Results 11 
16 of
16
Partial Morphisms in Categories of Effective Objects
, 1996
"... This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the seco ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the second part we dene the (partial) cartesian closed category GEN of generalized numbered sets, prove that it is a good extension of the category of numbered sets and show how it is related to the recursive topos. Introduction By data type one usually means a set of objects of the same kind, suitable for manipulation by a computer program. Of course, computers actually manipulate formal representations of objects. The purpose of the mathematical semantics of programming languages, however, is to characterize data types (and functions on them) in a way which is independent of any specic representation mechanism. So the objects one deals with are mostly elements of structures borrowed fro...
Uniqueness of Scott's Reflexive Domain in P omega
, 1994
"... Domains for the pure calculus, sometimes called reflexive domains, can be constructed in several ways, but the D1 domains found by Scott in 1969 still seem to be the ones with the best properties. The main disadvantage of D1 has been that the construction is considered somewhat artificial. On the o ..."
Abstract
 Add to MetaCart
Domains for the pure calculus, sometimes called reflexive domains, can be constructed in several ways, but the D1 domains found by Scott in 1969 still seem to be the ones with the best properties. The main disadvantage of D1 has been that the construction is considered somewhat artificial. On the other hand, a very concrete domain was found by Scott in 1976, in fact being a subset of P!, the set of subsets of the natural numbers, by means of retracts. The advantage of such a concrete domain is that one may more easily extract its properties, since one can analyse the domain in terms of natural numbers. The main results of this thesis are the following. ffl The reflexive domain in P! is unique, in the sense that it is the only nontrival solution to the retract equation d = dffi!d. ffl The P! domain is isomorphic to one of the D1 models. This implies that we may adopt results about the D1 domains for the P! domain. ffl The domain possesses no second least elements. ffl An algorithm...
An Abstract Look At Realizability
, 2000
"... This paper is about purely categorical approaches to realizability, and contrasts with recent work particularly by Longley [14] and Lietz and Streicher [13], in which the basis is taken as a typed generalisation of a partial combinatory algebra. We, like they, will be interested in when the construc ..."
Abstract
 Add to MetaCart
This paper is about purely categorical approaches to realizability, and contrasts with recent work particularly by Longley [14] and Lietz and Streicher [13], in which the basis is taken as a typed generalisation of a partial combinatory algebra. We, like they, will be interested in when the construction yields a topos, and hence gives a full interpretation of higherorder logic. This is also a theme of Birkedal's work, see [1, 2], and his joint work in [3]. Birkedal makes considerable use of the construction we study. We present realizability toposes as the product of two constructions. First one takes a category (which corresponds to the typed partial combinatory algebra), and then one glues Set to it in a variant of the comma construction. This, as we shall see, has the eect of improving the categorical properties of the algebra category. Then one takes an exact completion of the result. This also has the eect of improving the categorical properties. Formally the main result of the paper is that the result is a topos just (modulo some technical conditions) when the original category has a universal object. Early work on realizability (e.g.[12, 22], or see [23]) is characterised by its largely syntactic nature. The core denition is when a sentence of some formal logic is realised, and the main interest is in when certain deductive principles (such as Markov's rule) are validated. Martin Hyland's invention y The authors wish to acknowledge the support of the EPSRC, EU Working Group 26142 APPSEM, and MURST 1 2 of realizability toposes [10] advances on this, not only in the simplicity of the construction, but by providing a semantic framework in which the formal logics can naturally be interpreted. Hyland was strongly motivated in his work by a then recent approach...
Logic, Individuals and Concepts
, 2000
"... This extended abstract gives a brief outline of the connections between the descriptions and variable concepts. Thus, the notion of a concept is extended to include both the syntax and semantics features. The evaluation map in use is parameterized by a kind of computational environment, the index, g ..."
Abstract
 Add to MetaCart
This extended abstract gives a brief outline of the connections between the descriptions and variable concepts. Thus, the notion of a concept is extended to include both the syntax and semantics features. The evaluation map in use is parameterized by a kind of computational environment, the index, giving rise to indexed concepts. The concepts are inhabited into language by the descriptions from the higher order logic. In general the idea of objectasfunctor should assist the designer to outline a programming tool in conceptual shell style.
New Semantics for the Simply Typed lambdacalculus
, 2003
"... The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey ..."
Abstract
 Add to MetaCart
The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. In this thesis, I will describe explicitly how this more powerful completeness result follows from a result in [2]. As models of the form Sets for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for noncategory theorists to understand. We hope that the simpler semantics result in new applications of the simply typed calculus. We also describe how this gives a complete semantics of the simply typed calculus in a certain category of posets.
New Semantics for the Simply Typed
"... The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed  theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to loo ..."
Abstract
 Add to MetaCart
The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed  theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. For my thesis, I will describe explicitly how this more powerful completeness result follows from his recent paper. As models of the form for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for noncategory theorists to understand. We hope that the simpler semantics result in new applications of the simply typed calculus. We also describe how this gives a complete semantics of the simply typed calculus in Pos=P.