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Logic seminar

usually takes place each Monday at 13:30 temporarily on Zoom starting at 15:30, normally 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
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
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
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
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
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
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

temporarily on Zoom starting at 15:30, normally in IM, rear building, ground floor

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).

Propositional branching program proofs and logics for L and NL

Sam Buss
University of California, San Diego
Monday, 14. December 2020 - 15:30 to 17:00
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
We introduce systems of propositional logic for reasoning directly with decision trees, non-deterministic decision trees, branching programs and non-deterministic branching programs. These propositional systems allow reasoning about properties in non-uniform logarithmic space and non-deterministic logarithmic space. We also report on work-in-progress to use these propositional proof systems for the bounded arithmetic theories VL and VNL with proof theoretic strength corresponding to logarithmic space and non-deterministic logarithmic space. The talk will start with an overview of the propositional proof systems which are already known to have close correspondences with bounded arithmetic. The new results are joint work with Anupam Das and Alexander Knop.

Interactive theorem proving for the working logician

Jeremy Avigad
Carnegie Mellon University
Monday, 7. December 2020 - 15:30 to 17:00
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
Over the last few decades, computational proof assistants have made it possible to construct formal axiomatic derivations of increasing complexity. They are now used to verify that hardware and software designs meet their specifications, as well as to verify the correctness of mathematical proofs. The practice has taken root and promises to play an important role in mathematics and computer science.

In this talk, I will survey the technology, with an emphasis on formal mathematics. I will then discuss aspects of interactive theorem proving that may be of interest to the working logician, and places where a better theoretical understanding can lead to progress. Specifically, I'll discuss the need for practical foundations, search procedures, decision procedures, and proof systems.

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