Recursive definitions of partial functions and their computations

Abstract

The object of this paper is to present a syntactic and semantic model for recursive definitions, and to study the relation between their computed functions and their fixpoints. The recursive definitions that we consider are syntactic generalizations of those introduced in [2] by Kleene and in [5] by McCarthy. Each recursive definition yields two classes of fixpoint partial functions (“fixpoints over D υ {ω}” and “fixpoints over D”), and a class of computed partial functions obtained by applying different computation rules to the recursive definition. In this work we first describe the relations between the computed functions and the fixpoints over D υ {ω} (Theorem 1), and between the computed functions and the fixpoints over D (Theorem 2). Our main interest is in the class of fixpoints (over D or D υ {ω}) which are also computed functions of the recursive definition.