On (co)products of partial combinatory algebras, with an application to pushouts of realizability toposes

Collapsing Partial Combinatory Algebras

Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or re-cursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identiication of elements in a pca. We establish several facts concerning nal collapses (maximal identiication of elements). `En passan...

On Completability of Partial Combinatory Algebras

A Partial Combinatory Algebra is completable if it can be extended to a total one. Klop [11, 12] gave a sufficient condition for completability of a PCA M = (M, ·,K, S) in the form of ten axioms (inequalities) on terms of M . We prove that Klop’s sufficient condition is equivalent to the existence of an injective s-m-n function over M (that in turns is equivalent to the Padding Lemma). This is ...

Extending partial combinatory algebras

We give a negative answer to the question whether every partial combinatory algebra can be completed. The explicit counterexample will be an intricately constructed term model, the construction and the proof that it works heavily depending on syntactic techniques. In particular, it is a nice example of reasoning with elementary diagrams and descendants. We also include a domain-theoretic proof ...

Extending partial combinatory algebras

Partial Combinatory Algebras of Functions

We employ the notions of ‘sequential function’ and ‘interrogation’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results i...