Partial combinatory algebra and generalized numberings

TL;DR

This paper explores generalized numberings based on partial combinatory algebras, examining their algebraic properties and the role of extensionality, extending Ershov's classical notion of numberings.

Contribution

It introduces and analyzes algebraic properties of generalized numberings derived from partial combinatory algebras, highlighting the importance of extensionality.

Findings

Generalized numberings relate to properties of the underlying pca.

Extensionality plays a crucial role in the structure of generalized numberings.

Algebraic properties of numberings are characterized in terms of pca properties.

Abstract

Generalized numberings are an extension of Ershov's notion of numbering, based on partial combinatory algebra (pca) instead of the natural numbers. We study various algebraic properties of generalized numberings, relating properties of the numbering to properties of the pca. As in the lambda calculus, extensionality is a key notion here.