Expected Reachability-Price Games

Probabilistic timed automata(PTA) model real-time systems with non-deterministic and stochastic behavior. They extend Alur-Dill timed automata by allowing probabilistic transitions and a price structure on the locations and transitions. Thus, a PTA can be considered as a Markov decision process (MDP) with uncountably many states and transitions. Expected reachability-price games are turn-based games where two players, player \(\text {Min}\) and player \(\text {Max}\), move a token along the infinite configuration space of PTA. The objective of player \(\text {Min}\) is to minimize the expected price to reach a target location, while the goal of the \(\text {Max}\) player is the opposite. The undecidability of computing the value in the expected reachability-price games follows from the undecidability of the corresponding problem on timed automata. A key contribution of this work is a characterization of sufficient conditions under which an expected reachability-price game can be reduced to a stochastic game on a stochastic generalization of corner-point abstraction (a well-known finitary abstraction of timed automata). Exploiting this result, we show that expected reachability-price games for PTA with single clock and price-rates restricted to \(\{ 0, 1 \}\) are decidable.