Unary prime languages

I.R. Jecker, O. Kupferman, N. Mazzocchi, in:, 45th International Symposium on Mathematical Foundations of Computer Science, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.

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Author
Jecker, Ismael RIST Austria; Kupferman, Orna; Mazzocchi, Nicolas
Department
Series Title
LIPIcs
Abstract
A regular language L of finite words is composite if there are regular languages L₁,L₂,…,L_t such that L = ⋂_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of ℕ, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than L’s, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs.
Publishing Year
Date Published
2020-08-18
Proceedings Title
45th International Symposium on Mathematical Foundations of Computer Science
Acknowledgement
Ismaël Jecker: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. Nicolas Mazzocchi: PhD fellowship FRIA from the F.R.S.-FNRS.
Volume
170
Article Number
51:1-51:12
Conference
MFCS: Symposium on Mathematical Foundations of Computer Science
Conference Location
Prague, Czech Republic
Conference Date
2020-08-24 – 2020-08-28
ISSN
IST-REx-ID

Cite this

Jecker IR, Kupferman O, Mazzocchi N. Unary prime languages. In: 45th International Symposium on Mathematical Foundations of Computer Science. Vol 170. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.MFCS.2020.51
Jecker, I. R., Kupferman, O., & Mazzocchi, N. (2020). Unary prime languages. In 45th International Symposium on Mathematical Foundations of Computer Science (Vol. 170). Prague, Czech Republic: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2020.51
Jecker, Ismael R, Orna Kupferman, and Nicolas Mazzocchi. “Unary Prime Languages.” In 45th International Symposium on Mathematical Foundations of Computer Science, Vol. 170. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.MFCS.2020.51.
I. R. Jecker, O. Kupferman, and N. Mazzocchi, “Unary prime languages,” in 45th International Symposium on Mathematical Foundations of Computer Science, Prague, Czech Republic, 2020, vol. 170.
Jecker IR, Kupferman O, Mazzocchi N. 2020. Unary prime languages. 45th International Symposium on Mathematical Foundations of Computer Science. MFCS: Symposium on Mathematical Foundations of Computer Science, LIPIcs, vol. 170.
Jecker, Ismael R., et al. “Unary Prime Languages.” 45th International Symposium on Mathematical Foundations of Computer Science, vol. 170, 51:1-51:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:10.4230/LIPIcs.MFCS.2020.51.
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2020-09-21
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