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Filtrations of Formal Languages by Arithmetic Progressions

Mousavi H., Shallit J.

Fundamenta Informaticae

2013

Vol. 123, nr 2

135--142

A filtration of a formal language L by a sequence s maps L to the set of words formed by taking the letters of words of L indexed only by s. We consider the languages resulting from filtering by all arithmetic progressions. If L is regular, it is easy to see that only finitely many distinct languages result; we give bounds on the number of distinct languages in terms of the state complexity of L. By contrast, there exist CFL’s that give infinitely many distinct languages as a result. We use our technique to show that two related operations, including diag (which extracts the diagonal of words of square length arranged in a square array), preserve regularity but do not preserve context-freeness.

The Computational Complexity of Universality Problems for Prefixes, Suffixes, Factors, and Subwords of Regular Languages

Rampersad N., Shallit J., Xu Z.

Fundamenta Informaticae

2012

Vol. 116, nr 1-4

223-236

In this paper we consider the computational complexity of the following problems: given a DFA or NFA representing a regular language L over a finite alphabet Σ, is the set of all prefixes (resp., suffixes, factors, subwords) of all words of L equal to Σ*? In the case of testing universality for factors of languages, there is a connection to two classic problems: the synchronizing words problem of Černy, and Restivo's conjecture on the minimal uncompletable word.