鲁棒多维平均收益目标的有限记忆策略综合

Yaron Velner
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引用次数: 7

摘要

图上的两人博弈为反应系统的研究提供了数学基础。在定量框架中,目标为每次游戏赋值,玩家1的目标是最小化目标值。在这个框架中,有两个相关的综合问题需要考虑:定量分析问题是计算玩家1可以保证的最小(或最小)值,布尔分析问题是玩家1是否可以保证目标值最多为ν(对于给定的阈值ν)。平均收益表达博弈是在多维加权图上进行的。原子平均收益表达式目标是某一维度的平均收益值(长期平均权重),平均收益表达式类是原子平均收益表达式在最大、最小、数值补和代数运算下的闭包。在这项工作中,我们首次研究了具有鲁棒定量目标的博弈,即具有平均收益表达目标的博弈的策略综合问题。一般来说,这些游戏的最佳策略需要无限内存,而综合而言,我们通常对有限状态系统的构建感兴趣。因此,我们考虑玩家1受有限记忆策略限制的游戏,我们的主要贡献如下。我们证明了对于平均收益表达式,定量分析问题是可计算的,并且布尔分析问题与希尔伯特第十问题是可互约的,这是计算机科学和数学中一个长期存在的基本开放问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Finite-memory strategy synthesis for robust multidimensional mean-payoff objectives
Two-player games on graphs provide the mathematical foundation for the study of reactive systems. In the quantitative framework, an objective assigns a value to every play, and the goal of player 1 is to minimize the value of the objective. In this framework, there are two relevant synthesis problems to consider: the quantitative analysis problem is to compute the minimal (or infimum) value that player 1 can assure, and the boolean analysis problem asks whether player 1 can assure that the value of the objective is at most ν (for a given threshold ν). Mean-payoff expression games are played on a multidimensional weighted graph. An atomic mean-payoff expression objective is the mean-payoff value (the long-run average weight) of a certain dimension, and the class of mean-payoff expressions is the closure of atomic mean-payoff expressions under the algebraic operations of max, min, numerical complement and sum. In this work, we study for the first time the strategy synthesis problems for games with robust quantitative objectives, namely, games with mean-payoff expression objectives. While in general, optimal strategies for these games require infinite-memory, in synthesis we are typically interested in the construction of a finite-state system. Hence, we consider games in which player 1 is restricted to finite-memory strategies, and our main contribution is as follows. We prove that for mean-payoff expressions, the quantitative analysis problem is computable, and the boolean analysis problem is inter-reducible with Hilbert's tenth problem over rationals --- a fundamental long-standing open problem in computer science and mathematics.
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