fields. I also exhibit interesting properties of VCminimal theories, including eliminiation of imaginaries in one variable and interpretability of infinite linear orders. Some of this work is joint with Joseph Flenner.Pablo Cubides (Université de Paris VII, France)
Cminimal structuresCminimal structures constitute an interesting abstract setting in which one can test properties one would like to lift from stable and ominimal theories to the NIP context. Many properties shared by this two tame type of structures such as existence of prime models, algebraic closure having the exchange property, etc., are not in general true for Cminimal ones. Yet, by work of Haskell and Macpherson they have a cell decomposition theorem and a wellbehaved notion of dimension. We present a slightly different approach to these theorems and we show some applications to the study of definable types together with some examples.Hugo Luiz Mariano (University of São Paulo, Brazil)
Some preservation properties of the Profinite Hull Functor of Special Groups and applicationsProfinite RSGs appeared in the dual setting of orderings spaces. In this work we show that the profinite hull functor ($\cP$) of reduced special groups (RSG) preserves finite products (but does not preserves infinitary products, in general) and, in a certain way, it also preserves SG extensions. This allows: (i) to provide a description of the profinite fan that is the profinite hull of a given fan; (ii) to establish that $\cP$ preserves chain length (it has been proven that this functor preserves stability index). Thus the Profinite hull functor seems more “wellbehaved” than the Boolean hull functor, as the latter increases the chain length (strictly, in some cases) and often strictly decreases the stability index. We apply the universal property of $\cP$ and some results on the profinite hull of products, quotients and filtered colimits of RSGs, to provide a new modeltheoretic proof that the class of the RSGs satisfying a generalized localglobal principle is an elementary class axiomatizable by Horn sentences.Yevgeniy Vasilyev (University of Newfoundland, Canada)
Linearity and expansions of geometric theoriesA theory is called geometric if in all of its models algebraic closure satisfies the exchange property and the theory eliminates the infinity quantifier. Examples include ominimal, strongly minimal and SUrank 1 theories. In a joint work with Alexander Berenstein, we introduce several equivalent conditions, including weak local modularity, weak onebasedness and generic linearity, which are closed under reducts and provide a common generalization of the “classical” linearity notions used in the strongly minimal, SUrank 1, ominimal, as well as geometric Cminimal settings, to the general class of geometric theories. One of our main tools, the lovely pair expansion, allows us to find a connection between linearity and the presence of vector spaces over division rings, which, in the countably categorical case, turn out to be interpretable in the pair theory. I will also talk about other types of expansions of geometric theories, such as expansions by independent “dense” subsets.Pedro Zambrano (Universidad Nacional de Colombia)
A stability transfer theorem in dtame Metric Abstract Elementary ClassesIn discrete Abstract Elementary Classes, J. Baldwin, D. Kueker and M. VanDieren proved a stability transfer theorem, under tameness and omegalocality. We will talk about a stability transfer theorem in dtame Metric Abstract Elementary Classes, under epsilonlocal character of a suitable notion of independence which is wellbehaved in this setting.Andrés Caicedo (Boise State University, Indiana, USA)
Forcing with ${\mathbb P}_{max}$ over models of strong versions of determinacyHugh Woodin introduced ${\mathbb P}_{max}$, a definable poset, and showed that, when forcing with it over $L(\mathbb{R})$ (in the presence of determinacy), one recovers choice, and obtains a model of many combinatorial assertions for which simultaneous consistency was not known by traditional forcing techniques. ${\mathbb P}_{max}$ can be applied to larger models of determinacy. As part of joint work with Larson, Sargsyan, Schindler, Steel, and Zeman, we show how this allows us to calibrate the strength of different square principles.Natasha Dobrinen (University of Denver, USA)
The structure of the Tukey types of ultrafiltersThe Tukey ordering was first introduced in the study of MooreSmith convergence in topology, and soon proved useful in the more general study of ordered structures. Tukeytypes make possible classifications of certain ordered structures for which classification up to isomorphism is too fine to reveal any real information. When restricting attention to ultrafilters as directed posets, Tukey types turn out to be a coarsening of the wellstudied RudinKeisler equivalence classes. In this setting, Tukey equivalence is the same as cofinal equivalence. In this talk, we will give an overview of recent work on the Tukey types of ultrafilters. We will present some of the currently known structure theorems about Tukey types within the class of ppoints and iterated Fubini products of ppoints. Some new topological Ramsey spaces and Ramseyclassification theorems, generalizing the ErdösRado and PudlakRödl canonization theorems for the Ellentuck space, will be presented, as well as their applications to completely classifying the RudinKeisler (isomorphism) types within the Tukey types of the related ultrafilters. Most of the work in this talk is joint with Stevo Todorcevic.Lucia Junqueira (Universidad de Sao Paulo, Brazil)
There is more than one useful way to create a new topological space from a space and an elementary submodel containing it (slides)We have previously used a “subobject” created from a topological space and an elementary submodel. Recently a “quotient object” has been used instead. Given a topological space $X$, a base $\mathcal{B}$ for it, and an elementary submodel containing both, we define a set of equivalence classes $X/(M,\mathcal{B})$ with a topology that is weaker than the quotient topology. We discuss some applications and some subtleties.Dilip Raghavan (Kobe University, Japan)
Bounding, splitting, and almost disjointnessA famous (still) open problem in the theory of cardinal invariants of the continuum asks whether $\mathfrak{d} = \aleph_1$ implies $\mathfrak{a} = \aleph_1$. A variation of this question is “does $\mathfrak{b} = \mathfrak{s} = \aleph_1$ already imply that $\mathfrak{a} = {\aleph}_1$?”. We will shed some light on this question by examining when it is possible to destroy a MAD family without increasing either $\mathfrak{b}$ or $\mathfrak{s}$. This is joint work with J. Brendle.Victor Torres Perez (Vienna University, Austia)
Rado’s Conjecture and square sequencesRado’s Conjecture (RC) is the following statement: A family of intervals of a linearly ordered set is the union of countably many disjoint subfamilies ($\sigma$)disjoint if and only if every subfamily of size is ($\sigma$)disjoint.Todorcevic has shown that RC is independent with ZFC. He has also shown some interesting consequences. For example, RC implies that the continuum has size at most $\aleph_2$, a strong form of Chang’s conjecture holds; SCH holds; the negation of Jensen’s square principle $\Box_\kappa$ for every uncountable $\kappa$, etc. In this talk we will focus on the influence of Rado’s Conjecture over certain Aronszajn trees and some types of square principles in a two cardinal version. This a joint work with Stevo Todorcevic.
Carlos Uzcátegui (Universidad de los Andes, Mérida, Venezuela)
Borel selectors for families of setsLet $\mathcal H$ be a collection of infinite subsets of a countable set $X$ and let $\widehat{\mathcal H}$ be the upward closure of $\mathcal H$, that is to say,$$\widehat{\mathcal H}=\{A\subseteq X:\; (\exists H\in{\mathcal H})\; H\subseteq A\}.$$
A selector for $\mathcal H$ is a function $\Phi: X^{[\infty]}\rightarrow X^{[\infty]}$ (where $X^{[\infty]}$ is the collection of infinite subsets of $X$) such that
\[
\Phi(A)\subseteq A \;\;\&\;\;\Phi(A)\in{\mathcal H}\;\;\mbox{for all $A\in \widehat{\mathcal H}$}.
\]
We will present some results regarding the question of whether a family $\mathcal H$ admits a selector $\Phi$ which is a Borel function.The original motivation for this work is related to the classical Ramsey theorem: Given a coloring $c:\mathbb{N}^{[2]}\rightarrow 2$ of the 2elements subsets of $\mathbb{N}$, there is an infinite $B\subseteq \mathbb {N}$ homogeneous for $c$, that is to say, $c$ is constant on $B^{[2]}$. The question is whether there exists a Borel function that selects an infinite $c$homogeneous subset of a given infinite set $A\subseteq \mathbb {N}$. In other words, does the collection $\mathcal{H}_c$ of all $c$homogeneous infinite sets admit a Borel selector? We show that the answer is positive. We also present results about the existence of Borel selectors for the collection of homogeneous sets given by Galvin’s lemma, a well known extension of Ramsey’s theorem.
This kind of questions also fits naturally in the context of some well studied combinatorial properties like selectivity, $q^+$, $p$ and $p^+$. We will discuss several of these properties.
Contributed Talks (Set Theory)
Franqui Cárdenas (Universidad Nacional de Colombia)
Unfoldable cardinals and gap2 morassesIn this talk I present a consistency result for a certain type of large cardinal and gap2 morasses obtained with a forcing extension model where $GCH$ fails. Unfoldable cardinals and gap2 morasses exist in $L$ if there are unfoldable cardinals.Francisco Guevara (Universidad de los Andes, Mérida, Venezuela)
Uniformly Fréchet ideals.An ideal $I$ on a countable set $X$ is called Fréchet if for all $A\not\in I$ there is $B\in I^\perp$ infinite such that $B\subseteq A$, where $I^\perp$ is the orthogonal of $I$. When an ideal $I$ is Fréchet there is a function $F: 2^{X} \rightarrow 2^{X}$ such that $A\notin I \Leftrightarrow F(A)\subseteq A \textrm{, \, }
F(A) \textrm{ is infinite and } F(A)\in I^{\perp}.$We study ideals such that the function $F$ can be founded Borel. Such ideals will be called uniformly Fréchet ideals.
We can find results (different from the ones presented here) related to the extraction of converging sequences in a Borel way in separable Rosenthal compacta.
We present some examples of uniformly Fréchet ideals.
 • Let $I_c$ be the ideal on $\mathbb{N}^{<\mathbb{N}}$ generated by the chains $\overline{\alpha}=\{\alpha\upharpoonright m : m\in \mathbb{N}\}$, $\alpha\in \mathbb{N}^{\mathbb{N}}$. $I_c$ is Borel and uniformly Fréchet, while $I_{c}^{\perp}$ is $\mathbf{\Pi^{1}_{1}} $complete and it is not uniformly Fréchet.
 • $I_c$ is a particular case of a more general constructio. Let $\mathcal{A}\subseteq \mathbb{N}^{\mathbb{N}}$ be an analytic family and denote by $I(\mathcal{A})$ the ideal generated by $\{ \overline{\alpha} : \alpha \in \mathcal{A}\}$. Then $I(\mathcal{A})$ is Fr\’echet and when $\mathcal{A}$ is Borel, then $I(\mathcal{A})$ is also Borel and uniformly Fréchet.
David Meza Alcántara (Universidad Michoacana de San Nicolás de Hidalgo, México)
Hausdorff Ultrafilters in the Katetov orderAn ultrafilter $\mathcal{U}$ is Hausdorff if the topology $S$ on the ultrapower $\prod_{U}\mathbb{N}$ is Hausdorff. The main question about Hausdorff ultrafilters is if ZFC proves that they exist. There are some combinatorial equivalences for a $\mathcal{U}$ is Hausdorff. One of them concerns to the Katetov order and the $F_{\sigma}$ideal $\mathcal{G}_{fc}$ of all the finitely chromatic subsets of $[\mathbb{N}]^2$, what means that the Hausdorff ultrafilters are exactly the $\mathcal{G}_{fc}$ultrafilters (in the sense of Baumgartner). To know the position of $\mathcal{G}_{fc}$ in the Katetov order enable us to show some relations of the Hausdorff ultrafilters with other classes of ultrafilters, including those that satisfy a Fubinitype property, selective, Qpoint and nowhere dense ultrafilters.José Gregorio Mijares (Pontificia Universidad Javeriana, Colombia)
Nociones de Forcing presentadas como cocientesWe present an abstract version of Galvin’s lemma , within the framework of the theory of Ramsey spaces. Some instances of it are well known important results like Galvin’s lemma itself, Ramsey’s theorem, Ramsey’s theorem for $n$parameter sets or the GrahamLeebRothschild theorem. Nevertheless, some of them have been little explored before, as far as we are concerned. Some open problems will be discussed.Diana Montoya (Universidad de los Andes, Colombia)
Unfoldable cardinals and gap2 morasses(Trabajo conjunto con Joerg Brendle y Ramiro de La Vega.) Estudiamos nociones de forcing que se presentan como cocientes de un espacio polaco $X$ módulo un ideal sobre $X$, específicamente trabajamos con dos nociones: la primera es el cociente $P_{I} = B(X)/I$ de subconjuntos borelianos de $X$, módulo un $\sigma$ ideal $I$; la segunda es: dado un conjunto contable $Y$ , y $J$ un ideal sobre $Y$ , consideramos el cociente $Q_{J} = \wp(Y)/J$, en ambas nociones es la inclusión módulo el ideal $J$. Basados en el trabajo de Hru sákZapletal, estudiamos la relación entre los cocientes $P_{I}$ y $Q_{J}$ cuando $X = \omega^{\omega}$ y $J$ es el ideal traza de $I$.Finalmente, estudiamos dos casos particulares, cuando $I$ es el ideal de los conjuntos magros y cuando es el ideal de los conjuntos de medida $0$. En estos casos probamos que el forcing restante $R$ es $\sigma$cerrado, damos caracterizaciones para el ideal $tr(I)$ en los dos casos y, con el fin de estudiar quien es exactamente el forcing restante $R$ para el caso de conjuntos magros, estudiamos el cardinal invariante del continuo $\mathfrak{h}$, el n\’umero de distributividad, respondiendo afirmativamente una pregunta formulada por Balcar, HernandezHernandezHrusák
Uri Andrews (University of Wisconsin, USA)
Universal computably enumerable equivalence relationsI will discuss computably enumerable equivalence relations (ceers) under the reducibility $R\leq S$ if there exists a computable function $f$ such that, for every $x$,$y$, $x R y$ if and only if $f(x) S f(y)$. I will discuss how the uniformly effectively inseparable ceers coincide with the uniformly finitely precomplete ceers and the uniformly universal ceers, in particular showing that they are universal, but there are effectively inseparable ceers which are not universal. Also, I may discuss how the jump operator on ceers introduced by Gao and Gerdes is strictly increasing on nonuniveral ceers. (Joint work with Steffen Lempp, Joseph S. Miller, Keng Meng Ng, Luca San Mauro, and Andrea Sorbi.)Edward Hermann Haeusler (Pontifícia Universidade Catolica do Rio de Janeiro, Brazil)
On the Computational Complexity of Purely Implicational Logic: A ProofTheoretical Discussion (slides)The computational complexity of SAT for purely implicational propositional logic is known to be PSPACEcomplete. Intuitionistic Propositional Logic is known to be PSPACEcomplete also, while Classical Propositional Logic is CONPcomplete for Tautology checking and NPcomplete for Satisfiability checking. We show how prooftheoretical results on Natural Deduction help analysing the Purely Implicational Propositional Logic complexity regarded its polynomial relationship to Intuitionistic and Classical Propositional Logics. We provide a complete and sound deduction system for Implicational Propositional Logic that is able to provide countermodels and proofs. We hope that the analysis of the size of the objects provided by this tool from purely implicational formulas is worth of helping the understanding the relationship between the computational complexity of the logics involved in this study.Juan Andrés Montoya (Universidad Industrial de Santander, Colombia)
Paradoxes in discrete optimization and transportation sciencesIn this talk we survey the occurrence of paradoxes in discrete optimization and transportation sciences. We focus our work on some paradoxes arising in the context of Game Theory.Game theory was meant to be the philosophical/technical foundation of the liberal doctrine: it was supposed to become the final proof of the wisdom of Smith’s invisible hand. Short after the appearance of the classical monograph of Morgestern and Von Neumann some paradoxes (dilemmas) began to arise, the most famous of them being The Prisoner Dilemma. We study some classical paradoxes arising in this context. We focus our research on the paradoxes of Braess and Polymeris. This allows us to get into some topics of the history of theoretical computer science in general and the history of Informatics in Latin America in particular.
Carlos Ortiz (Arcadia University, USA)
Logics of Residue Classes in Descriptive ComplexityWe present a hierarchy of finitemodel logics that are extensions of first order logic with builtin order and modular quantifiers. We prove separation results among these logics using arguments and results from algebraic number theory and from model theory of finite fields.There are various motivations for studying these logics. They are closely related to circuit complexity classes, in particular to the classes such as $AC^{0}$ and $ACC^{0}$ contained within $NC^{1}$. Furthermore, the class of finite fields possess strong {\it asymptotic properties} (essentially this means that the definable sets behave nicely). These properties are obtained using classical methods of model theory and results of algebraic geometry. The celebrated result of Chatzidakis, van den Dries and Macintyre on the estimates of the cardinality of first order definable sets in finite fields is only the most recent example of these properties. The abovementioned logics provide a mechanism to use asymptotic properties on descriptive complexity problems.
This is ongoing work with Argimiro Arratia.
Ricardo Oscar Rodríguez (Universidad de Buenos Aires, Argentina)
Modal Extensions of Fuzzy Logics (slides)In the last years there has been a growing number of papers about combining modal and fuzzy logics (in the sense of P. Hajek as manyvalued residuated logics).The purpose of this talk is to give an overview about the problem of searching a syntactical characterization of modal manyvalued logics defined by fuzzy Kripke frames, i.e. structures $\langle W,R,e \rangle$ where $W$ is noempty set and both the accessibility relation $R$ and the interpretation of propositions $e$ are manyvalued over the same residuated algebra. Thus, the logics introduced either by manyvalued accessibility relations over classical logic or by classical accessibility relations over manyvalued logics could be considered as extensions of this doubly manyvalued semantics; indeed, the general logic is minimal.
Unfortunately, within this general framework nobody has been able to find a syntactic characterization of the notion of modal manyvalued logic that works in all the cases. The difficulties arise from the fact that it does not know a general method to axiomatize the modal manyvalued logics given by the class of all fuzzy Kripke frames (i.e., the minimal ones). It turns out that the only minimal logics axiomatized in the literature are the ones where the manyvalued counterpart is given by a finite Heyting algebra, the standard (infinite) Gödel algebra or a finite residuated algebra (in particular Lukasiewicz algebra).
Throughout the talk it will be shown the difficulties of our enterprise and we will try to specify which conditions should satisfy this syntactic notion. We will also review the works in the literature that fits inside the general proposed framework.
Contributed Talks (Recursion and Computer Science Logic)
Daniel Lima Ventura (Universidade Federal de Goiás, Brazil)
Intersection types and explicit substitution(Join with Mauricio AyalaRincón and Fairoz Kamareddine.) Intersection types (IT) provides polymorphism, a basic feature for modern computational typing systems, in a finitary way. IT systems satisfy important properties such as principal typings, providing modularity (data abstraction). Moreover, IT characterise strong normalisation, that is a term is strong normalising if and only if it is typable. Explicit substitution (ES) calculi are extensions of the $\lambda$calculus in which the operation of substitution, originally defined as a metaoperation, is specified in the calculus, filling the gap between theoretical systems and eventual implementations. The proposal of IT systems for ES calculi, besides the analysis of termination properties, can finally provide a theoretical typing system with good properties for computational systems and closer to real implementations. We present a brief overview about IT, ES and what was already done to put those
two disciplines together.Favio Ezequiel MirandaPerea (Universidad Autónoma de México)
Realizability for Haskelllike (Co)inductive DefinitionsGabriela Belen Ramírez Jiménez, and Lourdes del Carmen GonzálezHuesca) We present a sound realizability interpretation for an extension of the secondorder logic AF2 with a primitive mechanism of (co)inductive definitions similar to the data declaration of the Haskell programming language. This interpretation is nonreductive in the sense that the realizability of a (co)inductive definition is again (co)inductively defined. The intended meaning of our (co)inductive predicates is formalized by means of least and greatest fixed points of predicate transformers which are not neccesarily positive but are guaranteed to exist by the use of the Mendlerstyle, which involves a notion of monotonization of an arbitrary operator on a complete lattice. This feature of our logic allows for a direct generalization to nested or hetegoreneous (co)inductive predicates useful to implement (co)data types requiring strong invariants, like perfect trees or random access lists. The emphasis of this talk will be on the mechanism of program extraction from proofs provided by our interpretation proof as well as in some challenges posed by nested (co)inductive definitions.Manuela Busaniche (Universidad Nacional del Litoral, Argentina)
Strong semisimplicity in MValgebrasRecall that the $n$generated free MValgebra is the MValgebra of all McNaughton functions $f:[0,1]^n\rightarrow [0,1]$, with pointwise operations of negation $(\neg f)(x)=1f(x)$, truncated addition $(g\oplus f)(x)=min(1, g(x)+f(x))$, and the zero function as constant. A McNaughton function $f:[0,1]^n\rightarrow [0,1]$ is a piecewise linear continuous function, such that each linear piece has integer coefficients. Therefore one of the main tools to analyze MValgebras is that of polyhedral geometry. That is what we shall use to study strong semisimplicity in the class of finitely generated MValgebras.An MValgebra $A$ is called \textit{strongly semisimple} if for any $a\in A$ the intersection of all maximal ideals $J$ such that $a\in J$ is the ideal generated by $a$. This is equivalent to saying that every principal quotient of $A$ is semisimple.
We say that a finitely generated MValgebra $A$ is \textit{polyhedral} if there is a polyhedron $P\subseteq [0,1]^n$ such that $A$ is isomorphic to the MValgebra of restrictions to $P$ of McNaughton functions in the $n$cube. We will show that every polyhedral MValgebra is strongly semisimple. Then, to understand the failure of strong semisimplicity in semisimple MValgebras we present a finitely generated semisimple MValgebra that is not strongly semisimple. Lastly we study a sufficient condition for an $n$generated semisimple MValgebra not to be strongly semisimple.
The presentation is based on a joint work with Daniele Mundici.
Marcelo Coniglio (State University of Campinas, Brazil)
FirstOrder Paraconsistent LogicsOne of the basic features of a negation $\neg$ within a given logic L is the socalled ‘explosiveness’, which states that from a contradiction `$P$ and $\neg P$’ everything is derivable in L. Thus, classical logic (and many other logics) equate ‘consistency’ with ‘freedom from contradictions’. Paraconsistency is the study of logic systems with a negation $\neg$ such that not every contradiction of the form ‘$P$ and $\neg P$’ trivializes, that is, such that explosiveness is not always the case. This forces a reanalysis of the relationship between consistency, contradictoriness, inconsistency and triviality.The Logics of Formal Inconsistency (LFIs), proposed by W. Carnielli and J. Marcos, play an important role in the universe of paraconsistent logics, since they internalize in the object language the very notions of consistency and inconsistency by means of specific connectives (primitives or not). This generalizes the strategy of N.C.A. da Costa, which introduced the wellknown hierarchy of systems $C_n$, for $n > 0$. Dialetheism, mainly developed by G. Priest, is a different approach to paraconsistency based on the existence of ‘real contradictions’ (called dialetheias).
It is worth noting that most of the paraconsistent systems proposed in the literature are propositional. However, there exist several paraconsistent systems defined over firstorder languages. In this talk some proposals for firstorder paraconsistent logics will be analyzed and compared. The firstorder extension of several LFIs will be presented, as well as a Herbrand theorem for these systems, obtained by adapting the proof of S. Buss. Finally, a firstorder LFI with a 3valued semantics generalizing classical tarskian semantics will be introduced, showing its relations with evolutionary databases (due to W. Carnielli, J. Marcos and S. de Amo) and G. Priest’s firstorder logic LP.
Nikolaos Galatos (University of Denver, USA)
A category equivalence of varieties and the Beth definability propertyJoint work with J. Raftery. The Beth definability property for a consequence relation states that implicit definability implies explicit definability. For algebraizable logics this is equivalent to the demand that epimorphisms between algebraic models of the logic are all surjective. In order to establish this property for (odd) Sugihara monoids, the algebraic models of the relevance logic RM (relevance with the mingle axiom) and of the uninorm logic UIML, we show that they are categorically equivalent to a variety of enriched relative Stone/Goedel algebras (with a nucleus), making use of R. McKenzie’s characterization of category equivalence for varieties. The definition of the operations for the resulting algebras is related to multiplication of upper triangular matrices on modules over irings, and the appropriate modifications to these definitions are inspired by truth conditions for Kripkestyle semantics for relevant and paraconsistent logics. Apart from (a strong version of) the Beth definability for RM and UIML, other applications of the category equivalence include (hereditary) structural completeness and strong amalgamation for various varieties of Sugihara monoids.Vincenzo Marra (University of Milan, Italy)
Sheaftheoretic representations of MValgebras via Priestley DualityJoint work with Mai Gehrke (Paris) and Sam van Gool (Nijmegen and Paris).In 1958, C.C. Chang introduced MValgebras as the equivalent algebraic seman ticsof Lukasiewicz infinitevalued propositional logic. By now MValgebras sport a substantial array of connections with several other fields of mathematics, from piecewise linear topology to integration theory over Riesz spaces. Representation (or duality) theorems for classes of MValgebras often play a key role in fostering such connections. Useful representation theorems for the entire class of MValgebras, by contrast, have been much harder to come by. The best that can be done at present is to use sheaves. Two representation theorems for an arbitrary MValgebra as the algebra of global sections of a sheaf of MValgebras with special properties, over a base space which is either compact Hausdorff, or spectral, were obtained by Filipoiu and Georgescu in 1995, and by Dubuc and Poveda in 2010. We show that both representation theorems flow naturally from our analysis of the structure of the Priestley space dual to the underlying lattice of the MValgebra in question. In particular, we obtain in this manner a proof of the DubucPoveda PushoutPullback Lemma, i.e. the gluing axiom for sheaves, that uses a minimal amount of MValgebraic background. As a further consequence of interest, we establish a broad MValgebraic generalisation of a classical result by Kaplansky from 1947, stating that a compact Hausdorff space is uniquely determined to within a homeomorphism by its lattice of realvalued continuous functions.
Matias Menni (Universidad Nacional de La Plata, Argentina)
The manifestation of Hilbert’s Nullstellensatz in Lawvere’s Axiomatic CohesionSome of the interest of nonclassical logics stems from their use in clarifying and generalizing classical results. A source of examples is the role that the interpretation of intuitionistic logic in toposes has in independence results, in completeness theorems, and in the formalization of the use of infinitesimals in differential geometry.Another example, to be discussed in this talk, is the part that Hilbert’s Nullstellensatz plays in certain models of Axiomatic Cohesion (Lawvere, TAC 2007).
Any connected geometric morphism ${p^* \dashv p_*:\cal E \rightarrow \cal S}$ between toposes $\cal E$ and $\cal S$ such that ${p^*:\cal S \rightarrow \cal E}$ has a left adjoint ${p_!:\cal E \rightarrow \cal S}$ determines a natural transformation ${\theta:p_* \rightarrow p_!}$.
The (abstract) Nullstellensatz is said to hold if $\theta$ is epi. If $\Omega$ denotes the subobject classifier of $\cal E$ then Sufficient Cohesion holds if ${p_! \Omega = 1}$.
Experience with the intended examples suggests the following intuition: ${p^*:\cal S \rightarrow \cal E}$ is the inclusion of the category of `discrete spaces’ into that of all `spaces’, its right adjoint ${p_*:\cal E \rightarrow \cal S}$ assigns to each `space’ its associated `(discrete) space of points’, and the left adjoint ${p_!:\cal E \rightarrow \cal S}$ assigns the associated space of `pieces’. With this in mind, Sufficient Cohesion says that the space of truth values is connected (making the internal logic of $\cal E$ very nonclassical), and the Nullstellensatz says that every piece has a point.
We explain why, for a fixed field $k$, Hilbert’s Nullstellensatz implies the abstract one in a model of Axiomatic Cohesion ${p:\cal E \rightarrow \cal S}$ where $\cal S$ is the Galois topos of $k$.
Carles Noguera (Artificial Intelligence Research Institute IIIA – CSIC, Spain)
Nonassociative substructural logics: alternative axiomatization, algebraic and logical propertiesJoint work with Petr Cintula , Zuzana Haniková, Rostislav Horcík. Galatos and Ono have studied in a recent paper a nonassociative version of full Lambek calculus, which we call here $\mathrm{SL}$. They have introduced a Gentzenstyle and a Hilbertstyle calculus for $\mathrm{SL}$ and proved that it is algebraizable and its equivalent algebraic semantics is the variety of latticeordered residuated unital groupoids.On the other hand, Cintula and Noguera have developed (in Handbook of Mathematical Fuzzy Logic, College Publications, 2011) a general algebraic framework to deal with substructural logics with $\mathrm{SL}$ as the base logic. They introduced the notion of almost $\mathrm{MP}$based logic (a logic with a Hilbertstyle presentation where modus ponens is the only binary rule, there are no rules with more than two premises, and all unary rules are of the form $\varphi \vdash \delta(\varphi)$, for $\delta \in \mathrm{DT}$, where the set of terms $\mathrm{DT}$ satisfies some desired properties) and proved that every almost $\mathrm{MP}$based substructural logic enjoys a local deduction theorem and a certain form of proof by cases property. The main associative substructural logics are almost $\mathrm{MP}$based. However the problem was left open for $\mathrm{SL}$ and other nonassociative logics. In this talk we positively solve it. We present an alternative Hilbertstyle axiomatization of $\mathrm{SL}$ which allows to show that it is almost $\mathrm{MP}$based, and hence the same holds for all its axiomatic extensions. We obtain several interesting logical and algebraic consequences of this fact: a form of the local deduction theorem, a description of intersection of filters and of the filter generated by a set, an axiomatization of intersection of two axiomatic extensions of a given logic, and equational bases of varieties of $\mathrm{SL}$algebras generated by positive universal classes of $\mathrm{SL}$algebras.
Finally, as another byproduct of our alternative Hilbert system for $\mathrm{SL}$, we show how to axiomatize the logic of linearly ordered residuated unital groupoids.
Arnold Oostra (Universidad del Tolima, Colombia)
A lattice of Intuitionistic Existential Graphs systems (slides)Early in the 20th century Charles S. Peirce introduced a system of diagrams called by himself Existential Graphs which, among other features, provides a totally graphic method for Classical Logic at different levels: Propositional, First Order and some Modal Logics. A very slight modification of Peirce’s original system renders Existential Graphs for Intuitionistic Logic at the same levels. In this talk we show a series of Existential Graphs systems for various segments of Intuitionistic Propositional Calculus and also for all finitely axiomatizable Intermediate propositional logics. Besides the intrinsic interest of these graphic representations (and their First Order and Modal extensions), it is worth mentioning they provide novel axiomatizations for Boolean algebras, Heyting algebras and some of their generalizations.Ramón Pino Pérez (Universidad de los Andes, Venezuela)
An improvement to the revision operatorsThe aim of this talk is to give a presentation of improvement operators. These operators are close to belief revision operators introduced by Alchourrón Gärdenfors and Makinson in the seminal paper of Journal of Symbolic Logic of 1985. One important difference between our improvement operators and the classical ones in the AGM framework lies in the postulate of success: “the new piece of information has to belong to beliefs after revision”. Unlike the classical revision operators, our operators do not satisfy this postulate. However they incorporate the new piece of information in a mild manner. In some sense the “degree of plausibility” of the new piece of information improves. We will show that our operators are well adapted to treat the problem of minimal change and that their iterated version has a behavior like the classical AGM revision operators.Contributed Talks (Algebraic and NonClassical Logics)
Rodolfo C. Ertola Biraben (Universidade Estadual de Campinas, Brazil)
On a Problem posed by LópezEscobarIn 1985, LópezEscobar posed the following problem: “Suppose that $S$ is a schema, essentially involving quantifiers, such that $S$ is provable in the Classical Predicate Calculus, but not in the Intuituionistic Predicate Calculus, $IPC$. Then, is there a sentential connective $\oplus$ (with associated rules) so that the Intuituionistic Sentential Caclulus ($ISC$) $+ \oplus$ is a conservative extension of $ISC$ and $IPC + \oplus \vdash S$?” Moreover, he mentioned, as a possible example for $S$, the formula $\forall x \neg \neg P x \rightarrow \neg \neg \forall x P x$. This formula very well known not to be derivable in $IPC$, is sometimes known as the Kuroda Formula.We will talk about solutions of the given problem in the case of some formulas such as Kuroda’s and relate those solutions to the issue of intuitionistic connectives.
Hugo Luiz Mariano (University of São Paulo, Brazil)
Analysing categories of signatures(Joint work with Caio de Andrade Mendes.) We analyse two categories of finitary signatures underlying to (propositional) logics: the categories $\mathcal{S}_s$ and $\mathcal{S}_f$. $\mathcal{S}_s$ has a very simple notion of morphism, but it is too strict: $\mathcal{S}_s$ has good categorial properties (is a complete/cocomplete $\omega$accessible category), but does not allow a good treatment of the identity problem for logics. $\mathcal{S}_f$ has a more flexible notion of morphism: it allows a better treatment of the identity problem for logics but, on the other hand, $\mathcal{S}_f$ has serious categorial defects. We define a pair of (faithful) functors $(+) : \mathcal{S}_s \rightarrow \mathcal{S}_f$ and $(): \mathcal{S}_f \rightarrow \mathcal{S}_s$, such that $(+)$ is left adjoint to $()$. We consider the (endo)functor in $\mathcal{S}_s$, $T := () \circ (+)$ and we prove that $T$ preserves filtered colimits and reflects isos/epis/monos. We consider the monad (or triple) canonically associated to this adjunction, $\mathcal{T} = (T, \mu, \eta)$, and we prove that $\mathcal{S}_f = Kleisli(\mathcal{T})$: this result entails that the category of logics $\mathcal{L}_f$ built over $\mathcal{S}_f$ has: unconstrained fibrings, i.e.\ coproducts, and “constrained” fibrings, e.e.\ colimits with base diagram “in” $\mathcal{S}_s$ (i.e., obtained via $(+): \mathcal{S}_s \rightarrow \mathcal{S}_f$). We finish the work with some additional information on isomorphisms, sections, retractions and idempotents in $\mathcal{S}_f$.Ivan Martínez (Benemérita Universidad Autónoma de Puebla, México)
Semantics for some nonclassical possibilistic logics(Joint with Jose Arrazola y Rubén Vélez.) In the mid 80′s, Dubois and Prade introduced Possibilistic Logic, a logic in which a parameter in (0, 1] (or more generally, in a bounded and totally ordered set) is associated to each classical formula. Such a parameter represents the degree of certainty of the corresponding formula. This logic is based on classical logic and it has been useful in modeling problems where incom plete or partially contradictory information exists. Also, the semantics and syntactic aspects of Possibilistic Logic are established, as well as a theorem on soundness and completeness. Subsequently, other authors have extended these concepts involving different logics, for example, Description Logics and some intermediate logics. We consider relevant to develop Possibilistic Intuitionistic Logic and Possibilistic Cω; Logic, to study their syntactic properties and to give them suitable semantics, as well as the feasibility to develop the same study with elements of their corresponding lattices.In this talk, we first review an axiomatic system for Possibilistic Intuitionistic Logic and we present one for Possibilistic Cω Logic as well as some syntactic properties and we give them suitable semantics by extending
semantics that have already been defined for Intuitionism and Cω Logic.Petr Cintula (Academy of Sciences of the Czech Republic)
Semilinear nonassociative substructural logics: completeness properties and complexity(Joint with Zuzana Haniková, Rostislav Horcík, Carles Noguera.) As presented in Handbook of Mathematical Fuzzy Logic, fuzzy logics can be seen as a particular subfamily of substructural logics whose defining property is completeness with respect to linearly ordered algebras, also called chains. Following the terminology introduced in a previous paper by Cintula and Noguera, these logics and their algebraic counterparts are called semilinear, because the subdirectly irreducible members of the corresponding classes of algebras are chains. Some of these logics (e.g. Łukasiewicz or GödelDummett logic) even enjoy ‘better’ forms of completeness with respect to special classes of chains, such as those with domain being the real or rational unit interval or a finite set.In this talk we study semilinear nonassociative substructural logics. In another talk we show how these systems can be axiomatized; now we focus on their completeness properties and their complexity. Our base logic is the semilinear extension of the nonassociative full Lambek calculus $\mathrm{SL}$ studied by Galatos and Ono (Cut elimination and strong separation for substructural logics. We call it $\mathrm{SL}^\ell$, the logic of totally ordered residuated unital groupoids; moreover we consider for every set of axioms corresponding to structural rules, $\mathrm{S}\subseteq\{\mathrm{e},\mathrm{c},\mathrm{i},\mathrm{o}\}$, its associated semilinear logic $\mathrm{SL}^\ell_\mathrm{S}$. We prove that $\mathrm{SL}^\ell_\mathrm{S}$ is complete with respect to the class of all countably infinite dense $\mathrm{SL}_\mathrm{S}$chains, and with respect to the class of all $\mathrm{SL}_\mathrm{S}$chains on $[0,1]$. Moreover we prove the finite embeddability property for $\mathrm{SL}$chains and so we obtain completeness of $\mathrm{SL}$ with respect to the class of finite $\mathrm{SL}$chains. Finally, we show that the equational theory (and even the universal theory) of semilinear latticeordered residuated unital groupoids is coNPcomplete.
Hernando Gaitán (Universidad Nacional de Colombia)
Endomorfismos de álgebras de HilbertUn álgebra de Hilbert es una estructura $\mathbf{H} =\langle H, \rightarrow, 1 \rangle$ de tipo $(2,1,0)$ que satisface, para todo $a, b, c \in H$ lo siguiente: • $a \rightarrow (b \rightarrow c) = 1$;
 • $(a \rightarrow (b \rightarrow c)) \rightarrow ((a \rightarrow b) \rightarrow (a \rightarrow c)) = 1$;
 •$a \rightarrow b = 1$ and $b \rightarrow a = 1$ implica $a = b$.
Un álgebra de Tarski es un álgebra de Hilbert que satisface la identidad $(x \rightarrow y) \rightarrow x \approx x$: Las álgebras de Hilbert son la contraparte algebraica del fragmento implicativo de la lógica intuicionista de proposiciones así como las álgebras de Tarski son la contraparte algebraica del fragmento implicativo de la lógica de proposiciones clásica.
Está demostrado que las álgebras de Tarski finitas quedan completamente determinadas por el monoide de sus endomorfismos. En esta charla se discutirá la cuestión de que tanto tienen en común dos algebras de Hilbert finitas que tienen monoides de endomorfismos isomorfos.
Celimo Alexander Peña (Universidad del Valle, Colombia)
Espacios uniformes y MVálgebrasSe presentan dos resultados que relacionan los espacios uniformes y las MVálgebras:• Una representación dual entre la categoría de los espacios uniormes Hausdorff compactos y la categoría de las MVálgebras semisimples, para esto se hace uso de la correspondencia que existe entre la estructura topológica y la estructura uniforme en el caso Hausdorff compacto.
• La definición de una uniformidad para toda MVálgebra a partir de la función distancia, la cual está estrechamente ligada a la cantidad de ideales de la MVálgebra.
Yuri Poveda (Universidad Tecnológica de Pereira, Colombia)
The intimate relationship between the McNaughton and the Chinese theoremsWe will show the intimate relationship between McNaughton Theorem and the Chinese Theorem for MValgebras. We develop a very short and simple proof of McNaughton Theorem. The arguing is elementary and right out of the definitions. We exhibit the theorem as just an instance of the Chinese theorem. Since the variety of MValgebras is arithmetic, the Chinese theorem holds for MValgebras. However, to stress how elementary is our proof of McNaughton’s, we will also show a simple proof of the Chinese Theorem for MValgebras. We will not assume any prior knowledge of MValgebras by the audience.This is joint work with Eduardo J. Dubuc.
Juan Ricardo Prada (Universidad del Tolima)
Gráficos existenciales Alfa y teoría de categoríasLos gráficos existenciales, propuestos hace un poco más de cien años por el lógico norteamericano Charles Sanders Peirce, se analizan desde la perspectiva de la teoría de categorías considerando a los gráficos existenciales Alfa como un conjunto preordenado con el fin de estudiar sus propiedades categóricas y sus principales características, encontrando así una estrecha relación con las categorías monoidales, las categorías simétricas, las categorías cerradas, las categorías $*$autónomas y las categorías enriquecidas. Dichas categorías se estudian porque presentan una relación cercana con las reglas de transformación de los gráficos existenciales Alfa, con la definición de iteración para los gráficos en el contexto categórico, la definición de fuerza, y la definición de borramiento, nociones que son fundamentales para determinar las reglas de transformación Alfa generalizadas.Hernán San Martín (Universidad Nacional de La Plata, Argentina)
Remarks about the coproduct in some subvarieties of the variety of Heyting algebras with successor(Joint work with J.L. Castiglioni) Kuznetsov introduced an operation on Heyting algebras
as an attempt to build an intuitionistic version of the provability logic of GödelLöb, which formalizes the concept of provability in Peano Arithmetic. This unary operation, which we shall call successor, was also
studied by Caicedo, Cignoli and Esakia. In particular, it was considered as an example of an implicit compatible operation on Heyting algebras.A $S$algebra is a Heyting algebra endowed with its successor function, when it exists. Let $n$ be a natural number. We say that a $S$algebra $H$ has height $n$ if $S^{(n)}(0)= 1$ and $n$ is the minimum natural number with this property, and we write $\mathcal{S}\mathcal{H}_n$ for the class of $S$algebras of height less or
equal to $n$. This class is a variety with defining the equations of $S$algebras together with the additional equation $S^{(n)}(0)= 1$.In this talk we shall give a description of the coproduct in $\mathcal{S}\mathcal{H}_n$ between two algebras $H_1$ and $H_2$, with $H_1$ or $H_2$ satisfying prelinearity.
Pedro Sánchez Terraf (Universidad Nacional de Córdoba, Argentina)
Logics for Markov decision processes (slides)Larsen and Skou introduced a modal logic, based on HennessyMilner’s, to describe behavior for Probabilistic Transition Systems. Later Desharnais et al. proved that already a fragment of that logic is complete for bisimilarity on the broad class of labelled Markov processes (LMP) over analytic spaces, whereas Larsen and Skou only considered discrete systems.We extended the LMP model with nondeterminism and studied bisimulations. We developed a logic complete for bisimilarity over analytic spaces under the assumption of imagefiniteness.
In this talk I will survey these results, some (counter)examples and open problems concerning the new model and its logic.
This is joint work with Pedro D’Argenio and Nicolás Wolovick.
Mauricio Tellechea (Universidad Nacional de Córdoba, Argentina)
Algebraically expandable classes of $D_{01ab}$(Join with Pablo Celayes ) Lattices are algebraic structures ubiquitous in the field of logic. In this talk we will study some aspects of the structure of the variety $D_{01ab}$ of bounded distributive lattices with two constants in relation to the problem of characterizing the algebraically expandable subclasses of this variety.An algebraically expandable (AE) class is a class of algebras of the same type axiomatizable by sentences of the form $\forall \vec{x} \exists ! \vec{z} \bigwedge p_i(\vec{x}, \vec{z} = q_i (\vec{x}, \vec{z})$. These classes are interesting because it is possible to define new operations in the algebras of the class, apart from those given by the function symbols of the type.
Let $F$ be the subclass of all members of $D_{01ab}$ with two or three elements. Since sentences of the form $\forall \vec{x} \exists ! \vec{z} \bigwedge p_i(\vec{x}, \vec{z} = q_i (\vec{x}, \vec{z})$ are preserved by sheaves and every member of $D_{01ab}$ can be represented as a global subdirect product with factors in $F$, a first step in the above described characterization problem is describing the AE subclasses of $F$. We will give some advances in connection with this problem; in particular, we will give several interesting algebraically expandable subclasses of $D_{01ab}$ arising from the corresponding subclasses of $F$.