A Concrete Approach to Abstract Recursive Definitions

Abstract

We introduce a non-categorical alternative to Wagner's Abstract Recursive Definitions [Wg-1,2] using a generalization of the notion of clone called a u-clone. Our more concrete approach yields two new theorems: 1.) the free u-clone generated by a ranked set is isomorphic to the set of loop-representable flow diagrams with function symbols in the set, 2.) For every element of a u-clone there is an expression analogous to a regular expression. Several well-known theorems of language and automata theory are drawn as special cases of this theorem.