Mean-Payoff Pushdown Games

TL;DR

This paper studies the computational complexity of solving pushdown games with mean-payoff objectives, revealing decidability, undecidability, and NP-completeness results for various strategy types and game settings.

Contribution

First analysis of mean-payoff objectives in pushdown games, establishing complexity and decidability results for global and modular strategies, and exploring strategy memory requirements.

Findings

Polynomial-time decidability for one-player pushdown games with global strategies

Undecidability of two-player pushdown games with global strategies

NP-completeness of pushdown games with modular strategies

Abstract

Two-player games on graphs is central in many problems in formal verification and program analysis such as synthesis and verification of open systems. In this work we consider solving recursive game graphs (or pushdown game graphs) that can model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives, such as reachability and $\omega$-regular objectives, in this work we study for the first time such games with the most well-studied quantitative objective, namely, mean-payoff objectives. In pushdown games two types of strategies are relevant: (1) global strategies, that depend on the entire global history; and (2) modular strategies, that have only local memory and thus does not depend on the context of invocation, but only on the history of the current invocation of the module. Our main results are as follows (1) One-player pushdown games with mean-payoff objectives under global strategies is decidable in polynomial time. (2) Two-player pushdown games with mean-payoff objectives under global strategies is undecidable. (3) One-player pushdown games with mean-payoff objectives under modular strategies is NP-hard. (4) Two-player pushdown games with mean-payoff objectives under modular strategies can be solved in NP (i.e., both one-player and two-player pushdown games with mean-payoff objectives under modular strategies is NP-complete). We also establish the optimal strategy complexity showing that global strategies for mean-payoff objectives require infinite memory even in one-player pushdown games; and memoryless modular strategies are sufficient in two-player pushdown games. Finally we also show that all the problems have the same complexity if the stack boundedness condition is added, where along with the mean-payoff objective the player must also ensure that the stack height is bounded.