Qualitative Reachability in Stochastic BPA Games

TL;DR

This paper studies infinite-state stochastic games generated by pushdown automata with qualitative reachability objectives, providing polynomial-time algorithms for certain constraints and automata-based characterizations of winning regions.

Contribution

It introduces a framework for analyzing qualitative reachability in stochastic BPA games and offers algorithms for determining winners and synthesizing strategies.

Findings

Winner determination is in PTIME for '>0' constraints.

Winner determination is in NP ∩ coNP for '=1' constraints.

Winning regions are regular and can be represented by finite automata.

Abstract

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint `>0' or `=1'. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in PTIME for the `>0' constraint, and both in NP and coNP for the `=1' constraint. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.