We display an application of the notions of kernelization and data reduction from parameterized complexity to proof complexity: Specifically, we show that the existence of data reduction rules for a parameterized problem having (a) a small-length reduction chain, and (b) small-size (extended) Frege proofs certifying the soundness of reduction steps implies the existence of subexponential size (extended) Frege proofs for propositional formalizations of the given problem. We apply our result to infer the existence of subexponential Frege and extended Frege proofs for a variety of problems. Improving earlier results of Aisenberg et al. (ICALP 2015), we show that propositional formulas expressing (a stronger form of) the Kneser-Lovasz Theorem have polynomial size Frege proofs for each constant value of the parameter k. Previously only quasipolynomial bounds were known (and only for the ordinary Kneser-Lovasz Theorem). Another notable application of our framework is to impossibility results in computational social choice: we show that, for any fixed number of agents, propositional translations of the Arrow and Gibbard-Satterthwaite impossibility theorems have subexponential size Frege proofs.

This is joint work with Cosmin Bonchiș and Adrian Crăciun.

Dynamic epistemic logics are useful in reasoning about knowledge and acts of learning, seen as epistemic actions. However, not all epistemic actions are allowed to be executed in an initial epistemic model, and this is where the concept of a protocol comes in: a protocol stipulates what epistemic actions are allowed to be performed in a model. The aim of my presentation is to introduce the audience to two accounts of protocols in DEL: Hoshi’s [1], and Wang’s [2].

References:

We study the matching problem “∃σ:σ(L)⊆R?” where L and R are regular tree languages over finite ranked alphabets X and Σ respectively, and σ is a substitution such that σ(x) is a set of trees in T(Σ∪H)∖H for all x∈X. Here, H denotes a set of holes which are used to define a concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook Regular algebra and finite machines and showed that if L and R are regular, then the problem “∃σ:∀x∈X:σ(x)≠∅∧σ(L)⊆R?” is decidable. Moreover, there are only finitely many maximal solutions, which are regular and effectively computable. We extend Conway’s results when L and R are regular languages of finite and infinite trees, and language substitution is applied inside-out. We show that if L⊆T(X) and R⊆T(Σ) are regular tree languages then the problem “∃σ∀x∈X:σ(x)≠∅∧σ_io(L)⊆R?” is decidable. Moreover, there are only finitely many maximal solutions, which are regular and effectively computable. The corresponding question for the outside-in extension σ_oi remains open, even in the restricted setting of finite trees. Our main result is the decidability of “∃σ:σ_io(L)⊆R?” if R is regular and L belongs to a class of tree languages closed under intersection with regular sets. Such a special case pops up if L is context-free.
This is joint work with Carlos Camino, Volker Diekert, Besik Dundua and Géraud Sénizergues.

References:

[1] C. Camino, V. Diekert, B. Dundua, M. Marin, G. Sénizergues, Regular matching problems for infinite trees. arXiv:2004.09926 [cs.FL], 2021.

References:

In a previous exposition [1] we have seen that the solvability over ℚ is undecidable for systems of exponential Diophantine equations. We now show that the solvability of individual exponential Diophantine equations is also undecidable, and that this happens as well for some narrower families of exponential Diophantine equations.

References:

[1] M. Prunescu, The exponential Diophantine problem for ℚ. The Journal of Symbolic Logic, Volume 85, Issue 2, 671–672, 2020.

The unwinding of proofs program dates back to Kreisel in the fifties and rests on the following broad question: “What more do we know if we have proved a theorem by restricted means than if we merely know that it is true?”

In this talk, we set out to give a brief introduction to the proof mining program, focusing on the following points:

• functional interpretations in an introductory way;
• the bounded functional interpretation [3,4];
• a concrete translation example: the metric projection argument.

We finish with a brief discussion of some recent results [5,6].

References:

[1] U. Kohlenbach. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer-Verlag Berlin Heidelberg, 2008.
[2] U. Kohlenbach. Proof-theoretic methods in nonlinear analysis. In M. Viana B. Sirakov, P. Ney de Souza, editor, Proceedings of the International Congress of Mathematicians – Rio de Janeiro 2018, Vol. II: Invited lectures, pages 61–82. World Sci. Publ., Hackensack, NJ, 2019.
[3] F. Ferreira and P. Oliva. Bounded functional interpretation. Annals of Pure and Applied Logic, 135:73-112, 2005.
[4] F. Ferreira. Injecting uniformities into Peano arithmetic. Annals of Pure and Applied Logic, 157:122-129, 2009.
[5] B. Dinis and P. Pinto. Effective metastability for a method of alternating resolvents. arXiv:2101.12675 [math.FA], 2021.
[6] U. Kohlenbach and P. Pinto. Quantitative translations for viscosity approximation methods in hyperbolic spaces. arXiv:2102.03981 [math.FA], 2021.

 

References:

In 1958, Gödel introduced his functional interpretation as a method of reducing the consistency of first-order arithmetic to that of a quantifier-free system of primitive recursive functionals of higher type. His work has since enabled other advances in proof theory, notably the program of proof mining. We give a brief overview of Gödel’s System T and then explore a formalization thereof as a deep embedding in the Lean proof assistant.

References:

[1] mathlib Community, The Lean Mathematical Library, 2019.