Abstract
Given a strictly increasing sequence s of non-negative integers, filtering a word a 0 a 1 ⋯ a n by s consists in deleting the letters a i such that i is not in the set { s 0 , s 1 , … } . 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 regular language L, L [ s ] is regular. 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. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions.
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