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23
Dynamical systems, Measures and Fractals via Domain Theory
 Information and Computation
, 1995
"... We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L ar ..."
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Cited by 68 (19 self)
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We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L are respectively the Vietoris hyperspace, the upper hyperspace and the lower hyperspace functors. We show that if (X, f) is chaotic, then so is (UX, U f). When X is locally compact UX, is a continuous bounded complete dcpo. If X is second countable as well, then UX will be omegacontinuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of nondeterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of finite Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.
Domain Theory and Integration
 Theoretical Computer Science
, 1995
"... We present a domaintheoretic framework for measure theory and integration of bounded realvalued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilis ..."
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Cited by 58 (12 self)
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We present a domaintheoretic framework for measure theory and integration of bounded realvalued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilistic power domain of its upper space. Any bounded Borel measure on the compact metric space can then be obtained as the least upper bound of an !chain of linear combinations of point valuations (simple valuations) on the upper space, thus providing a constructive setup for these measures. We use this setting to define a new notion of integral of a bounded realvalued function with respect to a bounded Borel measure on a compact metric space. By using an !chain of simple valuations, whose lub is the given Borel measure, we can then obtain increasingly better approximations to the value of the integral, similar to the way the Riemann integral is obtained in calculus by using step functions. ...
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...
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
Power domains and iterated function systems
 Information and Computation
, 1996
"... We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniquene ..."
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Cited by 31 (10 self)
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We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.] 1996 Academic Press, Inc. 1.
Nondeterminism and Probabilistic Choice: Obeying the Laws
 In Proc. 11th CONCUR, volume 1877 of LNCS
, 2000
"... In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domaintheoretic techniques, we show how models ..."
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Cited by 25 (2 self)
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In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domaintheoretic techniques, we show how models can be devised using the "standard model" for probabilistic choice, and then applying modified domaintheoretic models for nondeterministic choice. These models are distinguished by the fact that the expected laws for nondeterministic choice and probabilistic choice remain valid. We also describe some potential applications of our model to aspects of security.
Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
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Domain of Computation of a Random Field in Statistical Physics (Extended Abstract)
 Theory and Formal Methods 1994: Proceedings of the second Imperial College Department of Computing Workshop on Theory and Formal Methods
, 1994
"... ) Abbas Edalat Department of Computing Imperial College 180 Queen's Gate, London SW7 2BZ UK. Abstract We present a domaintheoretic analysis of the invariant measure of the onedimensional Ising model in a random external magnetic field. The invariant measure is obtained as a fixed point of ..."
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Cited by 10 (7 self)
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) Abbas Edalat Department of Computing Imperial College 180 Queen's Gate, London SW7 2BZ UK. Abstract We present a domaintheoretic analysis of the invariant measure of the onedimensional Ising model in a random external magnetic field. The invariant measure is obtained as a fixed point of the Markov transition operator of an iterated function system with probabilities acting on the probabilistic power domain of the upper space of a closed real interval. This enables us to use the generalised Riemann integral in combination with Elton's ergodic theorem to obtain an algorithm to compute the free energy density of the system. We also develop the generalised double Riemann integral, which we use, together with a twodimensional version of Elton's theorem, to deduce algorithms to compute the magnetisation per spin and the EdwardsAnderson parameter of the system. 1 Introduction The Ising model was introduced by Ising as a model for ferromagnetism some seventy years ago; it also descri...
Domain theory in stochastic processes
 Proc. of the IEEE Annual Symposium on Logic in Computer Science (LICS
, 1995
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