Size, Cost, and Capacity: A Semantic Technique for Hard Random QBFs

TL;DR

This paper introduces a semantic technique based on size, cost, and capacity to establish exponential proof-size lower bounds for various QBF proof systems, including random and classical formulas.

Contribution

It formalizes a new lower bound technique for QBF proof systems using semantic measures, enabling the first high-probability exponential bounds for random QBFs.

Findings

Proves exponential lower bounds for equality QBFs.

Establishes high-probability exponential bounds for random QBFs.

Provides a simple proof of hardness for well-known formulas.

Abstract

As a natural extension of the SAT problem, an array of proof systems for quantified Boolean formulas (QBF) have been proposed, many of which extend a propositional proof system to handle universal quantification. By formalising the construction of the QBF proof system obtained from a propositional proof system by adding universal reduction (Beyersdorff, Bonacina & Chew, ITCS `16), we present a new technique for proving proof-size lower bounds in these systems. The technique relies only on two semantic measures: the cost of a QBF, and the capacity of a proof. By examining the capacity of proofs in several QBF systems, we are able to use the technique to obtain lower bounds based on cost alone. As applications of the technique, we first prove exponential lower bounds for a new family of simple QBFs representing equality. The main application is in proving exponential lower bounds with high probability for a class of randomly generated QBFs, the first `genuine' lower bounds of this kind, which apply to the QBF analogues of resolution, Cutting Planes, and Polynomial Calculus. Finally, we employ the technique to give a simple proof of hardness for the prominent formulas of Kleine B\"uning, Karpinski and Fl\"ogel.