Proof Complexity Modulo the Polynomial Hierarchy: Understanding Alternation as a Source of Hardness

TL;DR

This paper introduces a framework for analyzing proof complexity in quantified Boolean formulas using alternation-based lower bounds, and presents new results on proof system ensembles within the polynomial hierarchy.

Contribution

It develops the concept of proof system ensembles, especially relaxing QU-res, and proves exponential separation and lower bounds, advancing understanding of proof hardness related to alternation.

Findings

Exponential separation between tree-like and general relaxing QU-res

Exponential lower bounds for relaxing QU-res proof systems

Framework connects proof complexity with the polynomial hierarchy

Abstract

We present and study a framework in which one can present alternation-based lower bounds on proof length in proof systems for quantified Boolean formulas. A key notion in this framework is that of proof system ensemble, which is (essentially) a sequence of proof systems where, for each, proof checking can be performed in the polynomial hierarchy. We introduce a proof system ensemble called relaxing QU-res which is based on the established proof system QU-resolution. Our main results include an exponential separation of the tree-like and general versions of relaxing QU-res, and an exponential lower bound for relaxing QU-res; these are analogs of classical results in propositional proof complexity.