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The Continuous Functionals of Finite Types Over the Reals
, 1998
"... We investigate a hierarchy of domains with totality where we close some selected base domains, including domains for the reals, the natural numbers and the boolean values, under cartesian products and restricted function spaces. We show that the total objects will be dense in the respective doma ..."
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Cited by 16 (4 self)
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We investigate a hierarchy of domains with totality where we close some selected base domains, including domains for the reals, the natural numbers and the boolean values, under cartesian products and restricted function spaces. We show that the total objects will be dense in the respective domains, and that our construction is equivalent to the analogue construction in the category of limit spaces.
A Relationship between Equilogical Spaces and Type Two Effectivity
"... In this paper I compare two well studied approaches to topological semantics the domaintheoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both ..."
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Cited by 16 (0 self)
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In this paper I compare two well studied approaches to topological semantics the domaintheoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both locally cartesian closed extensions of countably based T 0 spaces. A natural question to ask is how they are related.
Computability Over the Partial Continuous Functionals
, 1998
"... We show that to every recursive total continuous functional there is a representative of in the hierearchy of partial continuous funcriohals such that is S1  S9 computable over the hierarchy of partial continuous functionals. Equivalently, the representative will be PCFdefinable over the parti ..."
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Cited by 13 (3 self)
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We show that to every recursive total continuous functional there is a representative of in the hierearchy of partial continuous funcriohals such that is S1  S9 computable over the hierarchy of partial continuous functionals. Equivalently, the representative will be PCFdefinable over the partial continuous functionals, where PCF is Plotkin's programming language for computable functionals.
Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations
, 1994
"... Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. ..."
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Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. The models we investigate are denotational in nature; we construct various categories, in which types (or formulae) are interpreted by objects, and terms (proofs) by morphisms. The results we investigate compare particular properties of the syntax and the semantics of a calculus, by trying to use syntax to characterise features of a model, or vice versa. There are four chapters in the thesis, one each on linear logic and the simply typed calculus, and two on inductive datatypes. In chapter one, we look at some models of linear logic, and prove a full completeness result for multiplicative linear logic. We form a model, the linear logical predicates , by abstracting a little the structure ...
Density Theorems for the DomainsWithTotality Semantics of Dependent Types
 Applied Categorical Structures
, 2000
"... . We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpre ..."
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. We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpretation of dependent types, and of D. Normann on universes of wellfounded types with density. Key words: Continuous functionals, Domains, Totality, Dependent types, Universes 1. Introduction In Mathematical Logic and Computer Science there is growing interest in constructive type theories as developed by Martin{Lof [8]. This paper is concerned with a semantics of such theories within the realm of Ershov{Scott domains [5] with totality [10]. Erik Palmgren and Viggo Stoltenberg{Hansen [15], [17] developed a semantics for a partial type theory (modelling partial functions and functionals) based on the notion of a parametrization, i.e. a domain depending on parameters. Since this semantics wa...
Equational Theories for Inductive Types
 Annals of Pure and Applied Logic
, 1997
"... This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain synt ..."
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This paper provides characterisations of the equational theory of the per model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model; a version of the KreiselLacombeShoenfield theorem allows us to transfer the result from the domain model to the per model. 0 Introduction This paper concerns a typed calculus with inductive types which correspond semantically to initial algebras of (covariant) functors; the calculus lies between Godel's T and Girard's F in prooftheoretic strength. The goal of the paper is to analyse the structure of the model of this calculus given by the category PER of partial equivalence relations over the natural numbers. We shall show that ...
Categories of Domains With Totality
 Preprint Series, Inst. Math. Univ. Oslo
, 2000
"... We investigate domains with totality where density in general does not hold. We define three categories of domains X with totality X satisfying certain structural properties. We then define the ordered set of evaluation structures. These will induce domains with totality. We show that the set of ..."
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We investigate domains with totality where density in general does not hold. We define three categories of domains X with totality X satisfying certain structural properties. We then define the ordered set of evaluation structures. These will induce domains with totality. We show that the set of evaluation structures in a natural way is closed under dependent sums and products and under direct limits.
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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Cited by 4 (2 self)
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
Limit Spaces and Transfinite Types
, 1998
"... We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types. ..."
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We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.
Representation Theorems for Transfinite Computability And Definability
, 1996
"... this paper we will pursue some of the ideas in Kreisel's approach and show that transfinite versions of the continuous functionals can be used to represent complex properties. We will be more precise later ..."
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this paper we will pursue some of the ideas in Kreisel's approach and show that transfinite versions of the continuous functionals can be used to represent complex properties. We will be more precise later