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A DomainTheoretic Approach to Computability on the Real Line
, 1997
"... In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and ..."
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Cited by 44 (10 self)
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In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of StoltenbergHansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turingmachine based approach which dates back to Grzegorczyk and Lacombe, is used by PourEl & Richards in their found...
Hereditarily Sequential Functionals: A GameTheoretic Approach to Sequentiality
, 1996
"... The aim of this thesis is to give a new understanding of sequential computations in higher types. We present a new computation model for higher types based on a game describing the interaction between a functional and its arguments. The functionals which may be described in this way are called hered ..."
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Cited by 16 (3 self)
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The aim of this thesis is to give a new understanding of sequential computations in higher types. We present a new computation model for higher types based on a game describing the interaction between a functional and its arguments. The functionals which may be described in this way are called hereditarily sequential. We show that this computation model captures exactly the notion of computability in higher types introduced by Kleene in his pioneering work starting 1959. We study the order structure of the hereditarily sequential functionals and discuss the occurring difficulties. These functionals form a fully abstract model for PCF and we discuss which problems remain still open for a satisfactory solution to the full abstraction problem of PCF. Zusammenfassung Ziel dieser Arbeit ist es, eine neue Beschreibung sequentieller Berechnungen in hoheren Typen zu geben. Wir stellen dazu ein neues Berechnungsmodell fur hohere Typen vor, in dem die Interaktion zwischen einem Funktional und se...
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 6 (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].
Complexity theory on real numbers and functions
 In Theoretical Computer Science
, 1982
"... Since Turing [17] introduced the concept of computable real numbers in 1937, many authors have studied computability of real numbers and functions (see for example, [1,2,6,10,11,12,13,14,16]). The next subject to be studied is now computational complexity. ..."
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Cited by 3 (1 self)
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Since Turing [17] introduced the concept of computable real numbers in 1937, many authors have studied computability of real numbers and functions (see for example, [1,2,6,10,11,12,13,14,16]). The next subject to be studied is now computational complexity.
An Enquiry Concerning Categories Effective Continuous Cpos
"... Introduction Consider the following easy game. Person A chooses an integer between I and 1000. Person B asks yesorno questions to A until she knows what number he is thinking of. The goal for B is to ask as few questions as possible. Soon after playing the game for a few rounds, B realises that ..."
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Introduction Consider the following easy game. Person A chooses an integer between I and 1000. Person B asks yesorno questions to A until she knows what number he is thinking of. The goal for B is to ask as few questions as possible. Soon after playing the game for a few rounds, B realises that it is always sucient to ask A 10 questions, but that 9 questions sometimes does not give B enough information. B gets bored and tries to find an easy set of questions to ask A. She finds that the following question can be asked over and over again: Is the number in the lower half of the interval that one can deduce it must be in after the previous question? The answers B gets to her questions is an approximating sequence of the number that A has chosen. She knows with certainty that the number is in the given interval, and each question gives her a better approximation of the number. Given enough questions, the approximation will consist of only one number. The sequence converges to that nu