Equality languages, fixed point languages and representations of recursively enumerable languages

Abstract

A considerable part of formal language'theory deals with mappings on free monoids. A way to measure the similarity of mappings a,a on the free monoid r* generated by an alphabet E is to consider the equality language of a and a denoted by Eq(a,8) and consisting of all words x in E* such that a(x) = 6(x). To measure the similarity of a mapping with the identity mapping on the same domain one considers the fixed point language of a de~oted by Fp(a) and consisting of all words x in E such that a(x) = x (if a is a relation in r* x E* then we take Fp(a) = {x £ E* : x £ a(x)}). Thus equality languages and fixed point languages are very natural from the mathematical point of view. If we consider homomorphisms of free monoids then their equality languages represent sets of instances of the Post correspondence Problem; in this sense considering equality languages of homomorphisms is a classical topic in formal language theory (and computability theory). A revival of interest in those languages was stimulated recently by research concerning some very challenging decision problems in formal language theory; it became apparent that in several cases equality languages of homomorphisms playa vital role in (positive!) solutions of some basic equivalence problems of L systems (see e.g. (2) and [4]). This paper is an attempt towards a systematic investigation of equality languages and fixed point languages of homomorphisms and dgsm mappings (i.e. mappings defined by deterministic generalized sequential machines with accepting states). Homomorphisms and dgsm mappings are certainly among the most important mappings in formal language theory and so they form a good departure point for building up a systematic theory. Related work appears in [3] and [8]. In this extended abstract we summarize" some of the results we have obtained in this direction. It is organized as follows. Section 2 provides basic language-theoretic properties of equality languages and fixed point languages of homomorphisms. In Section 3 we present some results on the equality languages and fixed point languages of dgsm mappings but we concentrate on the subclass of dgsm mappings (that we introduce) called symmetric dgsm mappings. The theorem on fixed point languages of these mappings seems to be quite central in our theory. Then Section 4 provides an illustration of the usefulness of the classes of languages we have considered to provide various