Optimal Strategies in Priced Timed Game Automata

Abstract

Priced timed (game) automata extend timed (game) automata with costs on both locations and transitions. In this paper we focus on reachability priced timed game automata and prove that the optimal cost for winning such a game is computable under conditions concerning the non-zenoness of cost. Under stronger conditions (strictness of constraints) we prove that in case an optimal strategy exists, we can compute a state-based winning optimal strategy.

On Optimal Timed Strategies

“…In this context we introduce the concept of winning strategy and the related cost problems as mentioned in [9]. We begin with the definition of weighted timed automaton.…”

“…given an initial state s of a weighted timed automaton A, a cost bound C and a set T of locations, determine if Player 1 has a strategy to enforce the game started in state s into a location of T within k rounds, while ensuring that the cost is bounded by C. Their algorithmic solution has an exponential-time worst case complexity. In [9], the authors study winning strategies to reach a set of target locations with an optimal cost in a weighted timed automaton A. To compute the optimal cost and to synthetize an optimal winning strategy, they provide a semi-algorithm for which they can guarantee the termination under a condition called strict non-zenoness of cost.…”

“…This condition imposes that every cycle in the region automaton of A has a cost bounded away from zero. The general case where this condition is not imposed, is left open in both papers [4] and [9].…”

“…In this paper, we consider timed games played on a weighted timed automaton as they are introduced in [4], and following the lines of [9] we study the two problems of the existence of a winning strategy with a bounded cost, and of the existence of a winning strategy with an optimal cost (Section 2). We prove the unexpected negative result that for weighted timed automata, the existence of a winning strategy with a cost bounded by a given value is undecidable (Section 3, Theorem 1).…”

“…On the positive side, we show that if we restrict the number of clocks to one and we limit the cost rate to 0 or d where d is a fixed integer, then the two problems mentioned above are decidable (Section 4, Corollary 3). The proof follows the approach of [9] but we can prove the termination of their semi-algorithm without the non-zenoness of cost hypothesis.…”