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Reducibility of Domain Representations and CantorWeihrauch Domain Representations
, 2006
"... We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a preorder on representatio ..."
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Cited by 8 (4 self)
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We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a preorder on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, nontriviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 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
Formalisation of Computability of Operators and RealValued Functionals via Domain Theory
 Proceedings of CCA2000
"... Based on an eective theory of continuous domains, notions of computability for operators and realvalued functionals dened on the class of continuous functions are introduced. Denability and semantic characterisation of computable functionals are given. Also we propose a recursion scheme which is a ..."
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Cited by 3 (3 self)
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Based on an eective theory of continuous domains, notions of computability for operators and realvalued functionals dened on the class of continuous functions are introduced. Denability and semantic characterisation of computable functionals are given. Also we propose a recursion scheme which is a suitable tool for formalisation of complex systems, such as hybrid systems. In this framework the trajectories of continuous parts of hybrid systems can be represented by computable functionals. 1
Semantic Characterisations of Secondorder Computability Over the Real Numbers
 LNCS
, 2001
"... We propose semantic characterisations of secondorder computability over the reals based on denability theory. Notions of computability for operators and realvalued functionals dened on the class of continuous functions are introduced via domain theory. We consider the reals with and without equal ..."
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Cited by 2 (2 self)
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We propose semantic characterisations of secondorder computability over the reals based on denability theory. Notions of computability for operators and realvalued functionals dened on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and realvalued functionals with validity of nite formulas. 1
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
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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Cited by 2 (1 self)
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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.
Cantor–Weihrauch domain representations
, 2007
"... We hope you have now downloaded a proof of your paper in pdf format, and also the Transfer of ..."
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We hope you have now downloaded a proof of your paper in pdf format, and also the Transfer of