Past webinars

10 September 2021, 7:00pm IST, Prof. Sankha S. Basu, IIIT-Delhi, India

Title: Left variable inclusion and restricted rules companions of logics

Abstract: Left variable inclusion and restricted rules companions of logics

Abstract: In this talk, I will describe our recent work on the left variable inclusion and the restricted rules companion of a logic. Given a logic \mathcal{S}=\langle\mathcal{L},\vdash\rangle, its left variable inclusion companion is the logic \mathcal{S}^l=\langle\mathcal{L},\vdash^l\rangle, where \vdash^l is defined as follows: for any \Gamma\cup\{\alpha\}\subseteq\mathcal{L}, \Gamma\vdash^l\alpha iff there is a \Delta\subseteq\Gamma such that \mathrm{var}(\Delta)\subseteq\mathrm{var}(\alpha) and \Delta\vdash\alpha. Here \mathrm{var}(*) denotes the set of variables in *. It can be proved that \mathcal{S}^l has the same theorems as \mathcal{S} and is paraconsistent regardless of whether \mathcal{S} is so. It is now well-known that the paraconsistent weak Kleene logic (PWK) is the left variable inclusion companion of classical propositional logic (CPC). Moreover, a Hilbert-style system for PWK can be obtained from that of CPC by just restricting the rule of inference, viz., modus ponens. We observed that a similar result is obtained if one starts with intuitionistic propositional logic (IPC) thus yielding the intuitionistic paraconsistent weak Kleene logic (IPWK). The question which then arose naturally was the following. Is it always the case that given a logic \mathcal{S} which has a Hilbert-style presentation, we can obtain a Hilbert-style presentation for \mathcal{S}^l by simply restricting the rules of inference? To answer this, we define the restricted rules companion of a Hilbert-style logic (a logic induced by a Hilbert-style presentation), \mathcal{S}=\langle\mathcal{L},\vdash\rangle, and denote it by \mathcal{S}^{re}=\langle\mathcal{L},\vdash^{re}\rangle. It turns out that while \vdash^{re}\subseteq\vdash^l, the reverse inclusion may not hold, and hence the answer to our question is "no". I will also mention a sufficient condition for the reverse inclusion to hold and hence for the two companion logics to coincide.

Slides and video are available here.

7 May 2021, 6:00pm IST, Prof. Papiya Bhattacharjee, Florida Atlantic University, USA

Title: M-Frames and Minimal Prime Spaces

Abstract: An M-Frame is an algebraic frame with FIP. Examples of M-Frames come from Topology, Commutative Rings, and Lattice-Ordered Groups. In this talk the speaker will discuss general M-Frames and different elements of the frame; in particular, the focus will be on minimal prime elements. The collection of minimal prime elements Min(L) of an M-Frame L can be endowed with two different topologies, namely the Zariski topology and the Inverse topology. The speaker will discuss various properties of the two topological spaces, Min(L) and Min(L)^{-1}. If time permits, there will be a discussion of another collection of prime elements, namely, the maximal d-elements of the frame.

Here are the slides and and the video of the talk.

23 April 2021, 7:00pm IST, Prof. Rohit Parikh, City University of New York, USA

Title: The Sorites paradox, Fuzzy Logic, and Wittgenstein's Language Games

Abstract: We look at the history of research on vagueness and the Sorites paradox. That search has been largely unsuccessful and the existing solutions are not quite adequate. But following Wittgenstein we show that the notion of a successful language game works.


Language games involving words like "small" or "red" can be successful and people can use these words to cooperate with others. And yet, ultimately these words do not have a meaning in the sense of a tight semantics. It is just that most of the time these games work. It works to say, the light is green and we can go," even though the color word green" does not actually have a semantics.


References:

Parikh, Rohit. "Vagueness and utility: The semantics of common nouns." Linguistics and Philosophy 17.6 (1994): 521-535.

Parikh, Rohit, Laxmi Parida, and Vaughan Pratt. "Sock Sorting: An Example of a Vague Algorithm." Logic Journal of the IGPL 9.5 (2001).

Parikh, Rohit. "Feasibility, sorites and vagueness." Bulletin of EATCS 3.132 (2020).

Here are the slide and the video of the talk.

16 April 2021, 7:00pm IST, Prof. Marcus Tressl, University of Machester, UK

Title: Classifying dcpo completions via topology

Abstract: A directed complete partially ordered set (dcpo) is a poset that has suprema for all up-directed subsets.

Dcpos play a central role in domain theory, see [1], which was developed after Dana Scott's characterization of embedding-injective T0-spaces

as continuous lattices, endowed with the so called Scott topology. (I will explain this in the talk).

There are two principal sources for dcpos: One is complete lattices the other one are sober topological spaces (i.e. spaces for which each closed and irreducible set has a generic point).

The talk addresses the question on how to complete a poset P to a dcpo Q, which means that P is a subposet of Q that generates Q as a dcpo.

This is done via "sobrification" of topologies on P and the dcpo completions are classified by certain nonempty intervals of topologies of P.


The classification will then be used to derive various consequences:

-- We attach a smallest dcpo completion to every poset and use it to study various Scott topologies on general posets.

-- Furnish the completions with algebraic operations inherited from operations on the generators (generalizing [2] )

-- Shed new light on the results of [3] on dcpo completions.

[1] Gierz, G. and Hofmann, K. H. and Keimel, K. and Lawson, J. D. and Mislove, M. and Scott, D. S. Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, vol. 93, p. xxxvi+591 (2003)

[2] Jung, Achim and Moshier, M. A. and Vickers, S. J., Presenting dcpos and dcpo algebras, Electronic Notes in Theoretical Computer Science, vol. 218, p. 209-229. (2008)

[3] Keimel, Klaus and Lawson, Jimmie D. D-completions and the d-topology, Ann. Pure Appl. Logic, vol. 159, no. 3, p. 292--306 (2009)

9 April 2021, 8:00pm IST, Prof. Hanamantagouda P Sankappanavar, State University of New York, New Paltz, USA

Title: Starting from Boolean Algebras...

Abstract: In this talk, I plan to present

• A quick survey of some known results on Boolean algebras, by way of motivation,

• Some known generalizations of Boolean algebras, including Regular Kleene Stone Algebras and Regular double Stone algebras.

Then I will introduce a new class of algebras, as a common generalization of regular Kleene Stone Algebras and Regular double Stone algebras, and give an explicit description of subdirectly irreducible algebras in this new variety.

I will conclude the talk with some open problems.

Here is the slide and the video of the talk.

2 April 2021, 7:00pm IST, Dr. Sam Van Gool, IRIF, Université de Paris

Title: What is an existentially closed Heyting algebra and what does it have to do with automata?

Abstract: Logical systems of deduction often resemble algebraic systems of equations. The simplest and original case of this is the correspondence between Boole's propositional logic and algebras over the 2-element field. When one changes the logical deduction system, the algebraic structures become less simple, and more interesting: Heyting algebras, modal algebras, and generalizations of such.

The aim of our work is to use model theory to better understand these algebraic structures coming from logic. Model completeness plays a central role in this study. In traditional model-theoretic algebra, model completeness provides the correct abstraction of the concept of an algebraically closed field. We show that model completeness also has an important role to play in logical algebra.

In particular, we will discuss two cases of model completeness in logical algebra: linear temporal logic and intuitionistic logic. In the former, automata on infinite words are the technical ingredient that leads to model completeness. In the latter, model completeness is shown to be closely related to a certain interpolation property of the logic, originally established by Pitts.

Here is the slide of the talk.

12 March 2021, 7:00pm IST, Prof. Sara L. Uckelman, Durham University, UK

Title: What Problem Did Ladd-Franklin (Think She) Solve(d)?

Abstract: Christine Ladd-Franklin is often hailed as a guiding star in the history of women in logic---not only did she study under C.S.\Peirce and was one of the first women to receive a PhD from Johns Hopkins, she also, according to many modern commentators, solved a logical problem which had plagued the field of syllogisms since Aristotle. In this paper, we revisit this claim, posing and answering two distinct questions: Which logical problem did Ladd-Franklin solve in her thesis, and which problem did she think she solved? We show that in neither case is the answer `a long-standing problem due to Aristotle'. Instead, what Ladd-Franklin solved was a problem due to Jevons that was first articulated in the 19th century.

Here is the slide and the video of the talk.

26 February 2021, 7:00pm IST, Prof. Eric Pacuit, University of Maryland, USA

Title: Using SAT solvers to find impossibility theorems about variable-candidate axioms in voting theory

Abstract: A key problem with many of the voting methods used throughout the world is that a candidate that has no chance to win the election can *spoil* the chances that another candidate can win. A famous example of this phenomenon is the 2000 US Presidential Election in Florida in which Ralph Nader (who had no chance to win) spoiled the election of Al Gore. One way to rule out undesirable spoiler effects is to use voting methods that satisfy an axiom of stability for winners: if a candidate a will be a winner when an election is run with a set X of candidates not including b, and a would beat b in a head-to-head majority vote, then a is still a winner when b is included in the election along with those in X. In this talk, I will discuss two impossibility theorems showing that there is no voting method satisfying natural strengthenings of stability for winners. Both strengthenings arise in light of the irresoluteness of voting methods: any method satisfying the fairness properties of anonymity and neutrality is such that for some profiles of voter preferences, several candidates tie for the win. I will explain how we used a SAT solver to prove our impossiblity theorems without any encoding of profiles in the SAT formalization or any guesswork after the SAT solver’s verification of unsatisfiability. A key tool is a construction of profiles from majority graphs and certain weighted majority graphs that differs from the classic constructions of McGarvey and Debord.

This is joint work with Wesley Holliday and Saam Zahedian.

Slides and video of the talk are available here.

12 February 2021, 7:00pm IST, Dr. Purbita Jana, Indian Institute of Technology, Kanpur (IITK), India

Title: Logic, Algebra, Topology,Topological System and their interconnections

Abstract: In this talk I will focus on the connection among logic, algebra, topology and topological system. Firstly, I will discuss the notion of topological system and its connection with geometric logic, frame and topology following the work by Steve Vickers. Then I will talk about our recent work on intuitionistic topological systems and its connection with intuitionistic logic, Heyting algebra and Esakia space.

Here is the slide and video of the talk

27 November 2020, 4:00pm IST, Prof. Jiri Rosicky, Masaryk University, Brno, Czech Republic

Title: From first order logic to accessible categories

Abstract: We will introduce accessible categories and explain their relations to model theory, in particular to abstract elementary classes of Shelah. We explain how it liberates model theory from a concrete presentation of structures. In particular, internal ranks replace cardinalities of underlying sets. We show that it makes possible to extend Shelah's Categoricity Conjecture beyond abstract elementary classes.

Here is the slide and the video of the talk.

20 November 2020, 4:30pm IST, Prof. Katsuhiko Sano, Hokkaido University, Japan

Title: Goldblatt-Thomason theorems for non-classical logics

Abstract: Many non-classical logics have Kripke semantics based on the notion of a graph or a Kripke frame (W,R) where W is a non-empty set and R is a binary relation on W. It is well-known that each of R's reflexivity, transitivity, symmetry, seriality, etc. is definable by a modal formula, while there is an undefinable property of R (e.g., irreflexivity) in terms of a set of modal formulas. Given a first-order property of Kripke frames, when is such a property definable by a set of modal formulas? The Goldblatt-Thomason theorem for modal logic answers this question as follows (Goldblatt and Thomason 1975): given a first-order definable class F of frames, the class F is definable by a set of modal formulas, if and only if, F is closed under taking generated subframes, disjoint unions and bounded morphic images and the complement of the class F is closed under taking ultrafilter extensions. That is, the modal definability of a first-order definable property of Kripke frames is characterized in terms of ``nice'' frame constructions. This talk overviews Goldblatt-Thomason-style characterizations for non-classical logics beyond modal logic, some of which are the author's own contributions with collaborators (S. and Ma 2010, S. and Virtema 2019, S. 2020). Such examples of non-classical logics may include: graded modal logic (Fine1972), modal logic with the universal modality (Goranko and Passy 1992), modal dependence logic (Vaeaenaenen 2008), intuitionistic logic (cf. Rodenburg1986), and intuitionistic inquisitive logic (Ciardelli et al. 2020).

Here is the slide and video of the talk.

13 November 2020, 4:00pm IST, Prof. R. Ramanujam, The Institute of Mathematical Sciences (IMSc), Chennai

Title: Decidable fragments of first order modal logic

Abstract: First Order Modal Logic (FOML) is "notoriously" undecidable, in the sense that even very weakly expressive fragments are undecidable. All the understanding of decidable fragments of first order logic and decidable extensions of propositional modal logics gained over 50 years seems to help little. The combination of modalities and quantifiers causes new problems that occur neither in the first-order case nor in the propositional modal case. Despite such discouragement, a small community has battled on, and this century has seen some small steps with positive results on new fragments, like the monodic fragment, bundled fragments, term-modal logics, and such. This line of work has opened up new research vistas with many interesting questions. The talk is an attempt to show you some highlights of this journey.

Here is the slide and the video of the talk.

6 November 2020, 4:00pm IST, Prof. Friedrich Wehrung, Universite de Caen, France

Title: Purity and freshness (in categorical model theory)

Abstract: The aim of this talk is to introduce the basic concepts of a technique enabling to prove that certain naturally defined classes of structures are ``intractable’’ in the sense that they cannot be described as classes of models of any infinitely formula (or more generally, of any class of $L_{\infty,\lambda}$ formulas, for any infinite cardinal $\lambda$).

The main idea is that for any suitably ``continuous’’ functor F, from the category of all subsets of some set X and one-to-one maps between those, to a category C of models, all large enough morphisms in the range of F are elementary embeddings with respect to large infinitary languages. This yields the concept of anti-elementarity, which entails intractability. In particular, this applies to classes such as (1) the class of all posets of finitely generated ideals in rings, (2) the class of all ordered $K_0$ groups of unit-regular rings, (3) the class of all lattices of principal l-ideals of abelian lattice-ordered groups (yields a negative answer to the so-called MV-spectrum problem).

Prerequisites would be: basic category theory (categories, functors, directed colimits, products), basic first-order logic (first-order languages, formulas, models, satisfaction). The very basics of infinitary logic ($L_{\infty,\lambda}$) would also be helpful, however I will anyway include a brief reminder about those.

Here is the slide and video of the talk.

30 October 2020, 7:30pm IST, Prof. Wesley H. Holliday, UC Berkley, USA

Title: Possibilities for Provability

Abstract: This talk will be based on a recent program of developing “possibility semantics” for modal and nonclassical logics, as a generalization of standard possible world semantics, based on partial possibilities instead of complete possible worlds. After introducing the key ideas of possibility semantics, I will discuss its connection to some incompleteness results for provability logics. The new material concerns semantics for polymodal provability logic with a “some theory proves” modality, inspired by the “someone knows” modality of epistemic logic. In this setting, we see that the phenomenon of Gödelian “inexhaustibility" conflicts with a semantics based on complete possible worlds but is consistent with a semantics based on ever-extendable partial possibilities.

Here is the slide and video of the talk

23 October 2020, 4:00pm IST, Prof. Kamal Lodaya, Retired from The Institute of Mathematical Sciences, (IMSc.), Chennai

Title: Phones, processes, programs

Abstract: Programs compute functions, given an input they produce an output. What kind of functions? There are many characterizations of the computable functions, all of them appear to use a programming notation or a definition of some kind of machine. Is there an independent way of talking about these functions? It is useful to think of operating systems. System programs are nonterminating, how does one think of their computation? When I chat on my mobile phone, I am only communicating, is there some computation going on?

Background: This talk only requires some familiarity with writing programs in a programming language, and points to papers where mathematical details can be found.

Here are the slides and the video

16 October 2020, 4:00pm IST, Prof. Frank Veltman, University of Amsterdam and Hunan University, Changsha

Title: Generics and Generality

Abstract: Most of us accept the statement `Birds lay eggs’, but not the statement `Birds are female’. Some of us think that Indians are good cricket players, but nobody thinks that Indians are cricket players. We were all taught that malaria mosquitoes transmit malaria, whereas in fact less than 5% of them really do. On the other hand, we don’t accept that prime numbers are odd while there is only one negative instance against infinitely many positive instances.

How come?

Apparently, the acceptability of a statement of the form `S’s have the property P’ does not depend on the percentage of S’s with this property P. But then, what does it depend on? In my talk I will sketch an answer to this question. It will appear that one needs a non-monotonic logic to get to grips with the capricious logical behaviour of these sentences.

Here are the slides and the video

9 October 2020, TBA, Prof. Alena Vencovska, Retired from University of Manchester, UK

Title: Pure Inductive Logic - Symmetry, Similarity and Invariance under Translations.

Abstract: Pure Inductive Logic (PIL) is a branch of logic which studies rational assignments of probabilities to sentences. This is achieved via focusing on probability functions on the set of all sentences of a predicate language, and via formulating principles that a probability function should satisfy to be `rational'. PIL assumes that given some information (as a sentence of the language), a probability function is updated to the corresponding conditional probability function, so it suffices to formulate principles on a rational probability function to be adopted on the basis of no information. A number of such principles have been proposed and investigated but many open questions remain.

I will give a historical introduction to PIL and then talk about some recent developments in the polyadic PIL. In particular, I will discuss the Permutation Invariance Principle, one of the latest principles that has been proposed for PIL, and its remarkable threefold justification.

Here are the slides and the video.

2 October 2020, 4pm IST, Dr. Amit Kuber, IIT Kanpur

Title: Definable combinatorics and Grothendieck rings

Abstract: What of elementary combinatorics holds true in a class of first order structures if sets, relations, and maps must be definable? Krajicek and Scanlon studied definable versions of various combinatorial principles like PHP (pigeonhole principle). The Ax-Grothendieck theorem which states that an injective polynomially defined endomorphism of a finite dimensional complex vector space is also surjective is in fact the statement that the field of complex numbers satisfies PHP! In fact the collection of definable subsets of a structure M modulo definable bijections can be equipped with a semiring structure, whose “ring completion”—the Grothendieck ring of the structure M—encodes some information about the definable combinatorics of the structure M. Certain Grothendieck rings are the value rings for some universal finitely additive measures. Specifically the Grothendieck ring for the complex number field plays an important role in Kontsevich's theory of motivic integration. The talk will be a gentle introduction to this area with plenty of motivation and examples. Only familiarity with FOL will be assumed.

Here are the slides and video.

25 September 2020, 7pm IST, Prof. Guram Bezhanishvili, New Mexico State University, USA

Title: Topological semantics of modal logic

Abstract: The aim of this talk is to provide an overview of topological semantics of modal logic, from the pioneering work of McKinsey and Tarski in 1940s to the present day. The talk is mainly aimed for graduate students, but it will also address the latest developments in the area.

Here you can find slides and video

18 September 2020, 3pm IST, Prof. Ulle Endriss, University of Amsterdam

Title: Logic and Social Choice

Abstract: Social choice theory is the formal study of mechanisms for collective decision making, such as voting rules. In the first part of this talk, I will give an introduction to the field and discuss one of its classical theorems in some detail. Logic is one of several formal tools used in research on social choice, and in the second part of this talk I will illustrate this connection with a number of examples of work coming out of my own research group.

Here you can find video and slides.

11 September 2020, 7pm IST, Prof. Frederick K. Dashiell, UCLA and Chapman University, USA

Title: THE CONSTRUCTIVE BOOLEAN ALGEBRA GENERATED BY A DISTRIBUTIVE LATTICE—A STRANGE OMISSION?

Abstract: It is widely known that every distributive lattice lives inside a Boolean algebra as a sublattice. Strangely, no standard book covering lattice theory gives a constructive proof of this. The smallest such Boolean algebra is called the free Boolean extension, or in category theory, the Boolean reflection. We discuss the history of this question and various solutions, one of which involves entailment relations (of Dana Scott), which generalize Gentzen’s multi-conclusion sequent calculus.

Here are the video and slides

4 September 2020, 4pm IST, Prof. Johan Van Benthem, University of Amsterdam, Stanford University, USA and Tsinghua University, Beijing

Title: Logic and information dynamics: a little tour

Abstract: Our knowledge and beliefs change all the time as new events produce fresh information. We show how to model simple informational scenarios with several agents, and then introduce a system of 'public announcement logic' for capturing basic laws governing this process. This adds a dynamic level to standard more 'statics-oriented' logics. After pointing out some closed and some open technical problems, we give an outlook on how these dynamics-oriented ideas can help us study richer informational scenarios, solutions of games, and even philosophical discussions of knowledge.

J. van Benthem, 2011, "Logical Dynamics of Information and Interaction", Cambridge University Press.

J. van Benthem, 2014, "Logic and Games", The MIT Press.

Here are the video and slides.

28 August 2020, 6pm IST, Prof. Anand Pillay, University of Notre Dame, USA

Title: Theories of free algebras and modules (Joint work with Phlipp Rothmaler)

Abstract: Given a variety in the sense of universal algebra, the free algebras on infinitely many generators are elementarily equivalent, so we obtain a complete theory (of these free algebras). We focus on the case of the variety of R modules for some ring R, and classify the rings R such that all models of the complete theory above are free, projective, flat,..I will give precise definitions and try to avoid too much technical stuff.

Links of the videos are here: Video1, Video2

21 August 2020, 4pm IST, Dr. Serafina Lapenta, University of Salerno, Italy

Title: Handling vagueness in logic via algebras: the case of Lukasiewicz logic

Abstract: In order to capture enough features of the real word in the language of mathematics one needs to represent vague concepts. Several approaches to vagueness can be found in literature, and one of them puts mathematical fuzzy logic at its core, interpreting vague predicates in truth-degrees ranging over [0; 1]. Such logics can be better understood via their semantics and in this seminar we shall briefly discuss vague predicates and their interpretation in Lukasiewicz logic, one of the most prominent among fuzzy logic. We shall introduce the basic facts on MV-algebras, the algebraic semantics for this logic, and discuss special classes of MV-algebras with an eye to duality theory.

Here is the link to slides and audio

14 August 2020, 4pm IST, Prof. Supratik Chakraborty, IIT Bombay, India

Title: Knowledge Compilation for Boolean Functional Synthesis

Abstract: Given a Boolean relation between inputs and outputs, Boolean Functional Synthesis seeks to represent the outputs as deterministic functions of the inputs. This problem, also known as Skolem function synthesis, has a wide range of applications. In this talk, we will first present (conditional) lower bounds on the hardness of the problem, thereby showing that it is unlikely that there are time or space efficient algorithms for Boolean Functional Synthesis. We will then present a knowledge-compilation approach to solving the problem. This involves compiling a given relational specification to a special negation normal form, called SynNNF (Synthesis Negation Normal Form), that ensures polynomial-time synthesis. SynNNF subsumes and can be exponentially more succinct than several other normal forms studied in the literature, like ROBDD, dDNNF, DNNF, wDNNF. Finally, we will discuss experimental results obtained with a preliminary implementation of CNF to SynNNF compiler, that illustrate the practical effectiveness of the algorithm.

This is based on joint work with S. Akshay, Jatin Arora, S. Krishna, Divya Raghunathan and Shetal Shah

Here is the link to slides and audio.

7 August 2020, 4pm IST, Dr. Anthony Peter Young, Accuity and King's college, London, UK

Title: Introduction to Argumentation Theory

Abstract: We construct and evaluate arguments on a daily basis, e.g. from discussing politics online to deciding what to eat for dinner. The study of argumentation dates back to Classical Indian and Greek philosophy, but has been relatively neglected due to the rise and successes of mathematical logic in the early 20th century. The rise of symbolic artificial intelligence (AI) in the mid-20th century has inspired intensive research in argumentation to understand how people and machines can argue about incomplete and contradictory information. Now, argumentation theory is actively studied by philosophers, mathematicians, computer scientists and psychologists.

In this talk, I will outline the formal techniques of argumentation theory, focussing on abstract argumentation, bipolar argumentation and structured argumentation with preferences, while also highlighting applications in, respectively, cooperative game theory, social media analytics, and non-monotonic logics.

You can find slides and audio of the talk.

31 July 2020, 4pm IST, Dr. Sreejith AV, IIT Goa

Title: Regular languages over countable linear orderings


Abstract: In this talk, we look at countable words which are mappings from countable linear orderings to a finite alphabet. Finite and omega words are proper subsets of countable words. We are interested in the expressive power of logics like first order logic, weak monadic second order logic etc. over languages of countable words. Especially, we are interested in whether these logics are expressively equivalent over countable words or not. To answer this, we design an algebraic structure called o-algebra. This also gives us decidability of certain problems. Next we present a block product principle and a couple of decomposition theorems.

The following works are included in the talk:

“Limited set quantifiers over countable linear orderings”, Colcombet and Sreejith, ICALP 15

“Two-Variable Logic over Countable Linear Orderings”, Amaldev and Sreejith, MFCS 16

“Block products for algebras over countable words and applications to logic”, Adsul, Sarkar, Sreejith, LICS19

Here are slides and audio.

24 July 2020, 7pm IST, Prof. John Harding, New Mexico State University, USA

Title: Canonical extensions and free completely distributive lattices

Abstract: Canonical extensions are a key algebraic ingredient in the treatment of Kripke completeness. In lattice theoretic terms, they have elegant descriptions, and pleasing properties. In this note, we make several simple observations about the connection between canonical extensions and a notion of freeness. These observations could have been made many decades ago, and provided a somewhat different path to various results.

You can find slides and audio of the talk.

17 July 2020, 11am IST, Prof. M. Andrew Moshier, Chapman University, Orange, CA, USA

Title: Frames and frame relations

Abstract: Frames come about in many different ways. Most well-known, they are explained as generalizing lattices of open sets of topological spaces. That is, the relevant lattice-theoretic property is that finite meets (intersections) distribute over all joins (unions). This motivates the idea that frames constitute “point-free” spaces. Coincidentally and less well-known, frames are precisely the injective meet semilattices. In the first part of this talk, I will try to convince you that this coincidence is not an accident.

To start, we look at a general setting of partially ordered structures and suitable binary relations (called _weakening relations_) between them. Weakening relations (a special case of profunctors) are best seen as abstract versions of sequent calculus entailment relations. These specialize to _semilattice relations_. In the category of all semilattice relations, all objects are injective. Frames come about by a suitable lifting operation that transforms a semilattice relation into a semilattice homomorphism. Armed with this analysis, we see that topological spaces arise as representations of injective semilttices.

With injective semilattices in mind, we continue the investigation of frames by specializing semilattice relations to frame relations. In the setting, we show that that lattice of reflexive, transitive frame relations on a given frame L also forms a frame, and satisfies the universal property of makes all elements of L complemented. That is, the reflexive, transitive frame relations on L constitutes what is known as the _assembly_ of L.

Through the talk, I will emphasize the motivations and connections to algebraic logic via weakening relations.

You can find slides and audio of the talk

10 July 2020, 4pm IST, Dr. Ashutosh Kumar, IIT Kanpur

Title: Transfinite recursion

Abstract: We'll discuss two applications of transfinite recursion—one in Complex analysis (Wetzel's problem) and the other in Euclidean Ramsey theory (Chromatic no. of the rational distance graph). The talk should be accessible to a general math audience.

Slides are available here.