On slender context-free languages

Abstract

In this paper we study slender context-free languages, i.e., those containing at most a constant number of words of each length. Recently, Ilie proved that every such language can be described by a finite union of terms of the form uv ⁱ wx ⁱ y [I]. We provide a completely different proof of this, using constructive methods. This enables us to prove that thinness and slenderness are decidable. Our proofs are based upon a novel characterization of languages in terms of the structure of the infinite paths in their prefix closure. This characterization seems to be interesting in itself, and can be expanded to more general families of languages.