分解置换自动机
Decomposing Permutation Automata
摘要：
一个确定性有限自动机（DFA）是复合的，如果它的语言可以分解成一个交集的语言的较小的DFA。否则，A是质数。这个素性概念是由Kupferman和Mosheiff在2013年提出的，虽然他们证明了我们可以决定DFA是否是复合的，但这个问题的精确复杂性仍然悬而未决，上界和下界之间存在双指数差距。在这项工作中，我们专注于置换DFA，即，其中，过渡幺半群是一个群。给出了判定置换DFA是否为复合的NP算法，并指出该问题的难点在于实例的非接受状态数：给出了以拒绝状态数为参数的固定参数易处理算法.此外，我们还研究了交换置换DFA类。它们的结构属性允许我们在NLOGSPACE中决定组合性，甚至在LOGSPACE中，如果字母表大小是固定的。尽管这种低复杂性，我们表明，复杂的行为仍然出现在这个类：我们提供了一个家庭的复合DFA，每个需要多项式的许多因素，其大小。我们还考虑了这个问题的变体，即对于给定的整数k，DFA是否是k因子复合的，也就是说，可分解为k个较小的DFA。我们表明，对于交换置换DFA，限制因子的数量使得决策在计算上更加困难，并产生一个具有严格边界的问题：它是NP完全的。最后，我们表明，在一般情况下，这个问题是在PSPACE，它是在LOGSPACE的DFA与单例字母表。
英文摘要：
A deterministic finite automaton (DFA) is composite if its language can be decomposed into an intersection of languages of smaller DFAs. Otherwise, A is prime. This notion of primality was introduced by Kupferman and Mosheiff in 2013, and while they proved that we can decide whether a DFA is composite, the precise complexity of this problem is still open, with a doubly-exponential gap between the upper and lower bounds. In this work, we focus on permutation DFAs, i.e., those for which the transition monoid is a group. We provide an NP algorithm to decide whether a permutation DFA is composite, and show that the difficulty of this problem comes from the number of non-accepting states of the instance: we give a fixed-parameter tractable algorithm with the number of rejecting states as the parameter. Moreover, we investigate the class of commutative permutation DFAs. Their structural properties allow us to decide compositionality in NLOGSPACE, and even in LOGSPACE if the alphabet size is fixed. Despite this low complexity, we show that complex behaviors still arise in this class: we provide a family of composite DFAs each requiring polynomially many factors with respect to its size. We also consider the variant of the problem that asks whether a DFA is k-factor composite, that is, decomposable into k smaller DFAs, for some given integer k. We show that, for commutative permutation DFAs, restricting the number of factors makes the decision computationally harder, and yields a problem with tight bounds: it is NP-complete. Finally, we show that in general, this problem is in PSPACE, and it is in LOGSPACE for DFAs with a singleton alphabet.
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