Expected Termination Time in BPA Games

Abstract

AbstractWe consider the problem of computing the value and finding the epsilon-optimal strategies for concurrent Basic Process Algebra games, which is a subclass of two-player infinite-state stochastic games with imperfect information. These games are played on the transition graph of stateless pushdown systems, or equivalently 1-exit recursive state machines, and can model recursive procedural program execution with probabilistic transitions. The objective of one player in these games is to minimise the expected termination time of such a program, while the objective of the other is to maximise it. We show that the quantitative decision questions regarding the value of the game as well as checking whether this value is infinite can be answered in PSPACE. We also show the latter problem to be as hard as the square root sum, whose containment even in the polynomial hierarchy is an open problem since the 1970s. Furthermore, an optimal strategy may require an infinite amount of memory in general, but we show that both player have epsilon-optimal stackless&memoryless strategies (i.e. strategies that do not use memory nor depend on the stack content). Finally, we show how to find such strategies using a strategy improvement algorithm.