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Logic seminar

usually takes place each Monday at 16:00 in IM, rear building, ground floor
Chair: Pavel Pudlak, Neil Thapen, Jan Krajíček
More information on the old seminar web page. The programme is announced via the mailing list.

Resolution, Heavy Width and Pseudorandom Generators

Dmitry Sokolov
St Petersburg State University and PDMI RAS
Monday, 21. February 2022 - 16:00 to 17:30
Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson we call a pseudorandom generator hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement that b is outside of the image for any string b \in {0, 1}^m.

In ABRW04 the authors suggested the "functional encoding" of the considered statement for Nisan--Wigderson generator that allows the introduction of "local" extension variables and gave a lower bound on the length of Resolution proofs if the number of extension variables is bounded by the n^2 (where n is the number of inputs of the PRG).

In this talk, we discuss a "heavy width" measure for Resolution that allows us to show a lower bound on the length of Resolution proofs of the considered statement for the Nisan--Wigderson generator with a superpolynomial number of local extension variables. It is a solution to one of the open problems from ABRW04.

A revision of the width method

Michal Garlik
St. Petersburg Department of Steklov Institute of Mathematics and Euler International Mathematical Institute
Monday, 10. January 2022 - 16:00 to 17:30
We consider a framework for proving lower bounds on the length of proofs in logical calculi that can be seen as a generalization of the game framework in which Pudlák presented Haken's lower bound for resolution. Our framework, while still requiring some notion of width, can be applied in a way that avoids the use of the random restriction technique. As an example, we demonstrate this for resolution, obtaining a proof in which the calculations are very simple.

Arithmetic under negated induction

Tin Lok Wong
National University of Singapore
Monday, 13. December 2021 - 13:00 to 14:30
Arithmetic generally does not admit any non-trivial quantifier elimination. I will talk about one exception, where the negation of an induction axiom is included in the theory. Here the Weak Koenig Lemma from reverse mathematics arises as a model completion.
This work is joint with Marta Fiori-Carones (Novosibirsk), Leszek Aleksander Kolodziejczyk (Warsaw) and Keita Yokoyama (Sendai).

Learning algorithms versus automatability of Frege systems

Jan Pich
University of Oxford
Monday, 6. December 2021 - 16:00 to 17:30
We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system P, we prove that the following statements are equivalent,

1. Provable learning: P proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries.

2. Provable automatability: P proves efficiently that P is automatable by non-uniform circuits on propositional formulas expressing p-size circuit lower bounds.

Here, P is sufficiently strong and well-behaved if I.-III. holds: I. P p-simulates Jeřábek's system WF (which strengthens the Extended Frege system EF by a surjective weak pigeonhole principle); II. P satisfies some basic properties of standard proof systems which p-simulate WF; III. P proves efficiently for some Boolean function h that h is hard on average for circuits of subexponential size. For example, if III. holds... more

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