Third-order functionals on partial combinatory algebras

TL;DR

This paper extends the theory of computability in partial combinatory algebras to third-order functionals, introducing a construction that makes such functionals computable within an algebraic framework.

Contribution

It develops a method to construct a partial combinatory algebra where a third-order functional becomes computable, generalizing previous second-order results.

Findings

Constructs a partial combinatory algebra A[Φ] where Φ is computable.

Shows the effect of making a third-order functional computable is equivalent to adding an oracle for a first-order function.

Establishes a lax factorization property for the constructed algebra.

Abstract

Computability relative to a partial function $f$ on the natural numbers can be formalized using the notion of an oracle for this function $f$. This can be generalized to arbitrary partial combinatory algebras, yielding a notion of `adjoining a partial function to a partial combinatory algebra $A$'. A similar construction is known for second-order functionals, but the third-order case is more difficult. In this paper, we prove several results for this third-order case. Given a third-order functional $\Phi$ on a partial combinatory algebra $A$, we show how to construct a partial combinatory algebra $A[\Phi]$ where $\Phi$ is `computable', and which has a `lax' factorization property. Moreover, we show that, on the level of first-order functions, the effect of making a third-order functional computable can be described as adding an oracle for a first-order function.