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 substitution edits on w. We consider this problem on the unit cost RAM model with logarithmic world length, where the problem always has a solution in O(log |w| / log log |w|). We show that the problem is in O(log log |w|) for languages in an algebraically-defined class QSG, and that it is in O(1) for another 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 n/ log log n) operations in the worst case; and that QSG languages not in QLZG admit a reduction from the prefix problem for the monoid U_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 n). 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 on the dynamic word problem, and additionally cover regular languages.

中文翻译：

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常规语言的动态成员资格

我们研究了常规语言的动态隶属度问题：修复语言L，读取单词w，在时间O（| w |）中建立表示w是否在L中的数据结构，并在对w进行替换编辑时有效地维护此结构。我们在具有对数世界长度的单位成本RAM模型上考虑此问题，该问题始终在O（log | w | / log log | w |）中有一个解。我们证明问题在于代数定义的类QSG中的语言在O（log log | w |）中，对于另一个类QLZG则在O（1）中存在问题。我们证明，不在QSG中的语言可以减少循环组的前缀问题，因此在最坏的情况下它们需要\ Omega（log n / log log n）操作；而且不在QLZG中的QSG语言也承认了针对单面体U_1的前缀问题的减少，我们推测这不能在O（1）中得到维持。这产生了有条件的三分法。我们还调查了O（1）和O（log log n）之间的中间情况。我们的结果通过针对半定词和半群的动态词问题显示，我们也对其进行了分类。因此，我们解决了Skovbjerg Frandsen，Miltersen和Skyum关于动态词问题的论文中存在的开放性问题，并另外涵盖了常规语言。