Dynamic Membership for Regular Languages

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 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 close gaps that were left open by the 1997 JACM paper of Skovbjerg Frandsen, Miltersen, and Skyum. Our results on monoids are summarized in Figure 1. This talk will present our results on the dynamic word problem for monoids and semigroups and on the dynamic membership problem for regular languages. It follows our work with Louis Jachiet and Charles Paperman which will appear at ICALP’21: http: //arxiv. org/ abs/2102. 07728.

Authors

• Antoine Amarilli

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