Events 2015-2016

Logic Colloquium

August 28, 2015, 4:10 PM (60 Evans Hall)

Antonio Montalban
Associate Professor of Mathematics, UC Berkeley

Analytic Equivalence Relations with 1-many Classes

We will survey the new results connecting Vaught’s conjecture and computability theory. We will look at some of these result from the more general viewpoint of analytic equivalence relations.

There are two types of results we will concentrate on. On the one hand are the ones about equivalence relations satisfying hyperarithmetic-is-recursive, which provide a purely computability theoretic equivalent to Vaught’s conjecture. On the other hand are the ones about equivalence relations on the natural numbers that are “intermediate,” which I hope will eventually provide another equivalent statement.

Slides

Logic Colloquium

September 11, 2015, 4:10 PM (60 Evans Hall)

Thomas Icard
Assistant Professor of Philosophy and Symbolic Systems, Stanford University

A Survey on Topological Semantics for Provability Logics

This will be a survey of ideas and results on topological interpretations of provability logics, especially polymodal provability logics. Esakia first proved completeness for the basic Gödel-Löb logic of provability with respect to scattered spaces. Abashidze and Blass (independently) proved completeness w.r.t. a particular scattered space defined on the ordinal omega^omega. I will discuss older work of my own that extended the Abashidze-Blass result to a polymodal provability logic, which is complete with respect to a polytopological space on epsilon_0. I will then discuss more recent developments, including work by Beklemishev, Fernandez, Joosten, Gabelaia, Aguilera, and others, generalizing and improving upon much of this.

Slides

Logic Colloquium

September 25, 2015, 4:10 PM (60 Evans Hall)

Dana Scott
University Professor, Emeritus, Carnegie Mellon University; Visiting Scholar, UC Berkeley

Can Modalities Save Naive Set Theory?

The late “Grisha” Mints once asked the speaker whether a naive set theory could be consistent in modal logic. Specifically he asked whether restricting the comprehension scheme to necessary properties was safe. Scott was working on a set theory in the Lewis system S4 of modal logic and Mints was happy to position his question in the same modal system. Obviously a very, very weak modality can avoid paradoxes, but such results are not especially interesting. At that time (2009) Scott could not answer the consistency question, and neither could Mints. Last November Scott noted that CMU Philosophy was hosting a seminar on a naive set theory by Harvey Lederman. Scott wrote him for his paper and said, “By the way, there is this question of Grisha Mints, and I wonder if you have an opinion?” Lederman sent back a sketch of a proof of the inconsistency of a strengthened version of comprehension. That proof at first did not quite work out, but was repaired in correspondence. Lederman mentioned the questions to two of his colleagues, and in March of 2015 Tiankai Liu suggested a possible model of a weaker comprehension scheme, which after a small correction gave a consistency proof. A few days later, Peter Fritz came up with essentially the same model. A paper has now been submitted for publication jointly by Fritz, Lederman, Liu, and Scott.

Slides

Paper

Logic Colloquium

October 09, 2015, 4:10 PM (60 Evans Hall)

Takayuki Kihara
JSPS Postdoctoral Fellow, Department of Mathematics, UC Berkeley

Recursion Theoretic Methods in Descriptive Set Theory and Infinite Dimensional Topology

The notion of degree spectrum of a structure in computable model theory is defined as the collection of all Turing degrees of presentations of the structure. We introduce the degree spectrum of a represented space as the class of collections of all Turing degrees of presentations of points in the space. The notion of point degree spectrum creates a connection among various areas of mathematics including computability theory, descriptive set theory, infinite dimensional topology and Banach space theory.

Through this new connection, for instance, we construct a family of continuum many infinite dimensional Cantor manifolds with property C in the sense of H aver/Addis-Gresham whose Borel structures at an arbitrary finite rank are mutually non-isomorphic. This provides new examples of Banach algebras of real valued Baire class two functions on metrizable compacta, and strengthen various theorems in infinite dimensional topology such as Pol’s solution to Alexandrov’s old problem.

To prove our main theorem, an invariant which we call degree co-spectrum, a collection of Turing ideals realized as lower Turing cones of points of a Polish space, plays a key role. The key idea is measuring the quantity of all possible Scott ideals (omega-models of WKL) realized within the degree co-spectrum (on a cone) of a given space. This is joint work with Arno Pauly (University of Cambridge, UK).

Slides

Logic Colloquium

October 23, 2015, 4:10 PM (60 Evans Hall)

Larry Moss
Professor of Mathematics, Indiana University, Bloomington

Natural Logic

Much of modern logic originates in work on the foundations of mathematics. My talk reports on work in logic that has a different goal, the study of inference in language. This study leads to what I will call “natural logic”, the enterprise of studying logical inference in languages that look more like natural language than standard logical systems. An early paper in this field is George Lakoff’s 1970 paper “Linguistics and Natural Logic.”

By now there is a growing body of work which presents logical systems that differ from first-order logic in various ways. Most of the systems are complete and decidable. Some are modern versions of syllogistic logic, but with additional features not present in syllogistic logics. And then there are flavors of logic which look rather far from either traditional or modern logic.

The talk will be programmatic and far-ranging rather than detailed. I hope to touch on computer implementations of natural logics, teaching materials on this topic, and interactions of logic and cognitive science. And keeping with most other talks in the Logic Colloquium, I’ll show that the whole subject of natural logic is full of questions with mathematical interest.

Slides

Logic Colloquium

November 06, 2015, 4:10 PM (60 Evans Hall)

Silvain Rideau
Morrey Visiting Assistant Professor, Department of Mathematics, UC Berkeley

Transferring imaginaries

Since it was first formulated by Poizat thirty years ago, the question of eliminating imaginaries, has become an important, if not necessary, step in understanding the model theory of given algebraic structures; it consists in classifying all the definable equivalence relations in a structure. An ever growing collection of such results is now available but this question remains open in many otherwise well understood structures.

Some of the recent elimination of imaginaries results are actually reductions to previously known elimination of imaginaries results. The goal of this talk will be to consider these reductions from an abstract standpoint and provide various settings in which one might derive elimination of imaginaires in a theory from elimination of imaginaries in some other theory.

Slides

Logic Colloquium

November 20, 2015, 4:10 PM (60 Evans Hall)

Edward Zalta
Senior Research Scholar, Center for the Study of Language and Information, Stanford University

Reflections on Foundations for Mathematical Structuralism

In a paper recently published in Mind (“Foundations for Mathematical Structuralism”), Nodelman and I offer a rigorous formulation of mathematical structuralism. A mathematics-free theory of structures and structural objects is presented in a background formalism and then the whole is applied to the analysis of mathematics. While the principal structuralist intuitions about structures and the elements of structures are preserved, distinctions formalized in the theory undermine a variety of arguments that have been put forward concerning the nature of structural elements. Arguments from Russell, Dedekind, Benacerraf, Shapiro, Linnebo, Awodey, Keranen, and others, are considered. For those interested in reading the paper in advance, a preprint is available at: http://mally.stanford.edu/Papers/structuralism.pdf

Logic Colloquium

December 04, 2015, 4:10 PM (60 Evans Hall)

Natarajan Shankar
Computer Scientist, SRI International

PVS and the Pragmatics of Formal Proof

A formal system establishes a trinity between language, meaning, and inference. Many domains of thought can be captured using formal systems so that sound conclusions can be drawn through correct inferences. Philosophers have long speculated about machines that can distinguish sound arguments from flawed ones, and with modern computing, we can realize this dream to a significant extent. A research program to mechanize formal proof inevitably confronts a range of foundational and pragmatic choices. Should the foundations be classical or constructive, and should they be based on set theory, type theory, or category theory? What definitional principles should the language admit? How can formalizations be constructed from reusable modules? Should proofs be represented in a Hilbert calculus, natural deduction, or sequent calculus? Which inference steps in the proof calculus can and should be effectively mechanized in order to close the gap between informal arguments and their formal counterparts?

SRI’s Prototype Verification System (PVS) is an interactive proof assistant aimed at capturing the pragmatic features of mathematical expression and argumentation. The PVS language augments a simply typed higher-order logic with subtypes, dependent types, and algebraic datatypes. The PVS interactive proof checker integrates a number of decision procedures into the sequent calculus proof system. We examine the implications of these choices for the original goal of constructing mathematically sound arguments.

Slides

Logic Colloquium

January 22, 2016, 4:10 PM (60 Evans Hall)

John R. Steel
Professor of Mathematics, UC Berkeley

Ordinality Definability in Models of the Axiom of Determinacy

Let HOD be the class of all hereditarily ordinal definable sets, and let M be a model of the Axiom of Determinacy. The inner model HOD^M consisting of all sets that are hereditarily ordinal definable in the sense of M is of great interest, for a variety of reasons. In the first part of the talk, we shall give a general introduction that explains some of these reasons.

We shall then describe a general Comparison Lemma for “hod mice” (structures that approximate HOD^M). Modulo still-open conjectures on the existence of iteration strategies, this lemma yields models M of the Axiom of Determinacy such that HOD^M can be analyzed fine structurally (for example, satisfies the GCH), and yet satisfies very strong large cardinal hypotheses (for example, that there are superstrong cardinals).

Model Theory Seminar

January 29, 2016, 4:10 PM (60 Evans Hall)

Pierre Simon
CNRS, Université Claude Bernard - Lyon 1

Decomposition of dependent structures

A structure is a set equipped with a family of predicates and functions on it, for example a group or a field. A structure is said to be dependent if it is not as complicated as a random graph (in some precise sense). The fields of complex numbers, real numbers and p-adic numbers are examples of dependent structures. If a dependent structure contains no definable order, then it is stable: a much stronger property that is now very well understood. Stability can be thought of as an abstraction of algebraic geometry. At the other extreme, are dependent structures which are very much controlled by linear orders. We call such structures distal. Distality is meant to serve as a general model for semi-algebraic (or analytic) geometry. In this talk, I will explain those notions and state a recent theorem saying that any dependent structure can be locally decomposed into a stable-like and a distal-like part.

Logic Colloquium

February 05, 2016, 4:10 PM (60 Evans Hall)

Vijay D’Silva
Google Research

Interpolant Construction and Applications

In the last decade, algorithms for the construction of interpolants from proofs have found numerous practical applications such as the analysis of circuits, generation of Floyd/Hoare proofs of program correctness, and parallelization of constraint solvers. The first part of this talk will trace how interpolation has spread from mathematical logic to commercial software via computational complexity theory and computer aided verification. The second part will delve into details of the algorithms for construction of interpolants in logics and theories. In the propositional case, the space of interpolant constructions has a semi-lattice structure that determines the logical strength and size of the interpolants obtained. In the case of first-order theories, this structure is still being unravelled. The talk will conclude with a review of open theoretical and empirical problems concerning interpolant construction.

Slides

Logic Colloquium

February 19, 2016, 4:10 PM (60 Evans Hall)

Christoph Benzmüller
Stanford University and Freie Universität Berlin

A Success Story of Higher-Order Theorem Proving in Computational Metaphysics

I will report on the discovery and verification of the inconsistency in Gödel’s ontological argument with reasoning tools for higher-order logic. Despite the popularity of the argument since the appearance of Gödel’s manuscript in the early 70’s, the inconsistency of the axioms used in the argument remained unnoticed until 2013, when it was detected automatically by the higher-order theorem prover LEO-II.

Understanding and verifying the refutation generated by LEO-II turned out to be a time-consuming task. Its completion required the reconstruction of the refutation in the Isabelle/HOL proof assistant. This effort also led to a more efficient way of automating higher-order modal logic S5 with a universal accessibility relation within the logic embedding technique we utilize in our work.

The development of an improved syntactical hiding for the logic embedding technique allows the refutation to be presented in a human-friendly way, suitable for non-experts in the technicalities of higher-order theorem proving. This should bring us a step closer to wider adoption of logic-based artificial intelligence tools by philosophers.

If time permits, I will also point to alternative, ongoing applications of the logic embeddings technique, such as the encoding of Zalta’s theory of abstract objects or the mechanization of Scott’s free logic.

Slides

Logic Colloquium

March 04, 2016, 4:10 PM (60 Evans Hall)

Katrin Tent
Professor of Mathematics and Mathematical Logic, Universität Münster

Describing finite groups by short sentences

We say that a class of finite structures for a finite first-order signature is r-compressible for an unbounded function r : N → N+ if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is log-compressible, and the class of all finite groups is log3-compressible. The result relies on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology.

Logic Colloquium

March 18, 2016, 4:10 PM (60 Evans Hall)

Guram Bezhanishvili
Professor of Mathematical Sciences, New Mexico State University

The algebra of topology: Tarski’s program 70 years later

In a slogan, Tarski’s program can be described as “creating an algebraic apparatus adequate for the treatment of portions of point-set topology” (McKinsey-Tarski, 1944). The “algebraic apparatus” can be developed in several ways. For example, we can look at the algebra of open sets of a topological space X or the powerset algebra of X equipped with topological closure (or interior). Both of these approaches are not only closely related, but are also widely used in logic. The first approach yields a topological representation of Heyting algebras, and hence provides topological completeness of intuitionistic logic (one of the first completeness results for intuitionistic logic, established by Tarski in the 1930s). On the other hand, the second approach provides a topological representation of closure algebras, thus yielding topological completeness of Lewis’s modal system S4. McKinsey and Tarski observed a close connection between these two approaches, which allowed them to prove that Gödel’s translation of intuitionistic logic into S4 is full and faithful. I will review this line of research of Tarski and his collaborators, discuss further advances in this direction, obtained in the second half of the twentieth century, and finish with the latest new developments.

Slides

Alfred Tarski Lectures

April 04, 2016, 4:10 PM (20 Barrows)

William W. Tait
Professor Emeritus, Department of Philosophy and CHSS, University of Chicago

On Skepticism about the Ideal

There is a historic skepticism about mathematics that hangs on the fact that the objects of mathematics, structures and their elements—numbers, functions, sets, etc., are ideal, i.e. that empirical facts, facts about the natural world, have no relevance to the truth of propositions about them.

Of course, the view that ‘existence’ simply means empirical existence, so that the term is misused as applied to ideal things can be countered only on pragmatic grounds, that we use the term in other contexts and that it is very useful there. The rejection of ideal existence becomes meaningful only if one has a transcendental ground on which to stand and judge applications of the term. I believe that the ground, at least implicitly, has been a wrong thesis about how language works, namely the view that genuine reference to objects presupposes a non-linguistic interaction with them. And that is what I want to talk about. My argument draws on a reading of Wittgenstein’s Philosophical Investigations, a reading which is more positive than a more common one according to which he, himself, was a skeptic.

Alfred Tarski Lectures

April 06, 2016, 4:10 PM (20 Barrows)

William W. Tait
Professor Emeritus, Department of Philosophy and CHSS, University of Chicago

Cut-Elimination for Subsystems of Classical Second-Order Number Theory: The Predicative Case

I will present the classical results of Gentzen and Schuette for first-order and ramified second-order number theory.

Alfred Tarski Lectures

April 08, 2016, 4:10 PM (20 Barrows)

William W. Tait
Professor Emeritus, Department of Philosophy and CHSS, University of Chicago

Cut-Elimination for Subsystems of Classical Second-Order Number Theory: Cut-Elimination for Π11 − CA with the ω-Rule—and Beyond(?)

I will present a simplification of the proof of cut-eliminability of Gaisi Takeuti and Mariko Yasugi for this system. In particular, I will avoid the use of Takeuti’s ordinal diagrams. (I do use the finite part of the Veblen hierarchy, though.) Given time and sufficient conviction, I may speak about the possibility of extending the result to Π21 − CA and beyond.

Logic Colloquium

April 22, 2016, 4:10 PM (60 Evans Hall)

Kai Wehmeier
Professor of Logic and Philosophy of Science, UC Irvine

The Role of Variables in Predicate Logic: Frege vs. Tarski

Standard presentations of predicate logic, following Tarski, segment a quantified statement such as x(P(x)∧R(x, a)) into the constituents x and P(x)∧R(x, a). At least on the face of it, Frege held a different view about the logical form of quantification; for him the same sentence decomposes into the constituents xxx and P( )∧R( , a). In order to adjudicate between these views, I will develop some of the details of Fregean predicate logic, with special emphasis on the compositionality of its semantics. I will then compare Tarskian and Fregean treatments of variables and draw some philosophical conclusions from this comparison. In particular, I will address the extensionality of predicate logic and Kit Fine’s so-called antinomy of the variable.

Logic Colloquium

May 06, 2016, 4:10 PM (60 Evans Hall)

Omer Ben Neria
Hedrick Assistant Adjunct Professor of Mathematics, UCLA

The distance between HOD and V

The pursuit of better understanding of the universe of set theory V motivated an extensive study of definable inner models M whose goal is to serve as good approximations to V.
A common property of these inner models is that they are contained in HOD, the universe of hereditarily ordinal definable sets. Motivated by the question of how “close” HOD is to V, we consider various related forcing methods and survey known and new results. This is a joint work with Spencer Unger.

Special Logic Seminar

May 10, 2016, 4:10 PM (234 Moses Hall)

Johan van Benthem
Amsterdam and Stanford

Decidable Versions of First-Order Predicate Logic

There are two ways of finding decidability inside first-order logic: one is by restricting attention to language fragments, the other is by generalizing the usual semantics. In this talk, I explain a recent generalized semantics by Aldo Antonelli, which leads to a decidable version of predicate logic that induces an effective translation into the Guarded Fragment. I will discuss this proposal and the resulting program.

Also, I make a comparison with existing decidable first-order semantics via general assignment models and via games, and with the move in modal logic from relational semantics to neighborhood semantics. Finally, I discuss the resulting landscape of weak decidable first-order logics, and its links with decidable fragments of the first-order language. This suggests many open problems that I will flag throughout.

Ref. H. Andréka, J. van Benthem & I. Németi, 2016, ‘On a New Semantics for First‐Order Predicate Logic’, Journal of Philosophical Logic, to appear.

Logic Colloquium

August 26, 2016, 4:10 PM (60 Evans Hall)

Martin Otto
Professor in Mathematical Logic, Department of Mathematics, Technische Universitaet Darmstadt (currently visiting the Simons Institute for Theory of Computing, Berkeley)

Back & Forth Between Malleable Finite Models

This talk attempts to survey some model-theoretic techniques, model transformations and constructions - especially for logics of a modal/guarded flavour - that can replace classical compactness arguments, which are not available in restriction to finite models or other non-elementary classes of interest. Even where these algebraic-combinatorial and game-oriented techniques concur with the usually smoother infinitary methods of classical model theory they occasionally offer additional constructive or quantitative information. Some of the underlying combinatorial constructions to be discussed involve rather generic finite coverings for graphs and hypergraphs and are of independent interest. They can also be used towards alternative proofs of powerful extension properties for partial isomorphisms, which also have applications for the finite model theory of related logics.

Logic Colloquium

September 09, 2016, 4:10 PM (60 Evans Hall)

Sam Buss
Professor of Mathematics and Computer Science, UC San Diego

Frege Proofs, Extended Frege Proofs, and Total NP Search Problems

This talk will discuss some of the fundamental problems of proof complexity and their connection to computational complexity. The talk focuses on the proof complexity of the Frege and extended Frege proof systems for propositional logic. Recent work includes new proofs for the pigeonhole principle (!), Frankl’s theorem, the AB=I theorem, and the Kneser-Lovasz theorem on chromatic numbers. These problems are closely related to Total NP Search Problems (TFNP), a complexity class that lies midway between P and NP. The consistency statements for exponentially large Frege and extended Frege proofs can be used to characterize the provably total functions of second-order theories of bounded arithmetic for polynomial space and exponential time.

Logic Colloquium

September 23, 2016, 4:10 PM (60 Evans Hall)

Andreas Blass
Professor of Mathematics, University of Michigan, and Visiting Scientist, Simons Institute for the Theory of Computing

Between countable and continuum

I plan to give a general introduction to cardinal characteristics of the continuum, including several examples of characteristics and of the sorts of results that the theory obtains. Afterward, if time permits, I’ll describe some recent work on a specific characteristic suggested by a familiar theorem from calculus. The talk will be aimed at people who are not set theorists.

Logic Colloquium

October 07, 2016, 4:10 PM (60 Evans Hall)

Dag Westerståhl
Professor of Theoretical Philosophy and Logic, Stockholm University

A Carnapian approach to the meaning of logical constants: the case of modal logic

Does a consequence relation in a language L, as a syntactic relation between sets of L-sentences and L-sentences, fix the meaning of the logical constants in L? In his 1943 book The Formalization of Logic, Carnap worried that this seems to fail even for classical propositional logic CL. However, by applying the viewpoint of modern formal semantics, which in particular requires meaning assignment to be compositional, Carnap’s worries about CL can be allayed. More importantly, it provides a precise framework for asking Carnap’s question about any logic. (To what extent) does classical first-order consequence determine the meaning of ? What about other (generalized) quantifiers? What about the intuitionistic meaning of the connectives? Or in (classical) modal logic? [1] answers the first of these questions. In this talk I focus on the last one. Despite obvious similarities between and , there are important differences. Roughly, while there is essentially just one first-order logic, there are innumerable modal logics. That makes answers to Carnap style questions about the meaning of more intricate, and more interesting, than in the case of . Although we in a sense cover extremely familiar ground, the perspective one gets, in particular of neighborhood semantics for modal logic, seems novel. This is joint work with Denis Bonnay.

[1] D. Bonnay and D. Westerståhl, ‘Compositionality solves Carnap’s Problem’, Erkenntnis 81 (4), 2016 (721–739).

Slides

Logic Colloquium

October 21, 2016, 4:10 PM (60 Evans Hall)

Erich Grädel
Professor of Mathematical Foundations of Computer Science, RWTH Aachen University

Back and Forth Between Team Semantics, Games, and Tarski Semantics

Team semantics, due to Wilfrid Hodges, is the mathematical basis of modern logics of dependence and independence in which, following a proposal by Väänänen, dependencies are considered as atomic statements, and not as annotations of quantifiers. In team semantics, a formula is evaluated not against a single assignment of values to the free variables, but against a set of such assignments.

Logics with team semantics have strong expressive power and some surprising properties. They can be analyzed in several different ways. First-order formulae with team semantics and dependencies can be translated into sentences of existential second-order logic, with an additional predicate for the team (and with classical Tarski semantics). We can thus understand the power of a specific logic of dependence or independence by identifying the fragment of existential second-order logic to which it corresponds. Further, logics can be understood through their games. In general the model-checking games for logics with team semantics turn out to be second-order reachability games.

We shall then have a closer look at the specific case of logics with inclusion dependencies, and reveal their connection with safety games and logics with greatest fixed-points.

Slides

Logic Colloquium

November 04, 2016, 4:10 PM (60 Evans Hall)

Mai Gehrke
Senior Research Director, CNRS & Université Paris Diderot - Paris 7

Canonical extensions and applications

In 1951-52 Jònsson and Tarski published a two part paper in which they cast Stone duality for Boolean algebras in algebraic form as so-called canonical extensions, and used this formulation to extend Stone duality to Boolean algebras with operators (BAOs). That is, Boolean algebras with additional operations which preserve join in each coordinate. In this way they obtained the relational semantics of modal logic. Since then the theory of canonical extensions, and thus duality for additional structure on lattices and Boolean algebras, has been generalised and better understood. These tools have played a central role in the discovery of the connection between formal language theory and Stone duality. In particular, the fact that the profinite completion of any abstract algebra is the Stone dual of the Boolean algebra of recognisable languages over the algebra equipped with certain residuation operations.

In this talk, I will give an introduction to these ideas and an overview of recent developments.

Working Group in the History and Philosophy of Logic, Mathematics, and Science

November 16, 2016, 6:00 PM (234 Moses Hall)

Juliette Kennedy
Associate Professor, Department of Mathematics and Statistics, University of Helsinki

Squeezing arguments and strong logics

G. Kreisel has suggested that squeezing arguments, originally formulated for the informal concept of first order validity, should be extendable to second order logic, although he points out obvious obstacles. We develop this idea in the light of more recent advances and delineate the difficulties across the spectrum of extensions of first order logics by generalised quantifiers and infinitary logics. This is joint work with Jouko Väänänen.

Logic Colloquium

November 18, 2016, 4:10 PM (60 Evans Hall)

Phokion G. Kolaitis
UC Santa Cruz & IBM Research - Almaden

Dependence Logic vs. Constraint Satisfaction

During the past decade, dependence logic has emerged as a formalism suitable for expressing and analyzing notions of dependence and independence that arise in different scientific areas. The sentences of dependence logic have the same expressive power as those of existential second-order logic, hence dependence logic captures the complexity class NP on the collection of all finite structures over a relational vocabulary. We identify a natural fragment of universal dependence logic and show that, in a precise sense, this fragment captures constraint satisfaction, a ubiquitous algorithmic problem encountered across different areas of computer science. This tight connection between dependence logic and constraint satisfaction contributes to the descriptive complexity of constraint satisfaction and elucidates the expressive power of universal dependence logic.

This is joint work with Lauri Hella, University of Tampere, Finland.

Slides

Logic Colloquium

December 02, 2016, 4:10 PM (60 Evans Hall)

Prakash Panangaden
Professor, School of Computer Science, McGill University

A Logical Characterization of Probabilistic Bisimulation

Labelled Markov Processes (LMPs) are a combination of traditional labelled transition systems and Markov processes. Discrete versions of such systems have been around for a while and were thoroughly explored by Larsen and Skou in the late 1980s and early 1990s. Our contribution has been to extend this study to systems with continuous state spaces.

The main technical contribution that I will discuss in this talk is a definition of probabilistic bisimulation for such systems and a logical characterization for it. The surprise is that a very simple modal logic with no negative constructs or infinitary conjunctions suffices to characterize bisimulation, even with uncountable branching. This is quite different from the traditional situation with Kripke structures where van Benthem and Hennessy and Milner characterized bisimulation using negation in an essential way.

Logic Colloquium

January 27, 2017, 4:10 PM (60 Evans Hall)

Ilijas Farah
Professor of Mathematics, York University

Model theory of operator algebras

Unexpected and deep connections between the theory of operator algebras and logic have been discovered in the past fifteen years. Even more remarkably, diverse areas of logic that are generally (and arguably) thought to have little connection to one another have found natural - and sometimes necessary - applications to operator algebras. In this talk I will emphasize applications of model theory to operator algebras.

Logic Colloquium

February 10, 2017, 4:10 PM (60 Evans Hall)

Clifton Ealy
Associate Professor of Mathematics, Western Illinois University

Residue field domination in real closed valued fields

In an algebraically closed valued field, as shown by Haskell, Hrushovski, and Macpherson, the residue field and the value group control the rest of the structure: tp(L/Ck(L)Γ(L) will have a unique extension to M ⊇ C, as long as the residue field and value group of M are independent from those of L. (Here k(L) and Γ(L) denote the residue field and value group, respectively, of L, and C is a maximal field.)

This behaviour is striking, because it is what typically occurs in a stable structure (where types over algebraically closed sets have unique extensions to independent sets) but valued fields are far from stable, due to the order on the value group.

Real closed valued fields are even further from stable since the main sort is ordered. One might expect the analogous theorem about real closed valued fields to be that tp(L/M) is implied by tp(L/Ck(L)Γ(L)) together with the order type of L over M. In fact we show that the order type is unnecessary, that just as in the algebraically closed case, one has that tp(L/M) is implied by tp(L/k(L)Γ(L)). This is joint work with Haskell and Marikova.

No knowledge of value fields will be assumed.

Slides

Logic Colloquium

February 24, 2017, 4:10 PM (60 Evans Hall)

Sean Walsh
Department of Logic and Philosophy of Science, University of California, Irvine

Realizability Semantics for Quantified Modal Logic

In 1985, Flagg produced a model of first-order Peano arithmetic and a modal principle known as Epistemic Church’s Thesis, which roughly expresses that any number-theoretic function known to be total is recursive. In some recent work ([1]), this construction was generalized to allow a construction of models of quantified modal logic on top of just about any of the traditional realizability models of various intuitionistic systems, such as fragments of second-order arithmetic and set theory. In this talk, we survey this construction and indicate what is known about the reduct of these structures to the non-modal language.

References: [1] B. G. Rin and S. Walsh. Realizability semantics for quantified modal logic: Generalizing Flagg’s 1985 construction. The Review of Symbolic Logic, 9(4):752–809, 2016.

Logic Colloquium

March 10, 2017, 4:10 PM (60 Evans Hall)

Caroline Terry
Brin Postdoctoral Fellow, University of Maryland, College Park, and Research Fellow, Simons Institute for the Theory of Computing

Structure and enumeration theorems for hereditary properties of -structures

The study of structure and enumeration for hereditary graph properties has been a major area of research in extremal combinatorics. Over the years such results have been extended to many combinatorial structures other than graphs. This line of research has developed an informal strategy for how to prove these results in various settings. In this talk we use tools from model theory to formalize this strategy. In particular, we generalize certain definitions, tools, and theorems which appear commonly in approximate structure and enumeration theorems in extremal combinatorics. Our results apply to classes of finite -structures which are closed under isomorphism and model-theoretic substructure, where is any finite relational language.

Slides

Logic Colloquium

March 24, 2017, 4:10 PM (60 Evans Hall)

Françoise Point
FNRS-FRS (UMons)

On expansions of (ℤ, +, 0)

In the special case of the structure (ℤ, +, 0), we will consider the following model-theoretic question: given a well-behaved first-order structure, which kind of predicates can one add and retain model-theoretic properties of the structure one started with, such as stability- like properties, quantifier elimination in a reasonable language, decidability. We will review recent results on stability properties of expansions of (ℤ, +, 0) by a unary predicate and we will make the comparison with former results on the decidability and model-completeness of the corresponding expansions of (ℤ, +, 0, <) (and if time permits, with the corresponding expansions of the field of real numbers).

Slides

Logic Colloquium

April 07, 2017, 4:10 PM (60 Evans Hall)

Pierre Simon
Assistant Professor of Mathematics, UC Berkeley

On omega-categorical structures and their automorphism groups

The study of omega-categorical structures and their automorphism groups has been for several decades a very active area of research and point of contact between model theory, group theory, combinatorics and descriptive set theory. A countable structure is omega-categorical if its automorphism group acts oligomorphically on it, that is has finitely many orbits on n-tuples, for all n. In this talk, I will introduce this subject by presenting the various angles of approach and talk about a selection of topics, in particular concerning countable subgroups of automorphism groups. I will present a recent work with Itay Kaplan on the existence of finitely generated dense subgroups and ask a number of questions.

Alfred Tarski Lectures

April 10, 2017, 4:10 PM (50 Birge Hall)

Lou van den Dries
Professor, Department of Mathematics, University of Illinois at Urbana-Champaign

Model Theory as a Geography of Mathematics

I like to think of model theory as a geography of mathematics, especially of its “tame” side. Here tame roughly corresponds to geometric as opposed to combinatorial-arithmetic. In this connection I will discuss Tarski’s work on the real field, and the notion of o-minimality that it suggested.

A structure M carries its own mathematical territory with it, via interpretability: its own posets, groups, fields, and so on. Understanding this “world according to M” can be rewarding. Stability-like properties of M forbid certain combinatorial patterns, thus providing highly intrinsic and robust information about this world.

Alfred Tarski Lectures

April 12, 2017, 4:10 PM (50 Birge Hall)

Lou van den Dries
Professor, Department of Mathematics, University of Illinois at Urbana-Champaign

Orders of Infinity and Transseries

The “orders of Infinity” of the title are du Bois Reymond’s growth rates of functions, put on a firm basis by Hardy and Hausdorff. This led to the notion of a Hardy field (Bourbaki). Other ways of describing these orders of infinity are transseries (formal series containing powers of the variable x as well as exponential and logarithmic terms like ex and log x), and Conway’s surreal numbers. These topics are all closely related, with the differential field 𝕋 of transseries taking center stage.

Alfred Tarski Lectures

April 14, 2017, 4:10 PM (50 Birge Hall)

Lou van den Dries
Professor, Department of Mathematics, University of Illinois at Urbana-Champaign

Model Theory of Transseries: Results and Open Problems

The book Asymptotic Differential Algebra and Model Theory of Transseries (arXiv:1509.02588) by Aschenbrenner, van der Hoeven, and myself will soon appear as the Annals of Mathematics Studies, number 195. It contains the main results of twenty years of investigating 𝕋. One of these results is easy to state: The theory of this differential field is completely axiomatized by the requirements of being a Liouville closed H-field with small derivation and the intermediate value property for differential polynomials. I will explain the meaning of these terms, further results on 𝕋 and some consequences for what is definable in 𝕋.

I will also discuss some of the many attractive open problems in this area.

Logic Colloquium

April 21, 2017, 4:10 PM (60 Evans Hall)

Jana Marikova
Assistant Professor of Mathematics, Western Illinois University

Convexly valued o-minimal fields

O-minimal structures are ordered structures in which the definable sets behave in a particularly tame fashion. For example, definable one-variable functions are piecewise continuous and differentiable. A prototypical example is the ordered real field in which the definable sets are precisely the semialgebraic sets.

Valuations yield a way of understanding an (expansion of) a field in terms of two, often simpler, associated structures, namely its residue field and value group.

We shall investigate o-minimal fields with valuations that are determined by convex subrings. In particular, we shall see that the class of o-minimal fields with onvex subrings that give rise to o-minimal residue fields have various desirable properties, including quantifier elimination in a suitable language.

Slides