Computation at levels beyond storage and transmission of information appears in physical systems at phase transitions. We investigate this phenomenon using minimal computational models of dynamical systems that undergo a transition to chaos as a function of a nonlinearity parameter. For period-doubling and band-merging cascades, we derive expressions for the entropy, the interdependence of-machine complexity and entropy, and the latent complexity of the transition to chaos. At the transition deterministic finite automaton models diverge in size. Although there is no regular or context-free Chomsky grammar in this case, we give finite descriptions at the higher computational level of context-free Lindenmayer systems. We construct a restricted indexed context-free grammar and its associated one-way nondeterministic nested stack automaton for the cascade limit language. This analysis of a family of dynamical systems suggests a complexity theoretic description of phase transitions based on the informational diversity and computational complexity of observed data that is independent of particular system control parameters. The approach gives a much more refined picture of the architecture of critical states than is available via

Defining structure and detecting the emergence of complexity in nature are inherently subjective, though essential, scientific activities. Despite the difficulties, these problems can be analyzed in terms of how model-building observers infer from measurements the computational capabilities embedded in nonlinear processes. An observer’s notion of what is ordered, what is random, and what is complex in its environment depends directly on its computational resources: the amount of raw measurement data, of memory, and of time available for estimation and inference. The discovery of structure in an environment depends more critically and subtlely, though, on how those resources are organized. The descriptive power of the observer’s chosen (or implicit) computational model class, for example, can be an overwhelming determinant in finding regularity in data. This paper presents an overview of an inductive framework — hierarchical-machine reconstruction — in which the emergence of complexity is associated with the innovation of new computational model classes. Complexity metrics for detecting structure and quantifying emergence, along with an analysis of the constraints on the dynamics of innovation, are outlined. Illustrative examples are drawn from the onset of unpredictability in nonlinear systems, finitary nondeterministic processes, and

In 1997 the most prestigious high school science fair in the United States—the Westinghouse Science Competition [Berger 94] —was won by Adam Cohen, then a senior at Hunter High School in New York City. Cohen's project, "Near-Field Photolithography", involved the construction of a home-built scanning tunneling microscope (or STM—a high-resolution

In this chapter, I review the main methods and techniques of complex systems science. As a

Complex Systems Science aims to understand concepts like complexity, self-organization, emergence and adaptation, among others. The inherent fuzziness in complex systems definitions is complicated by the unclear relation among these central processes: does self-organisation emerge or does it set the preconditions for emergence? Does complexity arise by adaptation or is complexity necessary for adaptation to arise? The inevitable consequence of the current impasse is miscommunication among scientists within and across disciplines. We propose a set of concepts, together with their information-theoretic interpretations, which can be used as a dictionary of Complex Systems Science discourse. Our hope is that the suggested information-theoretic baseline may facilitate consistent communications among practitioners, and provide new insights into the field.

Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in nature of what looks like computation, but for which there is no well-understood model. In many areas, we have to come to terms with emergence not being clearly algorithmic. The positive side of this is the growth of new computational paradigms based on metaphors for natural phenomena, and the devising of very informative computer simulations got from copying nature. This talk is concerned with general questions such as: • Can natural computation, in its various forms, provide us with genuinely new ways of computing? • To what extent can natural processes be captured computationally? • Is there a universal model underlying these new paradigms?

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Most of the recent literature on chaos and nonlinear dynamics is written either for popular science magazine readers or for advanced mathematicians. This paper gives a broad introduction to this interesting and rapidly growing field at a level that is between the two. The graphical and analytical tools used 'in the literature are explained and demonstrated, the rudiments of the current theory are outlined and that theory is discussed in the context of several examples: an electronic crcuit a chemical reaction and a system of satellites in the solar system.

this dissertation. 1.1 Radiosity, Complexity, and Information Theory The three fundamental pillars of this thesis are radiosity, complexity, and information theory: One of the most important topics in computer graphics is the accurate computation of the global illumination in a closed virtual environment (scene), i.e. the intensities of light over all its surfaces. "The production of realistic images requires in particular a precise treatment of lighting e#ects that can be achieved by simulating the underlying physical phenomena of light emission, propagation, and reflection"[82]. This type of simulation is called global illumination and is represented by the rendering equation [43], which is a Fredholm integral equation of the second kind. However obtaining an exact representation of the illumination is an intractable problem. Many di#erent techniques are used to obtain an approximate quantification of it [12, 82, 33]

It has been suggested that rising bubbles in dense fluids resemble an inverted dripping faucet and that they undergo analogous period-doubling bifurcations to chaos. We present experimental results demonstrating that this analogy is weak because the dominant source of instability in the bubble train is inherently different mutual interactions between spatially separated bubbles as opposed to nozzle dynamics. Unlike the dripping faucet, the initial instability in a bubble train develops at a location far from the injection nozzle and progresses toward the nozzle with increasing gas flow. From both qualitative and rigorous quantitative observations, we conclude that rising-bubble dynamics is best described as "small-box spatio-temporal chaos" with a flow instability. Such dynamics can superficially appear to be simple temporal chaos when considering spatially localized measurements. We show similarity between our experimental results and a bubbleinteraction model that accounts for drag a...