TY - GEN
T1 - Unary prime languages
AU - Jecker, Ismaël
AU - Kupferman, Orna
AU - Mazzocchi, Nicolas
N1 - Publisher Copyright: © Nathalie Bertrand; licensed under Creative Commons License CC-BY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020).
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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.
AB - 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.
KW - Deterministic Finite Automata (DFA)
KW - Primality
KW - Regular Languages
UR - http://www.scopus.com/inward/record.url?scp=85090505951&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.MFCS.2020.51
DO - https://doi.org/10.4230/LIPIcs.MFCS.2020.51
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
A2 - Esparza, Javier
A2 - Kral�, Daniel
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020
Y2 - 25 August 2020 through 26 August 2020
ER -