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18
Domain Theory in Logical Form
 Annals of Pure and Applied Logic
, 1991
"... The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and system ..."
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Cited by 228 (10 self)
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The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:
Comparing functional paradigms for exact realnumber computation
 in Proceedings ICALP 2002, Springer LNCS 2380
, 2002
"... Abstract. We compare the definability of total functionals over the reals in two functionalprogramming approaches to exact realnumber datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to ..."
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Cited by 14 (3 self)
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Abstract. We compare the definability of total functionals over the reals in two functionalprogramming approaches to exact realnumber datatype of real numbers; and the intensional approach, in which one encodes real numbers using ordinary datatypes. We show that the type hierarchies coincide up to secondorder types, and we relate this fact to an analogous comparison of type hierarchies over the external and internal real numbers in Dana Scott’s category of equilogical spaces. We do not know whether similar coincidences hold at thirdorder types. However, we relate this question to a purely topological conjecture about the KleeneKreisel continuous functionals over the natural numbers. Finally, although it is known that, in the extensional approach, parallel primitives are necessary for programming total firstorder functions, we demonstrate that, in the intensional approach, such primitives are not needed for secondorder types and below. 1
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.
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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Cited by 11 (5 self)
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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.
On sequential functionals of type 3
 Math. Structures Comput. Sci
, 2006
"... We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1 ..."
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Cited by 6 (0 self)
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We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1
Comparing hierarchies of total functionals
 Logical Methods in Computer Science
, 2005
"... In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem i ..."
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Cited by 5 (3 self)
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In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem is if these two hierarchies coincide. We will show that this coincidence problem is equivalent to the statement that the topology on the KleeneKreisel continuous functionals of a fixed type induced by all continuous functions into the reals is zerodimensional for each type. As a tool of independent interest, we will construct topological embeddings of the KleeneKreisel functionals into both the extensional and the intensional hierarchy at each type. The embeddings will be hierarchy embeddings as well in the sense that they are the inclusion maps at type 0 and respect application at higher types. 1
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.
Notions of computability at higher types II
 In preparation
, 2001
"... ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X ..."
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Cited by 2 (2 self)
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ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w
Applications of the KleeneKreisel Density Theorem to Theoretical Computer Science
, 2006
"... The KleeneKreisel density theorem is one of the tools used to investigate the denotational semantics of programs involving higher types. We give a brief introduction to the classical density theorem, then show how this may be generalized to set theoretical models for algorithms accepting real numbe ..."
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Cited by 2 (0 self)
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The KleeneKreisel density theorem is one of the tools used to investigate the denotational semantics of programs involving higher types. We give a brief introduction to the classical density theorem, then show how this may be generalized to set theoretical models for algorithms accepting real numbers as inputs and finally survey some recent applications of this generalization. 1