文章基本信息

标题：Dynamic Membership for Regular Languages
本地全文：下载
作者：Amarilli, Antoine ; Jachiet, Louis ; Paperman, Charles 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2021
卷号：198
DOI：10.4230/LIPIcs.ICALP.2021.116
语种：English
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We study the dynamic membership problem for regular languages: fix a language L, read a word w, build in time O(|w|) a data structure indicating if w is in L, and maintain this structure efficiently under letter substitutions on w. We consider this problem on the unit cost RAM model with logarithmic word length, where the problem always has a solution in O(log|w| / log log|w|) per operation.We show that the problem is in O(log log|w|) for languages in an algebraically-defined, decidable class QSG, and that it is in O(1) for another such class QLZG. We show that languages not in QSG admit a reduction from the prefix problem for a cyclic group, so that they require Ω(log|w| /log log|w|) operations in the worst case; and that QSG languages not in QLZG admit a reduction from the prefix problem for the multiplicative monoid U₁ = {0, 1}, which we conjecture cannot be maintained in O(1). This yields a conditional trichotomy. We also investigate intermediate cases between O(1) and O(log log|w|).Our results are shown via the dynamic word problem for monoids and semigroups, for which we also give a classification. We thus solve open problems of the paper of Skovbjerg Frandsen, Miltersen, and Skyum [Skovbjerg Frandsen et al., 1997] on the dynamic word problem, and additionally cover regular languages.
关键词：regular language;membership;RAM model;updates;dynamic