Complexity and Expressivity of Branching- and Alternating-Time Temporal Logics with Finitely Many Variables

We show that Branching-time temporal logics CTL and CTL, as well as Alternating-time temporal logics ATL and ATL, are as semantically expressive in the language with a single propositional variable as they are in the full language, i.e., with an unlimited supply of propositional variables. It follows that satisfiability for CTL, as well as for ATL, with a single variable is EXPTIME-complete, while satisfiability for CTL, as well as for ATL, with a single variable is 2EXPTIME-complete,--i.e., for these logics, the satisfiability for formulas with only one variable is as hard as satisfiability for arbitrary formulas.