1½-Player Stochastic StopWatch Games

Stochastic timed games (STGs), introduced by Bouyer and Forejt, generalize continuous-time Markov chains and timed automata. Depending on the number of players - 2, 1, or 0 - subclasses of stochastic timed games are classified as 2½-player, 1½-player, and ½-player games where the ½ symbolizes the presence of the stochastic player. The qualitative and quantitative reachability problem for STGs was studied in [Patricia Bouyer and Vojtech Forejt, 2009] and [S. Akshay et al., 2016]. In this paper, we introduce stochastic stopwatch games (SSG), an extension of (STG) from clocks to stopwatches. We focus on 1½-player SSGs and prove that with two variables which can be either a clock or a stopwatch, qualitative reachability is decidable, whereas, if we increase the number of variables to three, with at least one stopwatch, the problem becomes undecidable.