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...

We establish domain-theoretic models of finite-state discrete stochastic processes, Markov processes and vector recurrent iterated function systems. In each case, we show that the distribution of the stochastic process is canonically obtained as the least upper bound of an increasing chain of simple valuations in a probabilistic power domain associated to the process. This leads to various formulas and algorithms to compute the expected values of functions which are continuous almost everywhere with respect to the distribution of the stochastic process. We prove the existence and uniqueness of the invariant distribution of a vector recurrent iterated function system which is used in fractal image compression. We also present a finite algorithm to decode the image. 1 Introduction Domain theory was introduced by Dana Scott in 1970 [Sco70] as a mathematical theory of computation in the semantics of programming languages. It has, since then, developed extensively in various areas of seman...

Implicit surface techniques provide useful tools for modeling and rendering smooth surfaces. Deriving implicit formulations for fractal representations extends the scope of implicit surface techniques to rough surfaces. Linear fractals modeled by recurrent iterated function systems may be defined implicitly using either geometric distance or escape time. Random fractals modeled using Perlin's noise function are already defined implicitly when described as "hypertexture." Deriving new implicit formulae is only the first step. Unlike their smooth counterparts, rough implicit surfaces require special rendering techniques that do not rely on continuous differentiation of the defining function. Preliminary experiments applying blending operations to rough surfaces have succeeded in overcoming current challenges in natural modeling, including grafting a stem onto the base of a fractal leaf and continuously interpolating geometric bark across branching points in a tree. Keywords: bl...

: We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the well-known case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded non-empty sets not describable by IIFS. Keywords: Fractal geometry, iterated function systems, complete metric spaces, Baire space, Hausdorff measure, Hausdorff dimension, selfsimilarity. AMS classification: 28A80, 54E50, 54E52, 28A78, 54F45. 1. Introduction and Main Definitions IFS theory, starting out from Hutchinson's paper [14], gained more and more interest. Several books on this topic are available [3, 7, 5, 18, 19] which have become popular even amo...

The escape-time method was introduced to generate images of Julia and Mandelbrot sets, then applied to visualize attractors of iterated function systems. This paper extends it further to language-restricted iterated function systems (LRIFS's). They generalize the original definition of IFS's by providing means for restricting the sequences of applicable transformations. The resulting attractors include sets that cannot be generated using ordinary IFS's. The concepts of this paper are expressed using the terminology of formal languages and finite automata.

Interpolation of two-dimensional shapes described by iterated function systems is explored. Iterated function systems define shapes using self-transformations, and interpolation of these shapes requires interpolation of these transformations. Polar decomposition is used to avoid singular intermediate transformations and to better simulate articulated motion. Unlike some other representations, such as polygons, shaped described by iterated function systems can become totally disconnected. A new, fast and image-based technique for determining the connectedness of an iterated function system attractor is introduced. For each shape interpolation, a two parameter family of iterated function systems is defined, and a connectedness locus for these shapes is plotted, to maintain connectedness during the interpolation. Keywords: Fractal Geometry, Iterated Function System, Mandelbrot Set, Morphing, Shape Interpolation. 1 Introduction The basis of key-framed animation is shape interpolation, a....

In dieser Arbeit präsentieren wir eine verallgemeinerte Methode zur prozeduralen Erzeugung und Manipulation von Meshes, die im wesentlichen auf zwei verschiedenen Mechanismen beruht: generalized subdivision meshes und rule-based mesh growing. Herkömmliche Subdivision-Algorithmen beruhen darauf, dass eine genau definierte, spezifische Subdivision-Vorschrift in wiederholter Folge auf ein Mesh angewendet wird um so eine Reihe von immer weiter verfeinerten Meshes zu generieren. Die Vorschrift ist dabei so gewählt, dass die Ecken und Kanten des Basis-Meshs geglättet werden und die Reihe zu einer Grenzfläche konvergiert welche festgelegten Stetigkeitsansprüchen genügt. Im Gegensatz dazu erlaubt ein verallgemeinerter Ansatz die Anwendung verschiedener Vorschriften bei jedem Subdivision-Schritt. Konvergenz wird im wesentlichen dadurch erreicht, dass die absolute Größe der durchgeführten geometrischen Veränderungen von Schritt zu Schritt geringer wird. Bei genauerer Betrachtung stellt man jedoch fest, dass es in vielen Fällen von Vorteil wäre die

Although many artists have worked to create associations between music and animation, this has traditionally be done by developing one to suit the pre-existing other, as in visualization or sonification. The approach