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

##### Hubie Chen (University of the Basque Country et IKERBASQUE)

10 juin 2016 - 11:00If one looks at typical proof systems for QBF, such as Q-resolution, a dilemma is encountered: lower bounds for Q-resolution are implied immediately by lower bounds for resolution, yet this says nothing about Q-resolution’s ability to cope with quantifier alternation—and moreover clashes severely with the contemporary QBF view of SAT as “easy”. In this talk, we will discuss this dilemma and present a possible way to escape it.

In particular, 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 technical 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.

This talk will focus on a conceptual discussion of the work’s motivation, the framework and the main definitions.

This article will be presented at ICALP ’16; a version of this article is available at http://arxiv.org/abs/1410.5369.