The Complexity of Mean-Payoff Pushdown Games

K. Chatterjee, Yaron Velner
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引用次数: 3

Abstract

Two-player games on graphs are 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 model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives—such as reachability and ω-regular objectives—in this work, we study for the first time such games with the most well-studied quantitative objective, the mean-payoff objective. In pushdown games, two types of strategies are relevant: (1) global strategies, which depend on the entire global history; and (2) modular strategies, which have only local memory and thus do not depend on the context of invocation but rather 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 are decidable in polynomial time. (2) Two-player pushdown games with mean-payoff objectives under global strategies are undecidable. (3) One-player pushdown games with mean-payoff objectives under modular strategies are 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 are NP-complete). We also establish the optimal strategy complexity by 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.
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平均收益下推游戏的复杂性
图上的二人博弈是形式验证和程序分析中许多问题的核心,例如开放系统的综合和验证。在这项工作中,我们考虑求解递归博弈图(或下推博弈图),该图用递归对顺序程序的控制流进行建模。虽然之前已经用定性目标(如可达性和非规则目标)研究了下推游戏,但在这项工作中,我们第一次用研究得最充分的定量目标(平均收益目标)来研究这种游戏。在下推游戏中,两种类型的策略是相关的:(1)全局策略,它依赖于整个全局历史;(2)模块化策略,它只有局部内存,因此不依赖于调用的上下文,而只依赖于当前模块调用的历史。主要研究结果如下:(1)全局策略下具有平均收益目标的一人下推博弈在多项式时间内是可决定的。(2)全局策略下具有平均收益目标的二人下推博弈是不可确定的。(3)模块化策略下具有平均收益目标的单人下推博弈具有np难度。(4)模块化策略下具有平均收益目标的两人下推博弈可以在NP中求解(即模块化策略下具有平均收益目标的一人下推博弈和两人下推博弈都是NP完全的)。我们还通过证明平均收益目标的全局策略即使在一人下推博弈中也需要无限的内存,而在两人下推博弈中无内存模块化策略是足够的,从而建立了最优策略复杂性。最后,我们还表明,如果添加堆栈有界性条件,那么所有问题都具有相同的复杂性,其中除了平均收益目标外,玩家还必须确保堆栈高度有界。
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