Unary Prime Languages

Authors Ismaël Jecker, Orna Kupferman, Nicolas Mazzocchi

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Author Details

Ismaël Jecker
  • Institute of Science and Technology, Klosterneuburg, Austria
Orna Kupferman
  • School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel
Nicolas Mazzocchi
  • Université Libre de Bruxelles, Belgium

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Ismaël Jecker, Orna Kupferman, and Nicolas Mazzocchi. Unary Prime Languages. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Mathematics of computing → Number-theoretic computations
  • Deterministic Finite Automata (DFA)
  • Regular Languages
  • Primality


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