Embeddings between partial combinatory algebras

TL;DR

This paper investigates the conditions under which various partial combinatory algebras and lambda calculus models can be embedded into each other, revealing that embeddings correspond to specific reductions between oracles.

Contribution

It systematically characterizes embeddings between relativized models of computation and lambda calculus, linking them to reductions between oracles.

Findings

Embedding exists iff a particular oracle reduction exists

Lambda calculus cannot be embedded into Kleene's first model

Embeddings between models are characterized by oracle reductions

Abstract

Partial combinatory algebras are algebraic structures that serve as generalized models of computation. In this paper, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene's models, of van Oosten's sequential computation model, and of Scott's graph model, showing that an embedding between two relativized models exists if and only if there exists a particular reduction between the oracles. We obtain a similar result for the lambda calculus, showing in particular that it cannot be embedded in Kleene's first model.