Extensions of Scott's Graph Model and Kleene's Second Algebra

TL;DR

This paper explores extensions of partial combinatory algebras, focusing on Scott's Graph model and Kleene's second algebra, revealing hierarchies and computational properties related to equality and complement functions.

Contribution

It introduces a method to extend pcas to represent specific functions and analyzes the resulting hierarchy of models, including on Kleene's second algebra.

Findings

Equality is computable relative to the complement function in Scott's Graph model

The hierarchy of pcas relates to extensions on other models

Results differ for recursively enumerable sub pcas regarding complement function computation

Abstract

We use a way to extend partial combinatory algebras (pcas) by forcing them to represent certain functions. In the case of Scott's Graph model, equality is computable relative to the complement function. However, the converse is not true. This creates a hierarchy of pcas which relates to similar structures of extensions on other pcas. We study one such structure on Kleene's second model and one on a pca equivalent but not isomorphic to it. For the recursively enumerable sub pca of the Graph model, results differ as we can compute the (partial) complement function using the equality.