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Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Admissible Domain Representations of Topological Spaces
 DEPARTMENT OF MATHEMATICS, UPPSALA UNIVERSITY
, 2005
"... In this paper we consider admissible domain representations of topological spaces. A domain representation D of a space X is λadmissible if, in principle, all other λbased domain representations E of X can be reduced to D via a continuous function from E to D. We present a characterisation theorem ..."
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Cited by 7 (1 self)
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In this paper we consider admissible domain representations of topological spaces. A domain representation D of a space X is λadmissible if, in principle, all other λbased domain representations E of X can be reduced to D via a continuous function from E to D. We present a characterisation theorem of when a topological space has a λadmissible and κbased domain representation. We also prove that there is a natural cartesian closed category of countably based and countably admissible domain representations. These results are generalisations of [Sch02].
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.
Computing with functionals  computability theory or computer science
 Bulletin of Symbolic Logic
, 2006
"... We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject. ..."
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We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.
Natural nondcpo Domains and fSpaces Abstract
"... hereditarilysequential functionals is not ωcomplete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially (universal) model for PCF + = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natura ..."
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hereditarilysequential functionals is not ωcomplete (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a potentially (universal) model for PCF + = PCF + pif (parallel if). Here we will present an outline of a general approach to this kind of ‘natural ’ domains which, although being nondcpos, allow considering ‘naturally ’ continuous functions (with respect to existing directed ‘pointwise’, or ‘natural ’ least upper bounds). There is also an appropriate version of ‘naturally ’ algebraic and ‘naturally ’ bounded complete ‘natural’ domains which serves as the nondcpo analogue of the wellknown concept of Scott domains, or equivalently, the complete fspaces of Ershov. It is shown that this special version of ‘natural ’ domains, if considered under ‘natural ’ Scott topology, exactly corresponds to the class of fspaces, not necessarily complete. Key words: domain theory, dcpo and nondcpo domains, Scott topology,
Computability on topological spaces . . .
, 1997
"... Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using ScottErshov domains. Domain theory is a theory of approximations and incorporates a natural computability theor ..."
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Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using ScottErshov domains. Domain theory is a theory of approximations and incorporates a natural computability theory. This provides us with a uniform way to introduce computability on structures that have computable domain representations, by computations on the approximations of the structure. It is shown that large classes of topological spaces have satisfactory domain representations. In particular, all metric spaces are domain representable. It is also shown that the space of compact subsets of a complete metric space can be given a domain representation uniformly from a domain representation of the metric space. Several other classes of topological spaces are shown to have domain representations, although not all of them are suitable for introducing computability. Domain
Domains with Approximation Structure and Their Canonical QuasiMetrics
"... In this note continuous directedcomplete partial orders with least element (domains) are enriched by a family of projections, called approximation structure, that assigns to each point a sequence of canonical approximations. The morphisms are continuous maps which commute with the projections. Each ..."
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In this note continuous directedcomplete partial orders with least element (domains) are enriched by a family of projections, called approximation structure, that assigns to each point a sequence of canonical approximations. The morphisms are continuous maps which commute with the projections. Each approximation structure induces a quasiultrametric on the domain, the associated topology of which is finer than the Scott topology of the domain. It coincides with the Scott topology exactly if the subdomains generated by the canonical approximations at the various levels contain only compact elements, which is the case if and only if none of them contains infinite strictly increasing chains and if in each of them the lengths of the chains in any given set of finite strictly increasing chains with the same supremum are bounded. The condition implies that the subdomains as well as the domain must be algebraic. Similarly, the topology coming with the metric that is associated with the quasimetric is finer than the Lawson topology of the underlying domain and coincides with it if the condition is satisfied. For categories of domains that are closed under the construction of Cartesian products, function spaces and bilimits of ωchains, approximation structures on the composed domains can be defined from those coming with the components in a natural way. The quasimetrics induced by the newly constructed approximation structures are nicely related to those generated by the approximation structures of the components. 1
On Some Constructions in Quantitative Domain Theory
"... ) Dieter Spreen Theoretische Informatik, Fachbereich Mathematik Universitat Siegen, 57068 Siegen, Germany Email: spreen@informatik.unisiegen.de 1 ..."
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) Dieter Spreen Theoretische Informatik, Fachbereich Mathematik Universitat Siegen, 57068 Siegen, Germany Email: spreen@informatik.unisiegen.de 1
On Domains Witnessing Increase in Information
"... The paper considers algebraic directedcomplete partial orders with a semiregular Scott topology, called regular domains. As is well known, the category of Scott domains and continuous maps is Cartesian closed. This is no longer true, if the domains are required to be regular. Two Cartesian closed ..."
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The paper considers algebraic directedcomplete partial orders with a semiregular Scott topology, called regular domains. As is well known, the category of Scott domains and continuous maps is Cartesian closed. This is no longer true, if the domains are required to be regular. Two Cartesian closed subcategories of the regular Scott domains are exhibited: regular dIdomains with stable maps and strongly regular Scott domains with continuous maps. Here a Scott domains is strongly regular if all of its compact open subsets are regular open. If one considers only embeddings as morphisms, then both categories are closed under the construction of dependent products and sums. Moreover, they are #cocomplete and their object classes are closed under several constructions used in programming language semantics. It follows that recursive domains equations can be solved and models of typed and untyped lambda calculi can be constructed. Both kinds of domains can be used in giving meaning to progr...