Fixed point theorems for precomplete numberings

TL;DR

This paper explores fixed point theorems within the framework of precomplete numberings, extending classical results like Kleene's recursion theorem and Arslanov's completeness criterion to broader contexts including partial combinatory algebras.

Contribution

It generalizes key fixed point theorems for precomplete numberings and relates them to other results like the ADN theorem, also extending to partial combinatory algebras.

Findings

Kleene's recursion theorem holds for all precomplete numberings.

Arslanov's completeness criterion applies to every precomplete numbering.

Generalization of Ershov's theorem to partial combinatory algebras.

Abstract

In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.