On Quantified Propositional Logics and the Exponential Time Hierarchy

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.Comment: In Proceedings GandALF 2016, arXiv:1609.0364

Axiomatizations of team logics

“…We extend QPL to quantified propositional team logic QPTL [11,12] and PL to propositional team logic PTL [32] as follows. For clarity, in the following we reserve the letters α, β, γ, .…”

“…The modality-free fragments of EMDL, MIL, MInc and MIncEx are propositional dependence logic PDL [31], propositional independence logic PIL [12], propositional inclusion logic PInc [12], and propositional inclusion/exclusion logic PIncEx, respectively. Adding propositional quantifiers ∃p and ∀p to these fragments yields quantified propositional dependence logic QPDL [11], quantified propositional independence logic QPIL [11], quantified Figure 11 depicts the axiom system D that defines the above atoms in terms of propositional/modal team logic. Let us abbreviate 2 n := {0, 1} n , i.e., the set of all n-ary truth vectors.…”

“…Beside first-order logic, all these atoms have also been adapted for modal logic ML [27], and (quantified) propositional logic PL resp. QPL [11,25,31].…”

On the complexity of the quantified bit-vector arithmetic with binary encoding

“…We now introduce the SO 2 logic more formally. The definitions of the syntax and semantics of SO 2 used in this paper are due to Hannula et al [14]. We call function symbols of arity 0 propositions and all other function symbols proper functions.…”

“…SO 2 syntax[14]). Let F be a countable set of function symbols, where each symbol f ∈ F is given an arity ar(f ) ∈ N 0 .…”