One-Counter Stochastic Games

TL;DR

This paper investigates the computational complexity of decision problems in one-counter stochastic games, focusing on termination and reward objectives, and establishes complexity bounds for various cases including 1-player and 2-player scenarios.

Contribution

It provides complexity classifications for qualitative and quantitative problems in one-counter stochastic games, including NP∩coNP membership and polynomial-time results for 1-player cases.

Findings

Qualitative termination problem is in NP∩coNP.

1-player OC-SSGs can be solved in polynomial time.

Qualitative problems are at least as hard as known finite-state SSG problems.

Abstract

We study the computational complexity of basic decision problems for one-counter simple stochastic games (OC-SSGs), under various objectives. OC-SSGs are 2-player turn-based stochastic games played on the transition graph of classic one-counter automata. We study primarily the termination objective, where the goal of one player is to maximize the probability of reaching counter value 0, while the other player wishes to avoid this. Partly motivated by the goal of understanding termination objectives, we also study certain "limit" and "long run average" reward objectives that are closely related to some well-studied objectives for stochastic games with rewards. Examples of problems we address include: does player 1 have a strategy to ensure that the counter eventually hits 0, i.e., terminates, almost surely, regardless of what player 2 does? Or that the liminf (or limsup) counter value equals infinity with a desired probability? Or that the long run average reward is >0 with desired probability? We show that the qualitative termination problem for OC-SSGs is in NP intersection coNP, and is in P-time for 1-player OC-SSGs, or equivalently for one-counter Markov Decision Processes (OC-MDPs). Moreover, we show that quantitative limit problems for OC-SSGs are in NP intersection coNP, and are in P-time for 1-player OC-MDPs. Both qualitative limit problems and qualitative termination problems for OC-SSGs are already at least as hard as Condon's quantitative decision problem for finite-state SSGs.