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Partial Continuous Functions and Admissible Domain Representations
 the Journal of Logic and Computation
, 2007
"... It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial ..."
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Cited by 7 (2 self)
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It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison d’être for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that the category of admissibly representable spaces with morphisms functions which are representable by a partial continuous function is Cartesian closed. Finally, we consider the question of effectivity. Key words. Domain theory, domain representations, computability theory, computable analysis. 1
Partial Morphisms in Categories of Effective Objects
, 1996
"... This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the seco ..."
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Cited by 3 (0 self)
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This paper is divided in two parts. In the rst one we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction and show some of their properties. In the second part we dene the (partial) cartesian closed category GEN of generalized numbered sets, prove that it is a good extension of the category of numbered sets and show how it is related to the recursive topos. Introduction By data type one usually means a set of objects of the same kind, suitable for manipulation by a computer program. Of course, computers actually manipulate formal representations of objects. The purpose of the mathematical semantics of programming languages, however, is to characterize data types (and functions on them) in a way which is independent of any specic representation mechanism. So the objects one deals with are mostly elements of structures borrowed fro...
The Real Number Structure is Effectively Categorical
, 1997
"... On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show tha ..."
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On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. We also give further evidence for the wellknown nonappropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countab...
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
Two Categories of Effective Continuous Cpos
"... This paper presents two categories of effective continuous cpos. We define a new criterion on the basis of a cpo as to make the resulting category of consistently complete continuous cpos cartesian closed. We also generalise the definition of a complete set, used as a definition of effective bifinit ..."
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This paper presents two categories of effective continuous cpos. We define a new criterion on the basis of a cpo as to make the resulting category of consistently complete continuous cpos cartesian closed. We also generalise the definition of a complete set, used as a definition of effective bifinite domains in [HSH02], and investigate what closure results that can be obtained. 1
Archive for Mathematical Logic manuscript No. (will be inserted by the editor)
"... Abstract. A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a give ..."
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Abstract. A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice. In addition, Ershov’s completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic. This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices. 1.
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