In this paper, we are interested in automata over infinite words and infinite duration games, that we view as general transition systems. We study transformations of systems using a Muller condition into ones using a parity condition, extending Zielonka's construction. We introduce the alternating cycle decomposition transformation, and we prove a strong optimality result: for any given deterministic Muller automaton, the obtained parity automaton is minimal both in size and number of priorities among those automata admitting a morphism into the original Muller automaton. We give two applications. The first is an improvement in the process of determinisation of B\"uchi automata into parity automata by Piterman and Schewe. The second is to present alternative proofs unifying several results about the possibility of relabelling deterministic automata with different conditions.

中文翻译：

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穆勒条件的最佳变换

在本文中，我们对无限单词和无限持续时间游戏的自动机感兴趣，我们将其视为一般的过渡系统。我们研究使用穆勒条件将系统转换为使用奇偶条件的系统的转换，从而扩展了Zielonka的构造。我们引入了交替循环分解变换，并证明了强大的最优性结果：对于任何给定的确定性Muller自动机，在允许态射进入原始Muller自动机的那些自动机中，所获得的奇偶校验自动机的大小和优先级数量均最小。我们给出两个申请。第一个是Piterman和Schewe将Buchuchi自动机确定为奇偶自动机的过程的改进。