文章基本信息

标题：Unary Prime Languages
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作者：Isma{"e}l Jecker ; Orna Kupferman ; Nicolas Mazzocchi 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：170
页码：51:1-51:12
DOI：10.4230/LIPIcs.MFCS.2020.51
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：A regular language L of finite words is composite if there are regular languages Lâ,,Lâ,,,â¦,L_t such that L = â<,_{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 â"., 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 Lâs, 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.
关键词：Deterministic Finite Automata (DFA); Regular Languages; Primality