Complete Problems of Propositional Logic for the Exponential Hierarchy

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.