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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.

Compactness at small cardinals

Radek Honzik
Charles University
Monday, 12. April 2021 - 15:30 to 17:00
We will survey some results related to compactness principles at small cardinals which extend the usual first-order compactness to more complex structures.

More specifically, suppose kappa is an uncountable regular cardinal (typically kappa can be taken to be the size of the reals). We will review a variety of compactness principles, such as the tree property, stationary reflection, Rado's conjecture, etc., which claim that if all parts of size < kappa of a given structure of size kappa have some property, so does the whole structure.

We will discuss basic models in which such principles hold, consistency strength of these principles, implications between the principles and other hypotheses (such as CH), and some consequences.

Depth lower bounds in Stabbing Planes for combinatorial principles

Barnaby Martin
Durham University
Monday, 29. March 2021 - 15:30 to 17:00
We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in the case of the Pigeonhole principle are tight.

The technique known so far to prove depth lower bounds for Stabbing Planes is a generalization of that used for the Cutting Planes proof system. In this work we introduce two new approaches to prove length/depth lower bounds in Stabbing Planes: one relying on Sperner's Theorem which works for the Pigeonhole principle and Tseitin contradictions over the complete graph; a second proving the lower bound for Tseitin contradictions over a grid graph, which uses a result on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz.

(Joint work with Stefan... more

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.

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