Third-order functionals on partial combinatory algebras

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 Φ on a partial combinatory algebra A, we show how to construct a partial combinatory algebra A[Φ] where Φ 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.