Characterizations of recursively enumerable languages by means of insertion grammars

Abstract

An insertion grammar is based on pure rules of the form uv → uxv (the string x is inserted in the context (u, v)). A strict subfamily of the context-sensitive family is obtained, incomparable with the family of linear languages. We prove here that each recursively enumerable language can be written as the weak coding of the image by an inverse morphism of a language generated by an insertion grammar (with the maximal length of strings u, v as above equal to seven). This result is rather surprising in view of some closure properties established earlier in the literature. Some consequences of this result are also stated. When also erasing rules of the form uxv → uv are present (the string x is erased from the context ( u, v)), then a much easier representation of recursively enumerable languages is obtained, as the intersection with V ∗ of a language generated by an insertion grammar with erased strings (having the maximal length of strings u, v as above equal to two).