A representation of recursively enumerable languages by two homomorphisms and a quotient

Abstract

A new representation for recursively enumerable languages is presented. It uses a pair of homomorphisms and the left (or right) quotient: For each recursively enumerable language L one can find homomorphisms h1, h2: ∑∗A → ∑∗B, such that w ∈ ∑∗L is a word in L if and only if w =h1(α)h2(α) for some α∈∑+A. (Or, each recursively enumerable language can be given by L = O(h1h2) ∩ ∑∗L, where O(h1h2) is the so-called right overflow languaged defined as O(h1h2) = {h1(x)h2(x); x ∈ ∑∗A}.)