# 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 CSP on algebras with linear congruence

##### Azza Gaysin
Charles University
Monday, 4. April 2022 - 16:00 to 17:30
CSP (constraint satisfaction problems) is a class of problems deciding whether there exists a homomorphism from an instance relational structure to a target one. The CSP dichotomy is a profound result recently proved by Zhuk (2020, J. ACM, 67) and Bulatov (2017, FOCS, 58). It says that for each fixed target structure, CSP is either NP-complete or p-time. Zhuk's algorithm for the p-time case of the problem eventually leads to algebras with linear congruence.

We show that Zhuk's algorithm for algebras with linear congruence can be formalized in the theory V^1. Thus, using known methods of proof complexity, Zhuk's algorithm for negative instances of the problem can be augmented by extra information: it not only rejects X that cannot be homomorphically mapped into A but produces a certificate - a short extended Frege (EF) propositional proof - that this rejection is correct.

### A Walk with Goodstein

##### David Fernández-Duque
Ghent University and Czech Academy of Sciences
Monday, 21. March 2022 - 16:00 to 17:30
The classical Goodstein process is based on writing numbers in "normal form" in terms of addition and exponentiation with some base k. By iteratively changing base and subtracting one, one obtains very long sequences of natural numbers which eventually terminate. The latter is proven by comparing base-k normal forms with Cantor normal forms for ordinals, and in fact this proof relies heavily on the notion of normal form. The question then naturally arises: what if we write natural numbers in an arbitrary fashion, not necessarily using normal forms? What if we allow not only addition and exponentiation, but also multiplication for writing numbers?

A "Goodstein walk" is any sequence obtained by following the standard Goodstein process but arbitrarily choosing how each element of the sequence is represented. As it turns out, any Goodstein walk is finite, and indeed the longest possible Goodstein walk is given by the standard normal forms. In this talk we sketch a proof of this fact... more

### Nisan-Wigderson generators in Proof Complexity: New lower bounds

##### Erfan Khaniki
Institute of Mathematics
Monday, 14. March 2022 - 16:00 to 17:30
A map g:{0,1}^n --> {0,1}^m (m>n) is a hard proof complexity generator for a proof system P iff for every string b in {0,1}^m\Rng(g) the formula \tau_b(g), naturally expressing b \not \in Rng(g), requires superpolynomial size P-proofs. One of the well-studied maps in the theory of proof complexity generators is the Nisan-Wigderson generator. Razborov (Annals of Mathematics 2015) conjectured that if A is a suitable matrix and f is a NP \cap CoNP function hard-on-average for P/poly, then NW_{f, A} is a hard proof complexity generator for Extended Frege.

In this talk, we prove a form of Razborov's conjecture for AC0-Frege. We show that for any symmetric NP \cap CoNP function f that is exponentially hard for depth two AC0 circuits, NW_{f, A} is a hard proof complexity generator for AC0-Frege in a natural setting.

### On Semi-Algebraic Proofs and Algorithms

##### Robert Robere
McGill University
Monday, 7. March 2022 - 16:00 to 17:30
We discuss a new characterization of the Sherali-Adams proof system, a standard propositional proof system considered in both proof complexity and combinatorial optimization, showing that there is a degree-d Sherali-Adams refutation of an unsatisfiable CNF formula F if and only if there is an ε > 0 and a degree-d conical junta J such that viol(x) − ε = J, where viol(x) counts the number of falsified clauses of F on an input x. This result implies that the linear separation complexity, a complexity measure recently studied by Hrubes (and independently by de Oliveira Oliveira and Pudlak under the name of weak monotone linear programming gates), monotone feasibly interpolates Sherali-Adams proofs, sharpening a feasible interpolation result of Hakoniemi. On the lower-bound side, we prove a separation between the conical junta degree of viol(x) - 1 and Resolution width; since Sherali-Adams can simulate Resolution this also separates the conical junta degree of viol(x) - 1 and viol(x) -... more