Raising a Hardness Result

TL;DR

This paper introduces a technique for proving computational problems hard for the polynomial hierarchy and PSPACE by leveraging problem restrictions that simulate quantifiers, enabling incremental hardness proofs.

Contribution

The paper presents a novel method to decompose hardness proofs into simpler steps by using problem restrictions that mimic quantifiers, simplifying complexity class reductions.

Findings

Technique allows incremental hardness proofs for polynomial hierarchy levels.

Reductions from QBFs can be transformed into simpler, staged proofs.

Method facilitates proving problems are hard for PSPACE and the polynomial hierarchy.

Abstract

This article presents a technique for proving problems hard for classes of the polynomial hierarchy or for PSPACE. The rationale of this technique is that some problem restrictions are able to simulate existential or universal quantifiers. If this is the case, reductions from Quantified Boolean Formulae (QBF) to these restrictions can be transformed into reductions from QBFs having one more quantifier in the front. This means that a proof of hardness of a problem at level n in the polynomial hierarchy can be split into n separate proofs, which may be simpler than a proof directly showing a reduction from a class of QBFs to the considered problem.