Abstract

In 1973 E.W. Dijkstra introduced the notion of self-stabilization in the context of mutual exclusion. Considering the same problem on an array, we present a game theoretic analysis of self-stabilizing systems with three- ‎or four-state machines. We give a formalized definition of the problem as a game where each player’s strategy represents the state of its corresponding machine. For the three-state case, we prove the impossibility of any infinite self-stabilizing systems on an array. For the four-state case we consider two algorithms. For Ghosh’s solution [1] we prove the upper bound of (n – 1)(n – 3) steps and that this bound is tight. Also we present another four-state self-stabilizing system, ‎and prove that at most n² – 5n + 7 steps are required for the system to reach self-stabilization.

中文翻译：

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阵列自稳定系统的博弈论分析

摘要

1973年，EW迪克斯特拉（EW Dijkstra）在相互排斥的背景下提出了自我稳定的概念。考虑到阵列上的相同问题，我们提出了具有三态或四态机器的自稳定系统的博弈论分析。我们将问题的形式化定义为游戏，其中每个玩家的策略代表其相应机器的状态。对于三态情况，我们证明了阵列上任何无限的自稳定系统都是不可能的。对于四态情况，我们考虑两种算法。对于Ghosh的解[1]，我们证明了（n – 1）（n – 3）个步骤的上限，并且该界限是紧密的。此外，我们还提出了另一种四态自稳定系统，并证明最多n ² – 5系统要达到自我稳定，需要n + 7个步骤。