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22
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 57 (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. ...
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...
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 30 (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.
Domain Theory in Learning Processes
, 1998
"... We present applications of domain theory in stochastic learning automata and in neural nets. We show that a basic probabilistic algorithm, the socalled linear rewardpenalty scheme, for the binarystate stochastic learning automata can be modelled by the dynamics of an iterated function system on a ..."
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Cited by 12 (6 self)
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We present applications of domain theory in stochastic learning automata and in neural nets. We show that a basic probabilistic algorithm, the socalled linear rewardpenalty scheme, for the binarystate stochastic learning automata can be modelled by the dynamics of an iterated function system on a probabilistic power domain and we compute the expected value of any continuous function in the learning process. We then consider a general class of, socalled forgetful, neural nets in which pattern learning takes place by a local iterative scheme, and we present a domaintheoretic framework for the distribution of synaptic couplings in these networks using the action of an iterated function system on a probabilistic power domain. We then obtain algorithms to compute the decay of the embedding strength of the stored patterns. 1 Introduction The probabilistic power domain was introduced in [21] and developed in [20,14] for studying probabilistic computation, in order to provide semantics fo...
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 the M ..."
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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...
Signal Modeling With Iterated Function Systems
, 1993
"... this memory requirement issue may become a factor, in which case the Random Iteration Algorithm could be adapted to overcome the shortcomings mentioned here with some simple checks on the path of the calculations. 2.3 Conclusion ..."
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Cited by 9 (0 self)
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this memory requirement issue may become a factor, in which case the Random Iteration Algorithm could be adapted to overcome the shortcomings mentioned here with some simple checks on the path of the calculations. 2.3 Conclusion
Individual GP: an Alternative Viewpoint for the Resolution of Complex Problems.
"... An unususal GP implementation is proposed, based on a more "economic" exploitation of the GP algorithm: the "individual" approach, where each individual of the population embodies a single function rather than a set of functions. The nal solution is then a set of individuals. Examples are presented ..."
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Cited by 8 (3 self)
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An unususal GP implementation is proposed, based on a more "economic" exploitation of the GP algorithm: the "individual" approach, where each individual of the population embodies a single function rather than a set of functions. The nal solution is then a set of individuals. Examples are presented where results are obtained more rapidly than with the conventional approach, where all individuals of the nal generation but one are discarded.
Rigorous numerical estimation of Lyapunov exponents and invariant measures of iterated function systems and random matrix products
, 2000
"... We present a fast, simple matrix method of computing the unique invariant measure and associated Lyapunov exponents of a nonlinear iterated function system. Analytic bounds for the error in our approximate invariant measure (in terms of the Hutchinson metric) are provided, while convergence of the L ..."
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Cited by 6 (3 self)
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We present a fast, simple matrix method of computing the unique invariant measure and associated Lyapunov exponents of a nonlinear iterated function system. Analytic bounds for the error in our approximate invariant measure (in terms of the Hutchinson metric) are provided, while convergence of the Lyapunov exponent estimates to the true value is assured. As a special case, we are able to rigorously estimate the Lyapunov exponents of an iid random matrix product. Computation of the Lyapunov exponents is carried out by evaluating an integral with respect to the unique invariant measure, rather than following a random orbit. For low dimensional systems, our method is 1 considerably quicker and more accurate than conventional methods of exponent computation. An application to Markov random matrix product is also described. 1 Outline and Motivation This paper is divided into three parts. Firstly, we consider approximating the invariant measure of a contractive iterated function system (t...
The Storage Capacity of Forgetful Neural Networks
, 1995
"... In this report, we derive a two stage algorithm to evaluate the storage ca pacity of a forgetful neural network using any smooth learning scheme. ..."
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Cited by 6 (1 self)
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In this report, we derive a two stage algorithm to evaluate the storage ca pacity of a forgetful neural network using any smooth learning scheme.
Manipulation of nonlinear ifs attractors using genetic programming
 Congress on Evolutionary Computation (CEC'99
, 1999
"... Nonlinear Iterated Function Systems (IFSs) are very powerful mathematical objects related to fractal theory, that can be used in order to generate (or model) very irregular shapes. We investigate in this paper how Genetic Programming techniques can be efficiently exploited in order to generate rand ..."
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Cited by 6 (1 self)
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Nonlinear Iterated Function Systems (IFSs) are very powerful mathematical objects related to fractal theory, that can be used in order to generate (or model) very irregular shapes. We investigate in this paper how Genetic Programming techniques can be efficiently exploited in order to generate randomly or interactively artistic “fractal” 2D shapes. Two applications are presented for different types of nonlinear IFSs: interactive generation of Mixed IFSs attractors using a classical GP scheme, random generation of Polar IFSs attractors based on an “individual ” approach of GP. 1