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

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causal-state representation—an E-machine—is the minimal one consistent with

We have found a method to automatically extract the meaning of words and phrases from the world-wide-web using Google page counts. The approach is novel in its unrestricted problem domain, simplicity of implementation, and manifestly ontological underpinnings. The world-wide-web is the largest database on earth, and the latent semantic context information entered by millions of independent users averages out to provide automatic meaning of useful quality. We demonstrate positive correlations, evidencing an underlying semantic structure, in both numerical symbol notations and number-name words in a variety of natural languages and contexts. Next, we demonstrate the ability to distinguish between colors and numbers, and to distinguish between 17th century Dutch painters; the ability to understand electrical terms, religious terms, and emergency incidents; we conduct a massive experiment in understanding WordNet categories; and finally we demonstrate the ability to do a simple automatic English-Spanish translation.

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Abstract — We consider similarity distances for two types of objects: literal objects that as such contain all of their meaning, like genomes or books, and names for objects. The latter may have like “red ” or “christianity. ” For the first type we consider a family of computable distance measures corresponding to parameters expressing similarity according to particular features between pairs of literal objects. For the second type we consider similarity distances generated by web users corresponding to particular semantic relations between the (names for) the designated objects. For both families we give universal similarity distance measures, incorporating all particular distance measures in the family. In the first case the universal distance is based on compression and in the second case it is based on Google page counts related to search terms. In both cases experiments on a massive scale give evidence of the viability of the approaches. I.

ing applications based on different ways of approximating Kolmogorov complexity. 2. BEGINNINGS As we have already mentioned, the two main originators of the theory of Kolmogorov complexity were Ray Solomonoff (born 1926) and Andrei Nikolaevich Kolmogorov (1903-- 1987). The motivations behind their work were completely different; Solomonoff was interested in inductive inference and artificial intelligence and Kolmogorov was interested in the foundations of probability theory and, also, of information theory. They arrived, nevertheless, at the same mathematical notion, which is now known as Kolmogorov complexity. In 1964 Solomonoff published his model of inductive inference. He argued that any inference problem can be presented as a problem of extrapolating a very long sequence of binary symbols; `given a very long sequence, represented by T , what is the probability that it will be followed by a ... sequence A?'. Solomonoff assumed

The utility of a Kolmogorov complexity method in combinatorial theory is demonstrated by several examples. 1 Introduction Probabilistic arguments in combinatorial theory, as used by Erdos and Spencer [5], are usually aimed at establishing the existence of an object, in a nonconstructive sense. It is ascertained that a certain member of a class has a certain property, without actually exhibiting that object. Usually, the method proceeds by exhibiting a random process which produces the object with positive probability. Alternatively, a quantitative property is determined from a bound on its average in a probabilistic situation. The way to prove such `existential' propositions often uses averages. We may call this Supported by the NSERC operating grants OGP-0036747 and OGP-046506. Address: Computer Science Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email: mli@math.waterloo.edu y Partially supported by the NSERC International Scientific Exchange Award ISE...

We investigate to what extent finite binary sequences with high Kolmogorov complexity are normal (all blocks of equal length occur equally frequent), and the maximal length of all-zero or all-one runs which occur with certainty. 1 Introduction Each individual infinite sequence generated by a ( 1 2 ; 1 2 ) Bernoulli process (flipping a fair coin) has (with probability 1) the property that the relative frequency of zeros in an initial n-length segment goes to 1 2 for n goes to infinity. A related statement can be made for finite sequences, in the sense that one can say that the majority of all sequences will have about one half zeros. Supported by the NSERC operating grant OGP-046506. Address: Computer Science Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email: mli@math.uwaterloo.ca y Partially supported by the NSERC International Scientific Exchange Award ISE0046203 and by NWO through NFI Project ALADDIN under Contract number NF 62-376. Address: Centr...

We study the logistic map f(x) = λx(1 − x) on the unit square at the chaos threshold. By using the methods of symbolic dynamics, the information content of a given string is defined as the length of the string after it has been compressed by a compression algorithm, called CASToRe. The information content is then used to characterise the chaotic behaviour. From this analysis, it appears natural to introduce the notion of mild chaos to denote a chaotic behaviour in which the information content grows slower than any power law. 1