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325
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
PCF extended with real numbers
, 1996
"... We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (singlepoint intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be ..."
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Cited by 46 (14 self)
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We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (singlepoint intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be considered as “continuous words”. Concatenation of continuous words corresponds to refinement of partial information. The usual basic operations cons, head and tail used to explicitly or recursively define functions on words generalize to partial real numbers. We use this fact to give an operational semantics to the above referred extension of PCF. We prove that the operational semantics is sound and complete with respect to the denotational semantics. A program of real number type evaluates to a headnormal form iff its value is different from ⊥; if its value is different from ⊥ then it successively evaluates to headnormal forms giving better and better partial results converging to its value.
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 43 (8 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...
A Computational Model for Metric Spaces
 Theoretical Computer Science
, 1995
"... For every metric space X , we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and !continuous iff X is separable. The computational model BX is used to give domaintheoretic pro ..."
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Cited by 42 (8 self)
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For every metric space X , we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and !continuous iff X is separable. The computational model BX is used to give domaintheoretic proofs of Banach's fixed point theorem and of two classical results of Hutchinson: on a complete metric space, every hyperbolic iterated function system has a unique nonempty compact attractor, and every iterated function system with probabilities has a unique invariant measure with bounded support. We also show that the probabilistic power domain of BX provides an !continuous computational model for measure theory on a separable complete metric space X . 1 Introduction In this paper, we establish new connections between the theory of metric spaces and domain theory, the two basic mathematical structures in computer science. For every metric space X, we define a continuous poset (not necessar...
Logic Programs and Connectionist Networks
 Journal of Applied Logic
, 2004
"... One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic p ..."
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Cited by 42 (15 self)
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One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic programs can be computed by feedforward connectionist networks, and that the same semantic operators for firstorder normal logic programs can be approximated by feedforward connectionist networks. Turning the networks into recurrent ones allows one also to approximate the models associated with the semantic operators. Our methods depend on a wellknown theorem of Funahashi, and necessitate the study of when Funahasi's theorem can be applied, and also the study of what means of approximation are appropriate and significant.
The troublesome probabilistic powerdomain
 Proceedings of the Third Workshop on Computation and Approximation
, 1998
"... In [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerd ..."
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Cited by 40 (5 self)
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In [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerdomain construction can be restricted to RB. Inthispaper, wegiveacounterexampletoGraham’sproofanddescribe our own attempts at proving a closure result for the probabilistic powerdomain construction. We have positive results for finite trees and finite reversed trees. These illustrate the difficulties we face, rather than being a satisfying answer to the question of whether the probabilistic powerdomain and function spaces can be reconciled. We are more successful with coherent or Lawsoncompact domains. These form a category with many pleasing properties but they fall short of supporting function spaces. Along the way, we give a new proof of Jones ’ Splitting Lemma. 1
Information Hiding, Anonymity and Privacy: A Modular Approach
 Journal of Computer Security
, 2002
"... We propose a new specification framework for information hiding properties such as anonymity and privacy. The framework is based on the concept of a function view, which is a concise representation of the attacker's partial knowledge about a function. We describe system behavior as a set of function ..."
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Cited by 38 (0 self)
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We propose a new specification framework for information hiding properties such as anonymity and privacy. The framework is based on the concept of a function view, which is a concise representation of the attacker's partial knowledge about a function. We describe system behavior as a set of functions, and formalize different information hiding properties in terms of views of these functions. We present an extensive case study, in which we use the function view framework to systematically classify and rigorously define a rich domain of identityrelated properties, and to demonstrate that privacy and anonymity are independent.
A Mathematical Framework for the Study of Coevolution
 Foundations of Genetic Algorithms 7
, 2003
"... Despite achieving compelling results in engineering and optimization problems, coevolutionary algorithms remain difficult to understand, with most knowledge to date coming from practical successes and failures, not from theoretical understanding. Thus, explaining why coevolution succeeds is still ..."
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Cited by 35 (11 self)
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Despite achieving compelling results in engineering and optimization problems, coevolutionary algorithms remain difficult to understand, with most knowledge to date coming from practical successes and failures, not from theoretical understanding. Thus, explaining why coevolution succeeds is still more art than science. In this paper, we present a theoretical framework for studying coevolution based on the mathematics of ordered sets.
Foundation of a Computable Solid Modelling
 Theoretical Computer Science
, 2002
"... Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and g ..."
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Cited by 33 (13 self)
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Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and geometry can only be built in a framework with computable predicates and operations. In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable result that correct algorithms, when implemented, turn into unreliable programs. Here, we use a domaintheoretic approach to recursive analysis to develop the basis of an eective and realistic framework for solid modelling. This framework is equipped with a welldened and realistic notion of computability which reects the observable properties of real solids. The basic predicates and operations o...
An Extension Result for Continuous Valuations
, 1998
"... We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oealgebra of the dcpo with the Scott topology. It follows that every boun ..."
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Cited by 32 (4 self)
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We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oealgebra of the dcpo with the Scott topology. It follows that every bounded and continuous valuation on a continuous domain can be extended uniquely to a Borel measure. The result also holds for oefinite valuations, but fails for dcpo's in general. 1