The canonical pairs of bounded depth Frege systems

0535773 - MÚ 2022 RIV NL eng J - Journal Article
Pudlák, Pavel
The canonical pairs of bounded depth Frege systems.
Annals of Pure and Applied Logic. Roč. 172, č. 2 (2021), č. článku 102892. ISSN 0168-0072. E-ISSN 1873-2461
R&D Projects: GA ČR(CZ) GX19-27871X
EU Projects: European Commission(XE) 339691 - FEALORA
Institutional support: RVO:67985840
Keywords : circuits * complexity * games * proofs
OECD category: Pure mathematics
Impact factor: 0.776, year: 2021
Method of publishing: Limited access
https://doi.org/10.1016/j.apal.2020.102892

The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth d Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized by a number k, also called the depth. We show that the canonical pair of a depth d Frege system is polynomially equivalent to the pair (Ad+2,Bd+2) where Ad+2 (respectively, Bd+1) are depth d+1 games in which Player I (Player II) has a positional winning strategy. Although this characterization is stated in terms of games, we will show that these combinatorial structures can be viewed as generalizations of monotone Boolean circuits. In particular, depth 1 games are essentially monotone Boolean circuits. Thus we get a generalization of the monotone feasible interpolation for Resolution, which is a property that enables one to reduce the task of proving lower bounds on the size of refutations to lower bounds on the size of monotone Boolean circuits. However, we do not have a method yet for proving lower bounds on the size of depth d games for d>1.
Permanent Link: http://hdl.handle.net/11104/0313709