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arXiv · 2403.05088

Semidirect Product Decompositions for Periodic Regular Languages

Yusuke Inoue·Kenji Hashimoto·Hiroyuki Seki
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Abstract

The definition of period in finite-state Markov chains can be extended to regular languages by considering the transitions of DFAs accepting them. For example, the language $(ΣΣ)^*$ has period two because the length of a recursion (cycle) in its DFA must be even. This paper shows that the period of a regular language appears as a cyclic group within its syntactic monoid. Specifically, we show that a regular language has period $P$ if and only if its syntactic monoid is isomorphic to a submonoid of a semidirect product between a specific finite monoid and the cyclic group of order $P$. Moreover, we explore the relation between the structure of Markov chains and our result, and apply this relation to the theory of probabilities of languages. We also discuss the Krohn-Rhodes decomposition of finite semigroups, which is strongly linked to our methods.

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Yusuke Inoue, Kenji Hashimoto, Hiroyuki Seki. 2024-03-08. Semidirect Product Decompositions for Periodic Regular Languages. https://arxiv.org/abs/2403.05088

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