On optimal heuristic randomized semidecision procedures, with application to proof complexity

TL;DR

This paper explores the existence of optimal heuristic randomized semidecision procedures in proof complexity, showing that such optimal algorithms can exist when errors are permitted, contrasting with the open question of optimal proof systems.

Contribution

It introduces the concept of heuristic algorithms with bounded errors for proof systems, demonstrating their existence under certain conditions, and relates this to the broader question of optimal proof systems.

Findings

Optimal heuristic algorithms exist with bounded error rates.

Such algorithms can claim false theorems with small probability.

Results relate to generalized automatizable proof systems.

Abstract

The existence of a (p-)optimal propositional proof system is a major open question in (proof) complexity; many people conjecture that such systems do not exist. Krajicek and Pudlak (1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (2009) recently gave a conjecture implying that such algorithm does not exist. We show that in the presence of errors such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow the algorithm to claim a small number of false "theorems" (according to any samplable distribution on non-tautologies) and err with bounded probability on other inputs. Our result can also be viewed as the existence of an optimal proof system in a class of proof systems obtained by generalizing automatizable proof systems.