State Complexity of Multiple Concatenation
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917106220531712 |
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| author | Jirásek, Jozef Jirásková, Galina |
| author_facet | Jirásek, Jozef Jirásková, Galina |
| contents | We describe witness languages meeting the upper bound on the state complexity of the multiple concatenation of $k$ regular languages over an alphabet of size $k+1$ with a significantly simpler proof than that in the literature. We also consider the case where some languages may be recognized by two-state automata. Then we show that one symbol can be saved, and we define witnesses for the multiple concatenation of $k$ languages over a $k$-letter alphabet. This solves an open problem stated by Caron et al. [2018, Fundam. Inform. 160, 255--279]. We prove that for the concatenation of three languages, the ternary alphabet is optimal. We also show that a trivial upper bound on the state complexity of multiple concatenation is asymptotically tight for ternary languages, and that a lower bound remains exponential in the binary case. Finally, we obtain a tight upper bound for unary cyclic languages and languages recognized by unary automata that do not have final states in their tails. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | State Complexity of Multiple Concatenation Jirásek, Jozef Jirásková, Galina Formal Languages and Automata Theory 68QXX We describe witness languages meeting the upper bound on the state complexity of the multiple concatenation of $k$ regular languages over an alphabet of size $k+1$ with a significantly simpler proof than that in the literature. We also consider the case where some languages may be recognized by two-state automata. Then we show that one symbol can be saved, and we define witnesses for the multiple concatenation of $k$ languages over a $k$-letter alphabet. This solves an open problem stated by Caron et al. [2018, Fundam. Inform. 160, 255--279]. We prove that for the concatenation of three languages, the ternary alphabet is optimal. We also show that a trivial upper bound on the state complexity of multiple concatenation is asymptotically tight for ternary languages, and that a lower bound remains exponential in the binary case. Finally, we obtain a tight upper bound for unary cyclic languages and languages recognized by unary automata that do not have final states in their tails. |
| title | State Complexity of Multiple Concatenation |
| topic | Formal Languages and Automata Theory 68QXX |
| url | https://arxiv.org/abs/2511.03814 |