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

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.