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.
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.
R
substitution edits
letter substitutions
R
world
word
A
per operation
A
, decidable
A
such
R
\Omega(log n
\Omega
A
(log |w|
R
n
|w|
R
monoid U_1,
multiplicative monoid U 1 =
A
0, 1
A
,
R
n
|w|
A
30