Unary prime languages

Ismaël Jecker, Orna Kupferman, Nicolas Mazzocchi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations


A regular language L of finite words is composite if there are regular languages L1, L2, . . ., Lt such that L = Tti=1 Li and the index (number of states in a minimal DFA) of every language Li is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [9], 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.

Original languageAmerican English
Title of host publication45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
EditorsJavier Esparza, Daniel Kral�, Daniel Kral�
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771597
StatePublished - 1 Aug 2020
Event45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020 - Prague, Czech Republic
Duration: 25 Aug 202026 Aug 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
Country/TerritoryCzech Republic

Bibliographical note

Publisher Copyright:
© Nathalie Bertrand; licensed under Creative Commons License CC-BY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020).


  • Deterministic Finite Automata (DFA)
  • Primality
  • Regular Languages


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