Chair: Pavel Pudlak, Neil Thapen, Jan Krajíček

More information on the old seminar web page. The programme is announced via the mailing list.

Utrecht University

Monday, 26. April 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 present a uniform method to prove feasible disjunction property (DP) for various intuitionistic modal logics. More specifically, we prove that if the rules in a sequent calculus for a modal intuitionistic logic have a special form, then the sequent calculus enjoys feasible DP. Our method is essentially an adaptation of the method used by Hrubes in his lower bound proof for the intuitionistic Frege system. As a consequence, we uniformly prove that the sequent calculi for intuitionistic logic, the intuitionistic version of several modal logics such as K, T, K4, S4, S5, their Fisher-Servi versions, propositional lax logic, and many others have feasible DP. Our method also provides a way to prove negative results: we show that any intermediate modal logic without DP does not have a calculus of the given form. This talk is based on a joint work with Amir Tabatabi.

Charles University

Monday, 12. April 2021 - 15:30 to 17:00

Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join

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.

Durham University

Monday, 29. 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

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

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

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.

"Is there an optimal way to search for propositional proofs?"

I present a few proof complexity results, a new notion and some problems.

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.