Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We establish several facts concerning final collapses (maximal identification of elements). `En passant' we find another example of a pca that cannot be extended to a total one. 1

S. All local authors can be reached viae-mail at theaddress last-name@cs.unibo.it. Written requests and comments should be addressed to tr-admin@cs.unibo.it. UBLCS Technical Report Series 93-20 An Information Flow Security Property for CCS, R. Focardi, R. Gorrieri, October 1993. 93-21 A Classification of Security Properties, R. Focardi, R. Gorrieri, October 1993. 93-22 Real Time Systems: A Tutorial, F. Panzieri, R. Davoli, October 1993. 93-23 A Scalable Architecture for Reliable Distributed Multimedia Applications, F. Panzieri, M. Roccetti, October 1993. 93-24 Wide-Area Distribution Issues in Hypertext Systems, C. Maioli, S. Sola, F. Vitali, October 1993. 93-25 On Relating Some Models for Concurrency, P. Degano, R. Gorrieri, S. Vigna, October 1993. 93-26 Axiomatising ST Bisimulation Equivalence, N. Busi, R. van Glabbeek, R. Gorrieri, December 1993. 93-27 A Theory of Processeswith Durational Actions, R. Gorrieri, M. Roccetti, E. Stancampiano, December1993. 94-1 Further Modifications t...

In this note, we prove that having unique head-normal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CL-terms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras. 1.

Introduction Consider a structure A = hA; s; k; \Deltai, where A is some set containing the distinguished elements s; k, equipped with a binary operation \Delta on A, called application, which may be partial. Notation 1.1. 1 Instead of a \Delta b we write ab; and in writing applicative expressions, the usual convention of association to the left is employed. So for elements a; b; c 2 A, the expression aba(ac) is short for ((a \Delta b) \Delta a) \Delta (a \Delta c). 2 ab # will mean that ab is defined; ab " means that ab is not defined. Obviously, an applicative expression

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 proved by working with an alternative characterization of PCA’s, recently introduced by the authors (Effective Applicative Structures). As a corollary, we show that nine of Klop’s ten axioms are actually redundant (the so called Barendregt’s axiom is enough to guarantee completability). Moreover, we prove that any Uniformly Reflexive Structure [17, 18, 16] is completable. 1