Automata-theoretic decision of timed games

Abstract

The solution of games is a key decision problem in the context of verification of open systems and program synthesis. Given a game graph and a specification, we wish to determine if there exists a strategy of the protagonist that allows to select only behaviors fulfilling the specification. In this paper, we consider timed games, where the game graph is a timed automaton and the specification is given by formulas of the temporal logics Ltl and Ctl. We present an automata-theoretic approach to solve the addressed games, extending to the timed framework a successful approach to solve discrete games. The main idea of this approach is to translate the timed automaton A, modeling the game graph, into a tree automaton AT accepting all trees that correspond to a strategy of the protagonist. Then, given an automaton corresponding to the specification, we intersect it with the tree automaton AT and check for the nonemptiness of the resulting automaton. Our approach yields a decision algorithm running in exponential time for Ctl and in double exponential time for Ltl. The obtained algorithms are optimal in the sense that their computational complexity matches the known lower bounds.