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18
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Relations between varieties of Kolmogorov complexity
 Mathematical Systems Theory
, 1996
"... Abstract. There are several sorts of Kolmogorov complexity, better to say several Kolmogorov complexities: decision complexity, simple complexity, prefix complexity, monotonic complexity, a priori complexity. The last three can and the first two cannot be used for defining randomness of an infinite ..."
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Cited by 7 (3 self)
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Abstract. There are several sorts of Kolmogorov complexity, better to say several Kolmogorov complexities: decision complexity, simple complexity, prefix complexity, monotonic complexity, a priori complexity. The last three can and the first two cannot be used for defining randomness of an infinite binary sequence. All those five versions of Kolmogorov complexity were considered, from a unified point of view, in a paper by the first author which appeared in Watanabe’s book [23]. Upper and lower bounds for those complexities and also for their differences were announced in that paper without proofs. (Some of those bounds are mentioned in Section 4.4.5 of [16].) The purpose of this paper (which can be read independently of [23]) is to give proofs for the bounds from [23]. The terminology used in this paper is somehow nonstandard: we call “Kolmogorov entropy ” what is usually called “Kolmogorov complexity. ” This is a Moscow tradition suggested by Kolmogorov himself. By this tradition the term “complexity ” relates to any mode of description and “entropy ” is the complexity related to an optimal mode (i.e., to a mode that, roughly speaking, gives the shortest descriptions).
Orbit complexity, initial data sensitivity and weakly chaotic dynamical systems.Preprint
"... We give a definition of generalized indicators of sensitivity to initial conditions and orbit complexity (a measure of the information that is necessary to describe the orbit of a given point). The well known RuellePesin and BrinKatok theorems, combined with Brudno’s theorem give a relation betwee ..."
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Cited by 6 (3 self)
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We give a definition of generalized indicators of sensitivity to initial conditions and orbit complexity (a measure of the information that is necessary to describe the orbit of a given point). The well known RuellePesin and BrinKatok theorems, combined with Brudno’s theorem give a relation between initial data sensitivity and orbit complexity that is generalized in the present work. The generalized relation implies that the set of points where the sensitivity to initial conditions is more than exponential in all directions is a 0 dimensional set. The generalized relation is then applied to the study of an important example of weakly chaotic dynamics: the Manneville map. 1
The Manneville map: topological, metric and algorithmic entropy”, work in preparation
, 2001
"... We study the Manneville map f(x) = x + x z (mod 1), with z> 1, from a computational point of view, studying the behaviour of the Algorithmic Information Content. In particular, we consider a family of piecewise linear maps that gives examples of algorithmic behaviour ranging from the fully to the m ..."
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Cited by 5 (4 self)
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We study the Manneville map f(x) = x + x z (mod 1), with z> 1, from a computational point of view, studying the behaviour of the Algorithmic Information Content. In particular, we consider a family of piecewise linear maps that gives examples of algorithmic behaviour ranging from the fully to the mildly chaotic, and show that the Manneville map is a member of this family. 1
Compression and diffusion: a joint approach to detect complexity
 Chaos, Solitons & Fractals
, 2003
"... The adoption of the KolmogorovSinai (KS) entropy is becoming a popular research tool among physicists, especially when applied to a dynamical system fitting the conditions of validity of the Pesin theorem. The study of time series that are a manifestation of system dynamics whose rules are either u ..."
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Cited by 4 (3 self)
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The adoption of the KolmogorovSinai (KS) entropy is becoming a popular research tool among physicists, especially when applied to a dynamical system fitting the conditions of validity of the Pesin theorem. The study of time series that are a manifestation of system dynamics whose rules are either unknown or too complex for a mathematical treatment, is still a challenge since the KS entropy is not computable, in general, in that case. Here we present a plan of action based on the joint action of two procedures, both related to the KS entropy, but compatible with computer implementation through fast and efficient programs. The former procedure, called Compression Algorithm Sensitive To Regularity (CASToRe), establishes the amount of order by the numerical evaluation of algorithmic compressibility. The latter, called Complex Analysis of Sequences via Scaling AND Randomness Assessment (CASSANDRA), establishes the complexity degree through the numerical evaluation of the
Measure of uncertainty and information
 Imprecise Probability Project,1999 (http://ippserv.rug.ac.be/home/ipp.html
, 1999
"... Abstract. This contribution overviews the approaches, results and history of attempts at measuring uncertainty and information in the various theories of imprecise probabilities. The main focus, however, is on the theory of belief functions (or the DempsterShafer theory) [62] and the possibility th ..."
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Cited by 3 (0 self)
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Abstract. This contribution overviews the approaches, results and history of attempts at measuring uncertainty and information in the various theories of imprecise probabilities. The main focus, however, is on the theory of belief functions (or the DempsterShafer theory) [62] and the possibility theory [7] as most of the development so far has happened there. Due to the limited space I am focusing on the main ideas and point to references for details. There are
Vortex dynamics in evolutive flows: a weakly chaotic phenomenon
, 2003
"... We study the vortex dynamics in an evolutive flow. We carry out the statistical analysis of the resulting time series by means of the joint use of a compression and an entropy diffusion method. This approach to complexity makes it possible for us to establish that the time series emerging from the w ..."
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Cited by 1 (1 self)
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We study the vortex dynamics in an evolutive flow. We carry out the statistical analysis of the resulting time series by means of the joint use of a compression and an entropy diffusion method. This approach to complexity makes it possible for us to establish that the time series emerging from the wavelet analysis of the vortex dynamics is a weakly chaotic process driven by a hidden dynamic process. The complexity of the hidden driving process is shown to be located at the border between the stationary and nonstationary state. 1
Weakly Bounded Probabilistic Polytime is Contained in POLYSIZE
"... It is known that bounded error probabilistic polynomial time (BPP) languages are accepted by polynomial size circuit families (POLYSIZE). We sharpen and extend this result to WBPP for which the BPP error bound ffl ? 0 is weakened to ffl(n) =\Omega\Gamma3 =n O(1) ) for length n inputs. The WBPP re ..."
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It is known that bounded error probabilistic polynomial time (BPP) languages are accepted by polynomial size circuit families (POLYSIZE). We sharpen and extend this result to WBPP for which the BPP error bound ffl ? 0 is weakened to ffl(n) =\Omega\Gamma3 =n O(1) ) for length n inputs. The WBPP result is obtained by using Turing randomness to avoid involved counting arguments. 1 Introduction Complexity theory is the part of computer science that identifies computing resources and establishes quantitative relationships among them. In this way, one resource can be measured in terms of others. We will be concerned with measuring randomness in terms of Boolean circuit size. Additional details about the notions used here may be obtained from [2]. It will be convenient to express computation in terms of language acceptance. Languages will be subsets of f0; 1g + . The output of a Turing machine M on input x will be designated by M(x). If a Turing machine M with inputs over f0; 1g + ha...
Exploiting Equity Momentum with Symbolic Machine Learning
, 1998
"... This thesis investigates the use of machine learning to identify regularities in the phenomena of `momentum' within the equity market. Empirical results obtained here suggest that it is possible to obtain some degree of predictability in stock return movements. Results also suggest that investment s ..."
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This thesis investigates the use of machine learning to identify regularities in the phenomena of `momentum' within the equity market. Empirical results obtained here suggest that it is possible to obtain some degree of predictability in stock return movements. Results also suggest that investment strategies derived from machine learning can significantly outperform both the `S&P 500' and naive heuristic strategies. Finally, this study takes first steps towards the incorporation of firstorder machine learning techniques as embodied by Inductive Logic Programming into stock market prediction. Contents I
Looking From the Inside and From the Outside
, 1998
"... Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for ..."
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Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without "lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions.