A Hyperlogic for Strategies in Stochastic Games (Extended Version)

Abstract: We propose a probabilistic hyperlogic called HyperSt$<sup>2$ that can express hyperproperties of strategies in turn-based stochastic games. To the best of our knowledge, HyperSt$<sup>2$ is the first hyperlogic for stochastic games. HyperSt$<sup>2$ can relate probabilities of several independent executions of strategies in a stochastic game. For example, in HyperSt$<sup>2$ it is natural to formalize optimality, i.e., to express that some strategy is better than all other strategies, or to express the existence of Nash equilibria. We investigate the expressivity of HyperSt$<sup>2$ by comparing it to existing logics for stochastic games, as well as existing hyperlogics. Though the model-checking problem for HyperSt$<sup>2$ is in general undecidable, we show that it becomes decidable for bounded memory and is in EXPTIME and PSPACE-hard over memoryless deterministic strategies, and we identify a fragment for which the model-checking problem is PSPACE-complete.