On representing recursively enumerable languages by internal contextual languages

Abstract

We prove that each recursively enumerable language L can be written in the form L = cut c ( L 0 ∩ R), where L 0 is an internal contextual language, R is a regular language, and cut c is the operation which for a word x removes the prefix of x to the left of the unique occurrence of ± in x. As corollaries of this result we obtain representations of recursively enumerable languages as (1) weak codings of inverse morphic images, (2) left quotients by regular languages, and (3) images of gsm mappings of internal contextual languages. These representations imply that the family of internal contextual languages, which includes the family of regular languages and is strictly included in the family of context-sensitive languages, contains languages which cannot be generated by programmed grammars with arbitrary rules and empty failure fields. The case of grammars with one-sided contexts is also briefly investigated.