Logic Seminar

Location: MALL

Location: MALL
Title: Natural Deduction Proof for Substructural, Constructive and Classical Logics

Since the 1990s, we have seen how to understand a very wide range of logical systems (classical logic, intuitionistic logic, dual intuitionistic logic, relevant logics, linear logic, the Lambek calculus, affine logic, orthologic and more) by way of the distinction between operational and structural rules. We can have one set of rules for a connective (say, conjunction, negation, or the conditional) in a sequent calculus, and get different logical behaviour depending on the shape of the sequents allowed and the structural rules governing those sequents.

In this talk, I will consider the relationship between the “big four” traditional substructural logics—intuitionistic, relevant, affine and linear—corresponding to the four options for including or excluding the structural rules of weakening and contraction, in the setting of Gentzen/Prawitz-style natural deduction proofs for implication and the simply typed λ calculus. Such a natural deduction setting—in which proofs have any number of premises and a single conclusion—has a natural bias toward constructive, or intuitionistic logic.

I will show how the choice of whether to “go classical”, expanding the structural context to allow for more than one formula in positive position is orthogonal to the choice of the other structural rules, so that even in the context of natural deduction proofs, the familiar pair of traditional implication introduction and elimination rules gives rise to eight different propositional logics, four of which are “classical” and four of which are “constructive”. Furthermore, the familiar double-negation translation of classical logic inside intuitionistic logic generalises to the other three classical/constructive pairings.

I will explain these results, and, if there is time, I will end the talk with some reflections on what this might mean for the relationship between classical and constructive reasoning.

Location: MALL
Title: Peano arithmetic, games, and descent recursion

I will discuss a game semantics for classical (first-order) arithmetic due to Coquand. The real content of this semantics is a proof of cut elimination for infinitary propositional logic. As the title suggests, I will say something about descent recursive functions and how it all comes together.

Location: Roger Stevens LT 13 (10.13)
Title: Games, Comonads and Compositionality
NOTE location change: Roger Stevens LT 13 (10.13)

Recent work of Abramsky, Dawar and Wang, and subsequently Abramsky and Shah, provided a categorical abstraction of model comparison games such as the Ehrenfeucht-Fraïssé, pebble and bisimulation games, in the form of so-called game comonads. This work opened up new connections between disciplines associated with computational power and complexity such as finite model theory and combinatorics, and areas traditionally focussed upon the structural understanding of computation and logic, such as program semantics.

This talk will introduce the comonadic perspective upon model comparison games. I shall then describe more recent work, jointly with Tomáš Jakl and Nihil Shah, giving a categorical account of Feferman-Vaught-Mostowski type theorems with this categorical framework.

The talk will aim to be reasonably self-contained, assuming only a basic background in logic, and some understanding of the categorical notions of category, functor and natural transformation.

Location: MALL
Title: Definable groups in valued fields

I will discuss joint work with Gismatullin and Halupczok, giving the structure of definably (almost) simple groups definable in Henselian valued fields, possibly equipped with extra structure. I will also describe some other work on definable groups in valued fields.

Location: MALL
Title: The limit of Gödel's first incompleteness theorem
NOTE date and time change: Fri 4pm

In this talk, we discuss the limit of the first incompleteness theorem (G1). It is well known G1 can be extended to both extensions and weak sub-systems of PA. We examine the question: are there minimal theories for which G1 holds. The answer of this question depends on how we define the notion of minimality. We discuss different answers of this question based on varied notions of minimality.

The notion of interpretation provides us a general method to compare different theories in distinct languages. We examine the question: are there minimal theories for which G1 holds with respect to interpretability. It is known that G1 holds for essentially undecidable theories, and there are no minimal essentially undecidable theories with respect to interpretability. G1 holds for effectively inseparable (EI) theories and the notion of effective inseparability is much stronger than essential undecidability. A natural question is: are there minimal EI theories with respect to interpretability? We negatively answer this question and prove that there are no minimal effectively inseparable theories with respect to interpretability: for any EI theory T, we can effectively find a theory which is EI and strictly weaker than T with respect to interpretability. Moreover, we prove that there are no minimal finitely axiomatizable EI theories with respect to interpretability.

Finally, we give a summary of the similarities and differences between logical incompleteness and mathematical incompleteness based on technical evidences and philosophical reflections.

Location: Roger Stevens LT23 (8.23)
Title: Effective Pila–Wilkie bounds for Pfaffian sets with some diophantine applications
NOTE location change: Roger Stevens LT23 (8.23)

Following critical insights of Pila and Zannier, there are by now many applications of model theory to diophantine geometry arising from the celebrated counting theorem of Pila and Wilkie and its variants. The original Pila–Wilkie Theorem bounds the number of rational points of bounded numerator and denominator lying on (the transcendental parts of) sets definable in o-minimal expansions of the real field. However, the proof of this theorem (and that of its variants) does not provide an effective bound, which limits the precision of its applications. I will discuss some joint work with Gal Binyamini, Gareth O. Jones and Harry Schmidt in which we obtained effective forms of the Pila–Wilkie Theorem and its variants for sets definable in various structures described by Pfaffian functions (including an effective Yomdin–Gromov parameterization result for sets defined using restricted Pfaffian functions), and then used these effective estimates to derive several effective diophantine applications, including an effective, uniform Manin–Mumford statement for products of elliptic curves with complex multiplication.

Location: MALL
Title: Hilbert polynomials for finitary matroids

Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets $A$ and $B$ of a commutative semigroup, the size of the sumset $A+tB$ is eventually polynomial in $t$. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero.

Location: Social Sciences SR (14.33)
Title: Generalised measurability and bilinear forms
NOTE location change: Social Sciences SR (14.33)

In this talk I will go over measurable and generalised measurable structures, giving examples and non-examples. I will then go on to consider the two sorted structure $(V,F,β)$ where $V$ is an infinite dimensional vector space over $F$ an infinite field, and $β$ a bilinear form on this vector space. In particular I will consider the interaction of different notions of independence when this structure is pseudo finite. I will finish with some questions around generalised measurable structures.

Location: MALL
Title: Ordinal arithmetic and subgroups of Thompson's group

The class of finitely generated groups embeddable into Richard Thompson's group $F$ is both restrictive and rich at the same time. We show that there is a family of groups within this class which is pre-well-ordered in type $\epsilon_0$ by the embeddability relation. Moreover, the operations of addition and multiplication on the ordinals translate into natural group-theoretic operations—direct sum and a type of permutational wreath product. This talk will give a description of this correspondence. This is joint work with Collin Bleak and Matt Brin.