Results 1 
6 of
6
Collapsing Partial Combinatory Algebras
 HigherOrder Algebra, Logic, and Term Rewriting
, 1996
"... 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 ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
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
The Intensional Content of Rice’s Theorem
"... The proofs of major results of Computability Theory like Rice, RiceShapiro or Kleene’s fixed point theorem hide more information of what is usually expressed in their respective statements. We make this information explicit, allowing to state stronger, complexity theoreticversions of all these the ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
The proofs of major results of Computability Theory like Rice, RiceShapiro or Kleene’s fixed point theorem hide more information of what is usually expressed in their respective statements. We make this information explicit, allowing to state stronger, complexity theoreticversions of all these theorems. In particular, we replace the notion of extensional set of indices of programs, by a set of indices of programs having not only the same extensional behavior but also similar complexity (Complexity Clique). We prove, under very weak complexity assumptions, that any recursive Complexity Clique is trivial, and any r.e. Complexity Clique is an extensional set (and thus satisfies RiceShapiro conditions). This allows, for instance, to use Rice’s argument to prove that the property of having polynomial complexity is not decidable, and to use RiceShapiro to conclude that it is not even semidecidable. We conclude the paper with a discussion of “complexitytheoretic ” versions of Kleene’s
Completing Partial Combinatory Algebras with Unique HeadNormal Forms
, 1996
"... In this note, we prove that having unique headnormal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CLterms as well as the pca of natural numbers with partial recursive function application can be exte ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
In this note, we prove that having unique headnormal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CLterms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras. 1.
Extending Partial Combinatory Algebras
, 1999
"... 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 expression ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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
On Completability of Partial
"... 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 a ..."
Abstract
 Add to MetaCart
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 smn 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