• DOI:10.4230/LIPIcs.MFCS.2020.51
• Corpus ID: 221162235

Unary Prime Languages

@inproceedings{Jecker2020UnaryPL,
  title={Unary Prime Languages},
  author={Isma{\"e}l Jecker and Orna Kupferman and Nicolas Mazzocchi},
  booktitle={MFCS},
  year={2020}
}

• Ismaël Jecker, O. Kupferman, Nicolas Mazzocchi
• Published in MFCS 2020
• Computer Science, Linguistics

11 A regular language L of finite words is composite if there are regular languages L1, L2, . . . , Lt such 12 that L = ⋂t i=1 Li and the index (number of states in a minimal DFA) of every language Li is strictly 13 smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced 14 and studied in [9], where the complexity of deciding the primality of the language of a given DFA 15 was left open, with a doubly-exponential gap between the upper and lower bounds. We study 16 primality for unary regular languages, namely regular languages with a singleton alphabet. A unary 17 language corresponds to a subset of N, making the study of unary prime languages closer to that of 18 primality in number theory. We show that the setting of languages is richer. In particular, while 19 every composite number is the product of two smaller numbers, the number t of languages necessary 20 to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness 21 for a unary language L, namely a word that is not in L but is in all products of languages that 22 contain L and have an index smaller than L’s, may be of exponential length. Still, we are able 23 to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace 24 algorithm for primality checking of unary DFAs. 25 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; 26 Mathematics of computing → Mathematical analysis 27