Complete Problems of Propositional Logic for the Exponential Hierarchy

TL;DR

This paper explores a generalized propositional logic with higher-order quantifiers to identify complete problems within the exponential hierarchy, extending known results from polynomial hierarchy problems.

Contribution

It introduces Boolean higher-order quantifiers and generalizes existing complete problems to the exponential hierarchy, filling a gap in complexity theory.

Findings

Generalized propositional logic with higher-order quantifiers.

Complete problems for the exponential hierarchy are identified.

Hardness results remain robust across different normal forms.

Abstract

Large complexity classes, like the exponential time hierarchy, received little attention in terms of finding complete problems. In this work a generalization of propositional logic is investigated which fills this gap with the introduction of Boolean higher-order quantifiers or equivalently Boolean Skolem functions. This builds on the important results of Wrathall and Stockmeyer regarding complete problems, namely QBF and QBF-k, for the polynomial hierarchy. Furthermore it generalizes the Dependency QBF problem introduced by Peterson, Reif and Azhar which is complete for NEXP, the first level of the exponential hierarchy. Also it turns out that the hardness results do not collapse at the consideration of conjunctive and disjunctive normal forms, in contrast to plain QBF.