Complexity of finite-variable fragments of propositional temporal and modal logics of computation

Abstract

We prove that branching-time temporal logics CTL and CTL⁎ are polynomial-time embeddable into their single-variable fragments. It follows that satisfiability for CTL and CTL⁎, and therefore also for alternating-time temporal logics ATL and ATL⁎, in languages with one propositional variable is as algorithmically hard as satisfiability for the full logic: EXPTIME-complete for CTL and ATL, and 2EXPTIME-complete for CTL⁎ and ATL⁎. We discuss applicability of the technique used in the proofs to other modal and temporal logics, including propositional dynamic, epistemic, and temporal-epistemic logics, as well as many-dimensional and first-order modal and temporal logics.