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Dynamic Membership for Regular Languages

Antoine Amarilli 1, 2, 3 Louis Jachiet 1, 2, 3 Charles Paperman 4, 5 
3 DIG - Data, Intelligence and Graphs
LTCI - Laboratoire Traitement et Communication de l'Information
5 LINKS - Linking Dynamic Data
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
Abstract : 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 {\Omega}(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 U 1 = {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 [30] on the dynamic word problem, and additionally cover regular languages.
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https://hal.archives-ouvertes.fr/hal-03466453
Contributor : Charles Paperman Connect in order to contact the contributor
Submitted on : Sunday, December 5, 2021 - 6:19:34 PM
Last modification on : Wednesday, September 7, 2022 - 8:14:05 AM

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Antoine Amarilli, Louis Jachiet, Charles Paperman. Dynamic Membership for Regular Languages. ICALP 2021 - 48th International Colloquium on Automata, Languages, and Programming, Jul 2021, Glasgow, United Kingdom. pp.116:1--116:17, ⟨10.4230/LIPIcs.ICALP.2021.116⟩. ⟨hal-03466453⟩

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