We present a domain-theoretic framework for measure theory and integration of bounded real-valued 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 real-valued 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. ...

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...

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 domain-theoretic 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 everywh...

We present applications of domain theory in stochastic learning automata and in neural nets. We show that a basic probabilistic algorithm, the so-called linear reward-penalty scheme, for the binary-state 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, so-called forgetful, neural nets in which pattern learning takes place by a local iterative scheme, and we present a domain-theoretic 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...

) Abbas Edalat Department of Computing Imperial College 180 Queen's Gate, London SW7 2BZ UK. Abstract We present a domain-theoretic analysis of the invariant measure of the one-dimensional 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 two-dimensional version of Elton's theorem, to deduce algorithms to compute the magnetisation per spin and the Edwards-Anderson 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...

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

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.

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...

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.

This thesis describes a novel method for approximating digital images and similar data. The work is based on deterministic fractal geometry, which for some time has promised to provide new data compression techniques which may challenge existing industrial standards. However, few robust algorithms for fractal compression have hitherto been published. The present work provides enabling techniques for the eventual development of commercially applicable compression algorithms. A self-affine fractal is a geometrically complex shape which may be specified with relatively little information. The problem of constructing a fractal from its defining equations is identified as a forward problem, with the harder inverse of determining which equations define a given fractal. A new solution of the forward problem is proposed, which constructs an approximation to the fractal in optimal time. A novel scheme for assigning attributes to the elements of a fractal is then developed, which permits the re...