It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The sameholds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

中文翻译：

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常规语言的根源和力量：通过有限自动机识别非常规属性

众所周知，常规语言中任何给定阶次的幂集，例如正方形，都不必是常规的。但是，有限自动机可以通过它们的根来识别它们。更准确地说，我们记得，给定规则语言L，L的平方根集合是规则的。对于任何n的第n个根以及所有非平凡根的集合，均是如此。我们为所有人提供具体的结构。使用以上结果，我们获得了许多关于幂的自然问题的决策算法。例如，给定两种常规语言，可以确定它们在每个长度上是否包含相同数量的正方形。最后，我们给出了自动机在常规语言中识别能力的大小的指数下限。此外，我们将重点介绍采用常规语言的小数幂和采用小数长度的前缀之间的有趣行为差异。确实，常规语言中的分数词根通常无法通过有限自动机来识别。