ON CHARACTERIZATIONS OF RECURSIVELY-ENUMERABLE LANGUAGES

Geffert has shown that each recursively enumerable language L over SIGMA can be expressed in the form L = {h(x)-1g(x)\x in delta+}intersect-SIGMA* where delta is an alphabet and g, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations of L. For instance, L = rho(L-omicron)intersect-SIGMA* where L-omicron is a minimal linear language and rho is the Dyck reduction aaBAR --> epsilon, AABAR --> epsilon.