# Logic Seminar

The Logic seminar is held on Wednesdays, usually at 12-14. During the fall 2022 we will have both on-site and online talks, and we try to keep the page updated on when we have which kind.

The permanent Zoom room for the seminar is: https://helsinki.zoom.us/j/62891400777?pwd=UldCeThTaTJVQjUzUFo4S2ErcndNQT09 (Meeting ID: 628 9140 0777, Passcode: 164195)

**PLEASE NOTE: Due to the prevalence of zoom talks given by scholars based in various time zones, the times that the seminar meets will occasionally ****change from the usual 12-14 slot. **Also, if not separately specified, the talks always start at a quarter past.

The seminar is led by prof. Juha Kontinen and Juliette Kennedy.

## Schedule of the fall term 2022

Wed 7.9.2022 12-14, Zoom

Miguel Moreno: Indestructibility and characterization of filter reflection

Abstract: Filter reflection is a natural generalization of the stationary reflection. It was introduced by Fernandes-Moreno-Rinot when studying the generalized-Borel-reducibility. It was shown that it has deep implications in the generalized Baire space and Cantor space. It is known that filter reflection is independent from ZFC, it doesn't require large cardinals, and it is consistent that filter reflection holds at any uncountable regular cardinal. In this talk we will go through some recent results about filter reflection, including indestructibility results. We will also show that filter reflection is equivalent to the existence of continuous reductions in the generalized Baire spaces.

Wed 14.9.2022 14-16, Zoom **PLEASE NOTE TIME CHANGE**

Andres Villaveces

**Around Definability in AECs: Logics and Patterns**

Abstract: Definability in Abstract Elementary Classes has been both central and elusive. The possibility itself of the development of a good stability theory (now solidly established) was made possible early by the Shelah Presentation Theorem. Yet, in spite of the centrality of this theorem for the start of the theory, definability questions were never at the center of the model theory of AECs, at least not in a direct way. Later, results by Kueker et al. started capturing better definitions (not second order) of some AECs. In 2020, with Saharon Shelah, we provided a way to define arbitrary AECs with no expansion of the vocabulary. Leung improved the bound of the logic, at the expense of an infinitary game quantifier. We later improved Leung's bound, removing the use of that game quantifier. More recently, in joint work with Nájar, we have started doing classification theory using our newly gained definability; recasting older results and capturing issues around definability of types. I will discuss this line of research, as well as possible connections with Hrushovski's recent work on "definability patterns" aimed at balancing the Galois theory of (first order) model theory.

Wed 21.9.2022 12-14, C124

Matilda Häggblom

Title: Axiomatizing modal inclusion logic

Abstract: Modal inclusion logic is team-based modal logic extended with inclusion atoms. The talk will cover the main result of my master's thesis: A complete axiomatization of modal inclusion logic. We begin by recalling that modal inclusion logic is expressively complete for classes that have the empty team property, are closed under unions and closed under k-bisimulation for some k. Through the expressive completeness proof, we obtain a normal form for the logic. In the talk, we suggest a simplified version of the normal form compered to the one currently in the literature. We then introduce a natural deduction proof system and show completeness using the normal form.

Wed 28.9.2022 12-14, C124

cancelled

Wed 5.10.2022 12-14, ~~C124~~ **ONLINE (Zoom)**

Tuomas Hakoniemi: Ideal Refutations of Knapsack Contradictions

Abstract: The Ideal Proof System (IPS), introduced by Grochow and Pitassi, is a strong algebraic proof system with close connections to central questions in algebraic circuit complexity. In this talk we present superpolynomial lower bounds in a constant-depth subsystem of IPS for a variant of Knapsack. Our argument builds on the recent breakthrough lower bounds for constant-depth algebraic circuits by Limaye, Srinivasan and Tavenas.

This talk is based on joint work with Nashlen Govindasamy and Iddo Tzameret.

Wed 12.10.2022 14-16, Zoom

Jouko Väänänen: Inner models from extended logics Slides of the talk

Abstract:

If we replace first order logic by second order logic in the

original definition of Gödel's inner model L, we obtain HOD. In this

talk we consider inner models that arise if we replace first order

logic by a logic that has some, but not all, of the strength of

second order logic. Typical examples are the extensions of first

order logic by generalized quantifiers, such as the Magidor-Malitz

quantifier, the cofinality quantifier, or stationary logic. It can

be shown that both L and HOD manifest some amount of formalism

freeness in the sense that they are not very sensitive to the choice

of the underlying logic.

On the other hand, the cofinality quantifier gives rise to a new

robust inner model between L and HOD. Assuming a proper class of

Woodin cardinals the regular cardinals above aleph-1 of V are weakly

compact in the inner model arising from the cofinality quantifier

and the theory of that model is (set) forcing absolute and

independent of the cofinality in question. Assuming three Woodin

cardinals and a measurable above them, if the construction is

relativized to a real, then on a cone of reals the Continuum

Hypothesis is true in the relativized model.

A potentially bigger inner model C(aa) arises from stationary logic.

Assuming a proper class of Woodin cardinals, or alternatively MM-

plus-plus, the regular uncountable cardinals of V are measurable in

the inner model C(aa), the theory of C(aa) is (set) forcing

absolute, and C(aa) satisfies CH. We introduce an auxiliary concept

that we call club determinacy, which simplifies the construction of

C(aa) greatly but may have also independent interest. Based on club

determinacy, we introduce the concept of aa-mouse which we use to

prove CH and other properties of the inner model C(aa). Finally, we

discuss a delicate matter related to the Axiom of Choice in the

inner model C(aa) and in inner models of the same kind.

This is joint work with Juliette Kennedy and Menachem Magidor.

Wed 19.10.2022 12-14, C124

Davide Quadrellaro: Weak Dependence Logic

Abstract: We introduce and study a fragment of dependence logic which we call weak dependence logic. The interest in this fragment lays in the fact that, although it is much less expressive than full dependence logic, it has three desirable features:

(i) a strong form of compactness;

(ii) the De Jongh property w.r.t. propositional dependence logic;

(iii) it admits a notion of model-theoretic type which gives rise to Esakia spaces.

We show these properties and establish connections to the algebraic and topological semantics of propositional dependence logic.

Wed 26.10.2022 12-14, C124

Exam week

Wed 2.11.2022 12-14, C124

Miika Hannula: Dependencies and information inequalities

Abstract: Dependence and independence can be interpreted as inequalities over Shannon entropies. In fact, basic principles about dependencies are already at work at the more abstract level of polymatroids, which encapsulate the elementary properties of the entropy function. In this talk we investigate these connections and survey some basic facts about information-theoretic inequalities.

Wed 9.11.2022 12-14, C124 + Zoom (hybrid)

Vadim Weinstein: BLURRY FILTERS AND CLASSIFICATION BY COUNTABLE STRUCTURES

(Joint work with Martina Iannella)

Abstract: The Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras. The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this talk we introduce a weaker concept which we call the “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration from totally disconnected ones to a larger class. As an application of this method, we show that the following are completely classifiable by countable structures: the homeomorphism on 3-manifolds (also applicable to 2-manifolds; but this was known since 1971), and wild embeddings of Cantor sets in R³. By "classification" in this talk we mean classical Borel-reducibility.

Wed 16.11.2022 12-14, C124

Tobias Boege: Incidence geometry, conditional independence and the existential theory of the reals

Abstract: Deciding whether a system of polynomial equations and inequalities has a solution over the real numbers is a basic task in computational geometry, optimization and algebraic statistics. Under polynomial-time many-one reductions, this problem generates the complexity class $\exists\mathbb{R}$. I will briefly recall a geometric technique, due to von Staudt, for obtaining completeness results for this complexity class and apply it to the implication problem for conditional independence among jointly normal random variables.

Wed 23.11.2022 12-14, C124 + **Zoom (hybrid)**

Balthasar Grabmayr

Title: Fixing Montague's Problem

Abstract: Turing machines only operate directly on strings. Turing computation over any other domain therefore requires a notation system for the domain. Ever since Montague's (1960) observation that different notation systems in general yield different notions of Turing computability, the task of distinguishing those notation systems that are admissible for computation from those that are not continues to be a much debated and open problem in the philosophy of computation. In the first part of this talk, I will introduce a generalized version of Montague's problem. In the second part, I will formulate and defend a solution to this problem.

Wed 30.11.2022 12-14, C124

Miika Hannula: Dependencies and information inequalities II

Abstract: Linear inequalities over joint entropies of finitely distributed random variables (called information inequalities) are often characterized as the laws of information theory. Unfortunately, verifying such laws is not an easy task. The information inequality problem (IIP), which is to decide whether a given information inequality is valid, is only known to be co-recursively enumerable. The conditional variant of this problem is undecidable, as a direct corollary of a recent undecidability result for the implication problem of conditional independence over random variables.

In this talk we are interested in approximating IIP. For instance, as every entropic function is a polymatroid, one obtains a sound approximation of IIP by considering the validity of information inequalities over polymatroids. This problem can be decided in exponential time through linear programming, but the exact complexity seems to be unknown. The aim of this talk is to locate some tractability boundaries for sound (or complete) approximation of IIP.

Wed 7.12.2022 12-14, C124

Aleksi Anttila: A remark on the dual negation in propositional/modal team semantics

Wed 14.12.2022 12-14, C124

Sabrina Gaube: TBA

Wed 21.12.2022 12-14, C124

Nina Pardal: TBA

## Schedule of the spring term 2023

Wed 18.1.2023 12-14, C124

Vadim Weinstein: BLURRY FILTERS AND CLASSIFICATION BY COUNTABLE STRUCTURES (PART II: PROOFS)

(Joint work with Martina Lannella)

Abstract: This is a continuation of the talk given on Nov 9^{th} 2022. In the first talk we gave an overview of the field, main ideas, and results. In this, second, talk we dive into the details of the central proofs. The same abstract as for the first talk follows:

The Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras. The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this talk we introduce a weaker concept which we call the “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration from totally disconnected ones to a larger class. As an application of this method, we show that the following are completely classifiable by countable structures: the homeomorphism on 3-manifolds (also applicable to 2-manifolds; but this was known since 1971), and wild embeddings of Cantor sets in R³. By "classification" in this talk we mean classical Borel-reducibility.

Wed 25.1.2023 12-14, C124

TBA

Wed 1.2.2023 12-14, C124

TBA

Wed 8.2.2023 12-14, C124

TBA

Wed 15.2.2023 12-14, C124

TBA

Wed 22.2.2023 12-14, C124

TBA

Wed 1.3.2023 12-14, C124

TBA

Wed 8.3.2023 12-14, C124

Exam week

Wed 15.3.2023 12-14, C124

TBA

Wed 22.3.2023 12-14, C124

TBA

Wed 29.3.2023 12-14, C124

TBA

Wed 5.4.2023 12-14, C124

TBA

Wed 12.4.2023 12-14, C124

Easter

Wed 19.4.2023 12-14, C124

TBA

Wed 26.4.2023 12-14, C124

TBA

Wed 3.5.2023 12-14, C124

TBA

Wed 10.5.2023 12-14, C124

Exam week