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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.

Proof complexity of natural formulas via communication arguments

Dmitry Itsykson
Steklov Institute of Mathematics at St.Petersburg
Monday, 15. March 2021 - 15:30 to 17:00
A canonical communication problem Search(F) is defined for every unsatisfiable CNF F: an assignment to the variables of F is distributed among the communicating parties, they are to find a clause of F falsified by this assignment. Lower bounds on the communication complexity of Search(F) imply tree-size lower bounds, rank lower bounds, and size-space tradeoffs for the formula F in a large class of proof systems. All known lower bounds on Search(F) are realized on ad-hoc formulas F (i.e. they were introduced specifically for these lower bounds). We introduce a new communication complexity approach that allows establishing proof complexity lower bounds for natural formulas.

 

First, we demonstrate our approach for two-party communication and prove an exponential lower bound on the size of tree-like Res(+) refutations of the Perfect matching principle. Then we apply our approach to k-party communication complexity in the NOF model and obtain a lower bound on the randomized k-... more

Information in propositional proofs and algorithmic proof search

Jan Krajíček
Charles University
Monday, 1. March 2021 - 15:45 to 17:15
Motivated by the *informal* proof search problem:
 "Is there an optimal way to search for propositional proofs?"
I present a few proof complexity results, a new notion and some problems.

Feasible interpolation and (semi-)algebraic proof systems

Tuomas Hakoniemi
Universitat Politècnica de Catalunya
Monday, 15. February 2021 - 15:30 to 17:00
In this talk we discuss feasible interpolation for the algebraic and semialgebraic proof systems Polynomial Calculus, Sums-of-Squares and Sherali-Adams. We show that each system admits feasible interpolation in the following form: there is a poly-time algorithm that given two sets P(x,z) and Q(y,z) of polynomial constraints in distinct sequences x,y and z of variables; a refutation of the union of P(x,z) and Q(y,z) and an assignment a to the z variables, outputs either a refutation of P(x,a) or a refutation of Q(y,a). In each case the proof combines a semantic proof of the existence of a suitable resource-bounded refutation of either P(x,a) or Q(y,a) with an efficient proof search algorithm for the said refutations.

Quantified Boolean formulas: proof systems, circuit complexity, and solving

Olaf Beyersdorff
Friedrich Schiller University Jena
Monday, 11. January 2021 - 15:30 to 17:00
This talk will start with an overview of the relatively young field of QBF proof complexity, explaining QBF proof systems (including QBF resolution) and an assessment of which lower bound techniques are available for QBF proof systems. In the main part of the talk, I will explain hardness characterisations for QBF proof systems in terms of circuit complexity, yielding very direct connections between circuit lower bounds and QBF proof system lower bounds. The talk will also cover the relations between QBF resolution and QCDCL solving algorithms. Modelling QCDCL as proof systems we show that QCDCL and Q-Resolution are incomparable.

This talk is based on two recent papers, joint with Joshua Blinkhorn and Meena Mahajan (LICS'20) and with Benjamin Boehm (ITCS'21).

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