The Seminar on Applied Mathematical Logic (a.k.a. the Hájek Seminar) usually takes place on Wednesdays 4pm in the lecture room 318 of the Institute of Computer Science (Pod Vodárenskou věží 2, Prague 8. See the map). The seminar hosts talks on recent work in mathematical logic and applications by members of the Prague logic community and guests.

Due to the Covid-19 pandemic, the seminar will be held online via Zoom until further notice.

In spring 2021, the SAML will organize joint sessions with the seminar of the Bern Logic Group (logicians at the Institute of Compuer Science and the Mathematical Institute of the University of Bern). Note that the Zoom streaming details for the Bern-Prague joint sessions may differ from the usual SAML streaming details.

For more information on ICS seminars, take a look at the seminar section of the ICS webpages.

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Upcoming talks

• 16.06.2021
  Norihiro Yamada (University of Minnesota): Dependent types and finite limits in games

  Game semantics is a particular class of mathematical (or denotational) semantics of logic and computation, which interprets types (or formulas) by (debate) games between a prover and a disprover, and terms (or proofs) by strategies for the prover to win the games. By its distinctive intensionality, it has been a highly powerful tool for the study of logic and computation, and recently extended to a mathematical foundation of the dynamic aspects of logic and computation such as normalisation (or cut-elimination), algorithms and higher-order computability. One of the most challenging problems in game semantics is to interpret dependent types (or predicate logic). In fact, there has been no established solution for this problem for more than 25 years. Another, related problem is that existing game semantics does not have all finite limits. In this talk, I sketch my game semantics of dependent types and show that it has all finite limits. Rather than the technical details, I focus on: 1. Why game semantics is useful for the study of logic and computation; 2. Why game semantics of dependent types is difficult; 3. Main idea for the solution; 4. Ongoing and future work. A preprint is available: https://arxiv.org/pdf/1905.00993.pdf.

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Recent talks

• 26.05.2021
  Grigory Olkhovikov (Ruhr University Bochum): A maximality result for bi-intuitionistic propositional logic

  I will report on a recent Lindström theorem for bi-intuitionistic propositional logic (joint work with Guillermo Badia) showing that this logic is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property, and preservation under bi-asimulations. The result constitutes an extension of previous work done for the propositional intuitionistic logic in: G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11-30 (2020). However, due to the presence of a backwards-looking connective in bi-intuitionistic logic, the current result features a number of non-trivial modifications of the techniques and ideas employed in the previous work.



• 19.05.2021 [Bern-Prague Joint Seminar]
  Wesley Fussner (University of Nice Sophia Antipolis): Generalized basic logic from a modal point of view

  Generalized basic logic was introduced through its algebraic semantics (GBL-algebras) in order to provide a natural common generalization of lattice-ordered groups, Heyting algebras, and BL-algebras. When formulated with exchange, weakening, and falsum, generalized basic logic is a fragment of both Hájek's basic logic and propositional intuitionistic logic. In this formulation, the relationship between generalized basic logic and Łukasiewicz logic parallels the thoroughly-studied relationship between intuitionistic logic and classical logic. This talk explores several ways that the latter parallel manifests. First, we illustrate a relational semantics for generalized basic logic where worlds are valued in MV-algebras (analogous to the usual, Boolean-valued Kripke semantics for intuitionistic logic). Second, we present a translation of generalized basic logic into a modal Łukasiewicz logic that is analogous to the Gödel-McKinsey-Tarski translation of intuitionistic logic into the classical modal logic S4. All of these results are obtained with the help of the algebraic theory of GBL-algebras, and we also provide a brief survey of the latter.



• 05.05.2021
  Shawn Standefer (Slovak Academy of Sciences): Varieties of necessity in a non-classical setting

  In standard modal logics, there are three common conceptions of necessity:, the universal conception, the equivalence relation conception, and the axiomatic conception. theses provide distinct presentations of the modal logic S5, commonly used in metaphysics and epistemology. In standard settings, these presentations coincide, giving three views of a single, unified logic. I will explore these different conceptions in the context of the relevant logic R, explaining when they come apart and why that matters. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for relevant logics while the former is not. For the committed relevant logician, necessity cannot be truth in all possible worlds.



• 28.04.2021 [Bern-Prague Joint Seminar]
  Iris van der Giessen (Utrecht University): The admissible rules of Lax Logic

  Propositional Lax Logic is a fascinating intuitionistic modal logic with a non-standard modality that combines some properties of a ‘Box’ and some properties of a ‘Diamond’. In this talk I will present recent results about its admissible rules. The admissible rules are those rules under which the set of theorems of a logic is closed. Thereby they give insight in the structure of all possible inferences in a logic. Iemhoff (2001) showed that the so-called Visser rules form a basis for the admissible rules of IPC. Similarly, modal Visser rules are formulated for modal logics such as K4, S4 and GL (Jeřábek 2005). I will characterize a sequent-based proof system for the admissible rules of propositional Lax Logic, containing Visser-like rules. In this work we will see that the structure of the relational semantics will be of great importance.



• 21.04.2021 [Bern-Prague Joint Seminar]
  Federico Faroldi (Ghent University and University of Salzburg): The Structure of Reasons: Subtraction and Partiality

  Practical reasons are central both in everyday normative reasoning and in normative theorizing, but most accounts treat them as atomic and flat. In this talk, I investigate the structure of practical reasons in order to deal with aggregation, double counting, subtraction, and partiality. The ideal aim is to give a unified formal account that is able to serve as a semantic backdrop to construct natural logical systems to reason with reasons, based on a hyperintensional justification logic with a truthmaker semantics.



• 14.04.2021
  Hitoshi Omori (Ruhr University Bochum): Two applications of Herzberger’s semantics

  In his paper "Dimensions of truth", Hans Herzberger develops a semantic framework that captures both classical logic and weak Kleene logic through one and the same interpretation. The aim of this talk is to apply the simple idea of Herzberger to two kinds of many-valued semantics. This application will be led by the following two questions: (i) Is de Finetti conditional a conditional? (ii) What do CL, K3 and LP disagree about? Note: This is a joint work with Jonas R. B. Arenhart.



• 18.03.2021 (Thursday) [Joint session with the Logic Seminar of the Institute of Philosophy, CAS]
  Berta Grimau and Carles Noguera (The Czech Academy of Sciences): These Degrees go to Eleven: Fuzzy Logics and Graded Predicates

  In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness --largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature-- are logically nonclassical and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of ordinary language constructions such as the positive and comparative forms of gradable adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. In this talk we will argue that the fuzzy kind of theory can match the benefits of the classical kind but not vice versa. We develop a new version of the fuzzy logic approach that --unlike existing fuzzy theories-- yields compelling analyses of ordinary language constructions such as the positive and comparative forms of gradable adjectives, while retaining the advantage of genuinely solving the sorites paradox. At the same time we will argue that a bivalent, classical approach to vague predicates cannot form the basis for an equally convincing solution to the sorites. As an overall conclusion we will defend that the nonclassical, fuzzy kind of degree theory is superior. (Joint work of Petr Cintula, Berta Grimau, Carles Noguera, and Nicholas J.J. Smith)



• 10.03.2021 [Bern-Prague Joint Seminar]
  Brett McLean (Université Nice Sophia Antipolis): Temporal logic of Minkowski spacetime

  If we wish to do temporal logic on (flat) spacetime, special relativity suggests we should use an accessibility relation that is independent of the choice of inertial frame, and that there are a limited number of ways to do this. Two possible accessibility relations are 'can reach with a lightspeed-or-slower signal' and 'can reach with a slower-than-lightspeed signal'. We focus on the resulting frames in 2D spacetime (1 space + 1 time dimension). For both frames, validity of formulas in the basic temporal language is a PSPACE-complete problem. I will describe the proofs of these results and also how those proofs can be extended to obtain results on interval logics. This is joint work with Robin Hirsch. The lightspeed-or-slower case is due to Hirsch and Reynolds.



• 24.02.2021 [Bern-Prague Joint Seminar]
  Luca Reggio (University of Oxford): Counting Homomorphisms Between Finite Structures

  Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite relational structure C. Categorical generalisations of this result were proved independently in the early 1970s by Lovász and Pultr. I will present another categorical variant of Lovász' theorem and explain how it can be used, in combination with the game comonads recently introduced by Abramsky et al., to obtain homomorphism counting results in finite model theory. This is joint work with Anuj Dawar and Tomáš Jakl.



• 17.02.2021
  Igor Sedlár (The Czech Academy of Sciences): Changing the World, Constructively

  The finite tree property of intuitionistic logic entails completeness with respect to posets where each element, seen as a possibly partial situation, is under a maximal element, seen as a possible world containing the situation. This suggests a natural semantics for intuitionistic modal logic based on posets with a binary relation on the set of maximal elements. In this semantics, truth of modal formulas in a situation is determined by looking at worlds containing the situation and worlds accessible from them. In this paper we study modal logics arising from such a semantics. A general completeness-via-canonicty result is provided and various operations on such posets including filtrations are studied. Differences with respect to intuitionistic modal logics known from the literature are discussed. In the final part a completeness result for a version of intuitionistic propositional dynamic logic based on the framework is obtained and the logic is shown to be decidable.



• 03.02.2021
  Guillermo Badia (University of Queensland): 0-1 Laws in Mathematical Fuzzy Logic

  The 0-1 law for classical first-order logic states that, in a relational vocabulary, every formula is almost always true or almost always false on finite models. This theorem is due to Ronald Fagin and was proved in the 1970s building up on work started by Carnap. Given the failure of traditional model-theoretic properties such as compactness on finite models, it was quite remarkable to find a native of the finite setting using probabilistic techniques. In this talk, I will generalize the classical theorem to a many-valued context in the following form: for every formula there is a truth-value that the formula takes almost always or almost never on finite models. The new result will cover the cases of finitely-valued fuzzy logics such as Łukasiewicz, Gödel-Dummet and Product logics (and, of course, Boolean logic as an extreme case). This work also generalizes a theorem obtained in a more limited setting for the case of some Łukasiewicz logics by Robert Kosik and Chris Fermüller. (Joint work with Carles Noguera)



• 16.12.2020
  Jamie Wannenburg (University of Pretoria): Some algebraic (and topological) tools and their application to logic

  The talk will be a non-technical overview of the speaker’s research interests, for the purpose of introducing the researcher to the working group. The main areas of interest are Abstract Algebraic Logic, Universal Algebra, and Substructural Logic—with a focus on Relevance and Intuitionistic Logics. But the talk will focus on a handful of specific tools from these areas, ex. Birkhoff’s Theorem, Jónsson's Theorem, some Bridge theorems and Esakia Duality, emphasising their applications and giving examples from the speaker's previous research.



• 02.12.2020
  Adam Přenosil (Vanderbilt University): Logics of n-filters

  Classical logic is defined by truth-preservation in Boolean algebras: if the premises take the top value, then so does the conclusion. Dually, one might be interested in logics defined by the preservation of non-falsity: if the conclusion takes the bottom value, then so does one of the premises. Such logics are paraconsistent: in the four-element Boolean algebra both x and its negation may be non-false. In addition, the familiar rule of adjunction fails here: we may not infer that the conjunction of x and y is non-false given that x and y are non-false. In this talk, we shall study logics of non-falsity and other logics generated by upsets of Boolean and De Morgan algebras. In particular, we show that each finitely generated extension of the four-valued logic of Belnap and Dunn (FDE), admits a completeness theorem with respect to a finite Gentzen calculus, though not always a finite Hilbert calculus.



• 18. 11. 2020
  Kentaro Yamamoto (ICS CAS): The Combinatorics of Finite Heyting Algebras and the Topological Group of The Automorphisms of Their Limit
  

  There are several results linking combinatorial properties of Fraïssé classes, certain classes of structures having the amalgamation property, and topological properties of the automorphism groups of their limits. In this talk, the speaker's work in progress on one instance of research in this vein, which is on the Fraïssé class of finite Heyting algebras will be presented. This class is not uniformly locally finite unlike most other examples considered in this vein of research. It will be shown that our automorphism group is non-isomorphic to existing examples and that it is non-amenable. At the end of the talk, future research ideas pertaining to the Ramsey property of the class and, equivalently, the extreme amenability of the automorphism group will be discussed.



• 04.11.2020
  Tommaso Moraschini (Uni. of Barcelona): On Equational Completeness Theorems
  

  A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence of some class of algebras. Even if the vast majority of completeness theorems in the literature are of this form, it is known that there are logics lacking any equational completeness theorem. Despite the simplicity of this notion, intrinsic characterizations of logics with admitting an equational completeness theorem have proved elusive. This is partly because equational completeness theorems can take unexpected forms, e.g., in view of Glivenko's Theorem, classical propositional logic is related to the variety of Heyting algebras by a (certainly nonstandard) equational completeness theorem. As it happens, nonstandard equational completeness theorems of this form are ubiquitous.

  In this talk, we present a characterization of logics with at least one tautology (resp. locally tabular logics) admitting an equational completeness theorem. For a protoalgebraic logic, this amounts to the demand that there are two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem will be shown to be decidable for logics presented by a finite set of finite matrices and for locally tabular logics presented by a finite Hilbert calculus, we shall see that it becomes undecidable for arbitrary logics presented by finite Hilbert calculi. The undecidability result persists even if we restrict to equivalential logics.

  A draft collecting these observations is available online at http://uivty.cs.cas.cz/~moraschini/files/submitted/equational.pdf

• 21.10.2020
  Igor Sedlár (ICS CAS): Generalizing the Completeness Argument for Propositional Dynamic Logic
  The standard argument establishing weak completeness of Propositional Dynamic Logic with respect to Kripke models relies on the presence of Boolean negation in the language and produces a finite counter-model for each non-theorem. Both of these features make the standard argument inapplicable to versions of PDL based on specific non-classical logics, namely, those without Boolean negation and/or without the finite model property. For instance, the former applies to PDL based on intuitionistic and many superintuitionistic logics, the latter applies to PDL combined with Boolean Lambek calculus and both apply to PDL based on some relevant logics. In this talk, I will outline a generalization of the standard completeness argument that covers this wider class of propositional dynamic logics.


• 14.10.2020
  Andrew Tedder (ICS CAS): Decidability and Consistency in Some Paraconsistent Arithmetics
  Following on some joint work with Badia, Cintula, and Hájek, I'll show that two arithmetic theories couched in first order extensions of the three-valued paraconsistent logics A3 and RM3 are essentially undecidable (i.e. any $\Sigma_1$-consistent extension/expansion of these is undecidable). With this in hand, I'll consider questions of decidability concerning some $\Sigma_1$-inconsistent extensions of these theories. I'll conjecture that the theory in question has some $\Sigma_1$-inconsistent (but non-trivial) extensions which are decidable, and some which are undecidable, suggesting candidates for these extensions from among inconsistent models of arithmetic.


• 27.05.2020
  Vít Punčochář (FLU CAS and ICS CAS): Inquisitive Heyting algebras
  In my talk I will explore a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. I will show that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. I will explain how questions are represented in these structures (prime elements represent declarative propositions, non-prime elements represent questions, join is a question-forming operation) and provide several alternative characterizations of these algebras. For instance, it will be shown that a Heyting algebra is inquisitive if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum-Tarski algebra is an inquisitive Heyting algebra.


• 13.05.2020
  Petr Cintula (ICS CAS): How much Propositional Logic Suffices for Rosser’s Essential Undecidability Theorem?
  Abstract: In this talk we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It it well known for intuitionistic logic and Robinson's Q, and Petr H\'{a}jek proved it in fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.


• 29.04.2020
  Igor Sedlár (ICS CAS): Finitely-valued Propositional Dynamic Logic (online)
  Absstract: We study a many-valued generalization of Propositional Dynamic Logic where both formulas in states and accessibility relations between states of a Kripke model are evaluated in a finite residuated lattice. One natural interpretation of this framework is related to reasoning about costs of performing structured actions. We prove that PDL over any finite residuated lattice is decidable. We also establish a general completeness proof for a class of PDLs based on commutative integral residuated lattices.


• 26.02.2020
  Sajad Nazari (Laboratoire d'Informatique Fondamentale d'Orléans): Rough Concepts
  Abstract: In this talk, we discuss a logical and algebraic framework unifying Rough Set Theory and Formal Concept Analysis, two influential theories in information science. We also discuss some applications and extensions of this framework to the analysis of vagueness and uncertainty in categorization.


• 12.02.2020
  Michał Stronkowski (Warsaw University of Technology): Admissibility in the multi-conclusion setting
  Abstract: Let $\vdash$ be a (finitary, structural) consequence relation. An inference rule $\Gamma/\varphi$ is admissible for $\vdash$ if adding $\Gamma/\varphi$ to $\vdash$ does not change the set of theorems. The notion of admissibility in the single-conclusion setting is already well understood and its characterizations are known. The situation changes in the multi-conclusion setting. There inference rules with a finite set of formulas in the conclusion are allowed. It appears that then the notion of admissibility splits into a number of nonequivalent ones. I will discuss them and provide their algebraic characterizations. (This work is parallel to Iemhoff's proof-theoretic investigation on admissibility.)


• 22. 01. 2020
  Guillermo Badia (University of Queensland): Ehrenfeucht-Fraisse methods in the model theory of L-topological spaces over finite MTL-chains
  Abstract:Lattice-valued topological spaces were introduced by Goguen in the 1970s, as a generalization of Chang's fuzzy topological spaces. The intuitive idea is simply to study topologies where the open sets are "fuzzy" or lattice-valued, instead of crisp. Second-order languages to study topological spaces have been studied for classical logic in the past, so in this talk I'll introduce a second-order language over lattice-valued structures to study lattice-valued topological spaces. I will present some of the model-theoretic properties of said language, focusing on characterizations of expressivity via an Ehrenfeucht-Fraisse theorem. Some technical restrictions are necessary to get a well-behaved general model theory here: the lattice must be finite. In this talk, I consider only MTL algebras, but it is possible that this further restriction can be relaxed.


• 11. 12. 2019
  Amanda Vidal (ICS CAS): Axiomatic systems of Gödel Modal Logics
  Abstract: In this seminar we will go over the main results from the literature concerning Gödel modal logics, understood as the logics arising from prominent distinguished classes of Kripke models valued over the standard Gödel algebra. Known results cover axiomatizing the box and the diamond fragments of the logic both over the most general class of valued models (MG), and over the ones where the underlying frame is classical, and the logic with both modal operators over the class MG. We will then approach the problem of axiomatizing the bi-modal logic over the class of models with underlying classical frames, and propose an axiomatization relying in Dunn's axiom of positive modal logic. We will see some consequences that arise from the technical details of the completeness proof, and if time allows, the relation of these logics with some modal Intuitionistic logics from the literature.


• 04. 12. 2019
  Martin Blicha (Charles University), Craig interpolation in software verification
  Abstract: Craig interpolation is a logical concept popularized in formal verification community by McMillan (2003). Since then, several successful verification techniques based on interpolation have been developed and interpolation remains one of the dominant approaches for proving safety. We will outline the use of interpolants for computing invariants of program loops, show how interpolants are computed in modern SMT solvers from proof of unsatisfiability, and finally present our recent result regarding computation of interpolants for systems of linear inequalities over rational numbers.


• 20. 11. 2019
  Andrew Tedder (ICS CAS): Information flow in logics in the vicinity of BB
  Abstract: B is the basic logic of the Routley-Meyer semantic framework, and BB that of the `neighbourhood style' RM semantics. The RM semantics invites a number of motivational stories, the one of interest here being that of situation semantics along with a channel-theoretic reading of the central ternary relation - according to this reading, a situation supports some conditional information when it facilitates information flow between situations supporting the antecedent of the conditional and situations supporting the consequent. With this story it is easy to make sense of the use of the ternary relation in the truth conditions of conditional formulas in terms of the application of a channel to some input, yielding some output. Part and parcel with the channel-theoretic reading of the RM semantics, however, are some natural actions performable on channels, and some constraints on those actions. The key action of interest here is the serial composition of channels. Elsewhere I have motivated a particular method for cooking up composites out of pairs of channels, and have shown that a simple fragment of B is complete w.r.t. the class of models in which such points exist (so the logic supports a robust channel-theoretic reading). In this talk, I refine that argument a bit by working in the neighbourhood style RM semantics, and discuss some extensions of my previous results, and some (more or less severe) limitations on the use of the method for further logics. I end with some further questions, and some reflections on situation theoretic interpretations of the RM semantics more generally.


• 13. 11. 2019 [Room 419]
  Sabine Frittella (Laboratoire d’Informatique Fondamentale d’Orléans): Probabilistic Dynamic Epistemic Logic
  Abstract: First, I will present a paper on Probabilistic Epistemic Updates on Algebras (joint work with Willem Conradie, Alessandra Palmigiano, Apostolos Tzimoulis and Nachoem Wijnberg, link: https://arxiv.org/abs/1612.05267). This paper contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. Thanks to this construction, an interpretation of the language of PDEL can be defined on algebraic models based on Heyting algebras. This justifies our proposal for the axiomatization of the intuitionistic counterpart of PDEL, of which we prove soundness and completeness with respect to algebraic probabilistic epistemic models.
  Then, I will present a starting research project on Dynamic Epistemic Logic applied to privacy. In a nutshell, we aim at characterising the knowledge an attacker can infer on a user/client/citizen given its prior knowledge on the world and the personal data he collects on the user. Since a large part of the relevant knowledge the attacker can infer is based on statistical knowledge on the population, one big challenge is to describe statistical knowledge and reasoning based on that type of knowledge.


• 06. 11. 2019
  Neil Thapen (Math. CAS): Random resolution and approximate counting
  Abstract: Resolution is a propositional proof system that is bad at formalizing arguments involving counting, even approximately. I will discuss some upper and lower bounds on random resolution, a system which, roughly speaking, adds to resolution the ability to use any axiom, as long as it is true most of the time. This is connected to a result about a related first-order proof system of bounded arithmetic with approximate counting.


• 30. 10. 2019
  Tommaso Moraschini (ICS CAS): Profinite Heyting algebras and the representation problem for Esakia spaces
  Abstract: A poset is said to be "representable" if it can be endowed with an Esakia topology. Gratzer's classical representation problem asks for a description of representable posets which - unfortunately - is not expected to take a simple form, as these do not form an elementary class. As at the moment a solution to the representation problem seems out of reach, we address a simpler version of the problem which, roughly speaking, asks to determine the posets that may occur as top parts of Esakia spaces. Crossing the mirror between algebra and topology, this task amounts to characterize the profinite Heyting algebras that are also profinite completions. We shall report on the on-going effort to solve this problem by understanding the structure of varieties of Heyting algebras whose profinite members are profinite completions. This talk is based on joint work with G. Bezhanishvili, N. Bezhanishvili, and M. Stronkowski.


• 23. 10. 2019
  Michał M. Stronkowski (Warsaw Uni. of Technology): Profiniteness and finitely generated varieties
  Abstract: Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety V of algebras is standard if every Boolean topological algebra with the algebraic reduct in V is profinite. We show that there is no algorithm that takes as input a finite algebra A (of a finite type) and decides whether the variety generated by A is standard. We also show the undecidability of some related properties. In particular, we solve the problem posed by Clark, Davey, Freese, and Jackson about finitely determined syntactic congruences in finitely generated varieties. Our work is based on Moore's theorem about undecidability of having definable principal subcongruences for finitely generated varieties. Joint work with Anvar M. Nurakunov.


• 9. 10. 2019
  Tadeusz Litak (Friedrich-Alexander Uni. Erlangen-Nürnberg): Loeb constructively meets μ
  Abstract: A famous result of Solovay identifies the classical modal Loeb system (KL) as the provability logic of Peano Arithmetic. Under this perspective, Goedel’s Second Incompleteness Theorem and Loeb’s Theorem leave an algebraic trace in the form of explicit definability of fixpoints of modalized formulas in KL. Van Benthem has observed that this definability of fixpoints entails that adding standard fixpoint operators of positive formulas does not increase the expressive power of KL. Visser has analyzed this observation syntactically and extended it to a proof that KL is a retract of the modal μ-calculus in a suitably defined category of interpretations. A fully polished theory of such embeddings and translations involving the fixpoint operator and similar binders is yet to be developed (it would have to marry AAL with HOAS and/or HRS).
  Furthermore, we face some interesting problems when studying intuitionistic analogs of such results. First, while the fixpoint theorem for modalized formulas works in intuitionistic KL (iKL), the propositional reasoning underlying derivations of van Benthem and Visser is essentially classical. Deep results about the intuitionistic propositional calculus (IPC) established by Pitts or Ruitenburg can be used to overcome this limitation.
  Second, over IPC (and sometimes even classically) it is for several reasons natural to look at a more general syntactic setup replacing the box with Lewis arrow, which arithmetically is standardly interpreted in terms of preservativity (although it does admit many other interpretations). As it turns out, there are two major incompatible ways of computing explicit fixpoints for formulas whose principal connective is the strict implication, each requiring a different base logic. Our approach here can be described as the first study of reverse mathematics of explicit fixpoints. This is obviously a joint work with Albert Visser (Utrecht).

Archive

The seminar archive can be found here.