Representations of Language Families by Homomorphic Equality Operations and Generalized Equality Sets

Abstract In this paper the operations of homomorphic equality and inverse homomorphic equality are introduced and are studied in all detail as part of formal language theory. These operations are defined from n -tuples of homomorphisms by a matching of the homomorphic images. They model a simple and powerful equality mechanism and incorporate the notion of generalized equality sets. Single homomorphisms on free monoids are natural and simple operations. However, combining n -tuples of homomorphisms into a homomorphic equality or inverse homomorphic equality may result in very complex morphic mappings, which map, e.g., σ ∗ to the set of solutions of an instance of the Post Correspondence Problem. In this paper particular emphasis is laid upon strong representation theorems. In the general case every recursively enumerable set can be obtained from the trivial set {$} by applying an inverse homomorphism, a (inverse) homomorphic equality of two nonerasing homomorphisms and an erasing homomorphism. This is a uniform morphic representation of the recursively enumerable sets with special regular sets as a basis, and it is minimal in its constituents. Hence, the recursively enumerable sets are the smallest class containing all regular sets of the form σ ∗ $σ ∗ and closed under homomorphic equality or under inverse homomorphic equality and homomorphism. Every nonempty language is a generator of the recursively enumerable sets, when the full trio operations are extended to include homomorphic equality or inverse homomorphic equality. In a similar manner, the regular sets can be characterized in terms of homomorphic equality and inverse homomorphic equality operations with bounded balance. Finally, in the nonerasing case, the target are classes of the form H( L ⋀ BNP ). For a class of languages L , H( L ⋀ BNP ) is the smallest class containing L and closed under nonerasing homomorphic equality or under inverse homomorphic equality and nonerasing homomorphism. It includes the intersection closure of L . Every language in H( L ⋀ BNP ) can be obtained from a language in L by a nonerasing homomorphic equality or an inverse homomorphic equality of three homomorphisms and followed by a nonerasing homomorphism, provided that L is closed under inverse homomorphism and endmarking. This representation goes through to trio generators of L , which are the generators of H( L ⋀ BNP ), if the trio operations are extended to include nonerasing homomorphic equality or inverse homomorphic equality. The operations of homomorphic equality and inverse homomorphic equality are based on generalized equality sets. These are equality sets of finite sets of homomorphisms, which are used forwards or backwards. We investigate basic language-theoretic properties of these sets including the star event property, commutativity and hardest sets, and provide a complete classification of the generalized equality sets according to the numbers of homomorphisms that are used forwards or backwards.