Semidirect Product Decompositions for Periodic Regular Languages
Abstrak
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.
Topik & Kata Kunci
Penulis (3)
Yusuke Inoue
Kenji Hashimoto
Hiroyuki Seki
Akses Cepat
- Tahun Terbit
- 2024
- Bahasa
- en
- Sumber Database
- arXiv
- Akses
- Open Access ✓