Operations preserving recognizable languages

Abstract

Given a subset S of ℕ, filtering a word α0α1 ⋯ αn by S consists in deleting the letters αi such that i is not in S. By a natural generalization, denote by L[S], where L is a language, the set of all words of L filtered by S. The filtering problem is to characterize the filters S such that, for every recognizable language L, L[S] is recognizable. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. There are two main ingredients on our approach: the first one is the notion of residually ultimately periodic sequences, and the second one is the notion of representable transductions. © Springer-Verlag Berlin Heidelberg 2003.