游戏关系和参数

L. D. Alfaro, R. Majumdar, Vishwanath Raman, Marielle Stoelinga
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引用次数: 42

摘要

我们考虑在无限回合的有限状态空间中进行的两人博弈。在每个状态下,玩家同时选择移动;这些举动决定了一个继承国。对于玩家来说,选择概率分布而不是单一的移动通常是有利的。给定一个目标(例如,“达到一个目标状态”),获胜问题就变成了一个概率问题:“在给定状态下获胜的最大概率是多少?”在这些游戏结构中,有两个基本概念是等价性和参数。给定一组获胜条件,如果玩家在两种状态下都能以相同的概率赢得同一场比赛,那么两种状态就是等效的。指标提供了各州获胜概率差异的界限,捕捉了各州“相似性”的定量概念。我们引入了双玩家博弈结构的等价和度量,并展示了它们表征了用定量微积分表达目标的获胜博弈概率的差异。定量微积分可以表达大量的目标,包括可达性、安全性和ω -正则性。因此,我们声称我们的关系和参数为游戏提供了典型的扩展,为过渡系统提供了经典的双模拟概念。我们开发了等效和度量的结果,它们推广了双模拟,以及不对称版本,它们推广了模拟。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Game Relations and Metrics
We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal (e.g., "reach a target state"), the question of winning is thus a probabilistic one: "what is the maximal probability of winning from a given state?". On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state "similarity". We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative mu-calculus. The quantitative mu- calculus can express a large set of goals, including reachability, safety, and omega-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation.
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