On Quantified Propositional Logics and the Exponential Time Hierarchy

TL;DR

This paper explores the complexity of quantified propositional logics, introducing ADQBF and analyzing their computational hardness, revealing their placement within the exponential time hierarchy and extending to team-based logics.

Contribution

It introduces ADQBF, generalizes existing propositional logics, and establishes their complexity classifications within the exponential hierarchy.

Findings

Truth evaluation for ADQBF is AEXPTIME(poly)-complete.

Certain fragments are complete for levels of the exponential hierarchy.

Extended propositional team logic with dependence atoms is AEXPTIME(poly)-complete.

Abstract

We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the exponential hierarchy. Second we study propositional team-based logics. We show that DQBF formulae correspond naturally to quantified propositional dependence logic and present a general NEXPTIME upper bound for quantified propositional logic with a large class of generalized dependence atoms. Moreover we show AEXPTIME(poly)-completeness for extensions of propositional team logic with generalized dependence atoms.