During the talk we discuss the lengths of proofs of arithmetical sentences in canonical truth theories. We focus mainly on extensions of the canonical compositional truth theory CT^-, introduced in the first talk of Bartosz Wcisło. In particular, we present a proof of the existence of a PTIME computable function f such that for every CT^- proof p of an arithmetical sentence A, f(p) is a PA-proof of A. Our proof relies on the arithmetization and uniformity of the Enayat-Visser model-theoretic construction of models with satisfaction classes. If time permits we shall comment on the speed-up phenomenon for various other truth theories, such as Kripke-Feferman (KF^-) and Friedman-Sheard (FS^-).
Logic seminar
usually takes place each Monday at 13:30 temporarily on Zoom starting at 15:30, normally in IM, rear building, ground floorChair: Pavel Pudlak, Neil Thapen, Jan Krajíček
More information on the old seminar web page. The programme is announced via the mailing list.
Proof theoretic properties of truth predicates II
Mateusz Lelyk
Department of Philosophy, University of Warsaw
Monday, 17. May 2021 - 15:30 to 17:00
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
Proof theoretic properties of truth predicates I
Bartosz Wcislo
Institute of Mathematics, Polish Academy of Sciences
Monday, 10. May 2021 - 15:30 to 17:00
Zoom meeting 472 648 284 - https://cesnet.zoom.us/j/472648284 - contact thapen@math.cas.cz to join
Axiomatic theory of truth is an area of logic which studies this notion in the following way: To a fixed base theory B (which in this talk will be Peano Arithmetic, PA), we add a fresh predicate T(x) with the intended reading "x is (a Goedel code of) a true sentence" and axioms which guarantee that the predicate T has a truth-like behaviour. For instance, we can stipulate that T satisfies Tarskian compositional conditions for arithmetical sentences (obtaining a theory called CT^-) or that it satisfies various forms of induction.
It is a classical result that the theory CT^- with the full induction scheme for the formulae containing the truth predicate, called CT, is not conservative over PA. In fact, it proves so called global reflection principle which says that an arbitrary arithmetical sentence provable in PA is true (in the sense of the truth predicate). On the other hand, a theorem by Kotlarski, Krajewski, and Lachlan shows that CT^- is conservative.
In the... more
Feasible disjunction property for intuitionistic modal logics
Raheleh Jalali
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.
Compactness at small cardinals
Radek Honzik
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.
Depth lower bounds in Stabbing Planes for combinatorial principles
Barnaby Martin
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
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