Almost Optimal Strategies in One Clock Priced Timed Games

Abstract

Abstract. We consider timed games extended with cost information, and prove computability of the optimal cost and of ε-optimal memoryless strategies in timed games with one clock. In contrast, this problem has recently been proved undecidable for timed games with three clocks.

Weighted o-minimal hybrid systems

“…Cost( ) < c) is definable. 18 Moreover, if such a run does not exist, there is a computable index n 0 , not depending on (q, y), such that every run of length greater than n 0 is such that Cost( ) > c (resp. Cost( ) ≥ c).…”

“…However, more involved properties like cost-optimal reachability control (find the minimum cost that can be ensured for reaching a given state, regardless of the behaviour of the environment in which the system is embedded) or WCTL model-checking (WCTL extends the branching-time temporal logic CTL with cost constraints on modalities [11,13]) have been proved undecidable for weighted timed automata with three clocks or more, see [11,12,10]. Though both problems have recently been proved decidable for one-clock weighted timed automata [18,17], these undecidability results are nevertheless disappointing, because the one-clock assumption is rather restrictive.…”

Model-Based Verification, Optimization, Synthesis and Performance Evaluation of Real-Time Systems

“…Later undecidability results [33,38] show that for weighted timed games with three or more clocks this condition (or a similar one) is necessary. Lately [36] proves that optimal reachability strategies are computable for one-clock weighted timed games, though there is an unsettled (large) gap between the known lower bound complexity P and an upper bound of 3EXPTIME.…”

Infinite Runs in Weighted Timed Automata with Energy Constraints

“…Examples include cost-optimal reachability problem in a single-cost setting [2,3], as well as in a multi-cost setting [20], or the computation of (mean-cost or discounted) costoptimal infinite schedules [6,16]. Also games where players try to optimize the cost have been considered [1,7,13,5,10], and several model-checking problems have been studied [12,5,9,11]. It is worth noticing that in these last frameworks (model-checking and games), most of the problems are undecidable, and they can be solved algorithmically only for the (restricted) class of one-clock automata.…”