334 research outputs found
Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness
We describe an alternative method (to compression) that combines several
theoretical and experimental results to numerically approximate the algorithmic
(Kolmogorov-Chaitin) complexity of all bit strings up to 8
bits long, and for some between 9 and 16 bits long. This is done by an
exhaustive execution of all deterministic 2-symbol Turing machines with up to 4
states for which the halting times are known thanks to the Busy Beaver problem,
that is 11019960576 machines. An output frequency distribution is then
computed, from which the algorithmic probability is calculated and the
algorithmic complexity evaluated by way of the (Levin-Zvonkin-Chaitin) coding
theorem.Comment: 29 pages, 5 figures. Version as accepted by the journal Applied
Mathematics and Computatio
On the Algorithmic Nature of the World
We propose a test based on the theory of algorithmic complexity and an
experimental evaluation of Levin's universal distribution to identify evidence
in support of or in contravention of the claim that the world is algorithmic in
nature. To this end we have undertaken a statistical comparison of the
frequency distributions of data from physical sources on the one
hand--repositories of information such as images, data stored in a hard drive,
computer programs and DNA sequences--and the frequency distributions generated
by purely algorithmic means on the other--by running abstract computing devices
such as Turing machines, cellular automata and Post Tag systems. Statistical
correlations were found and their significance measured.Comment: Book chapter in Gordana Dodig-Crnkovic and Mark Burgin (eds.)
Information and Computation by World Scientific, 2010.
(http://www.idt.mdh.se/ECAP-2005/INFOCOMPBOOK/). Paper website:
http://www.mathrix.org/experimentalAIT
Image Characterization and Classification by Physical Complexity
We present a method for estimating the complexity of an image based on
Bennett's concept of logical depth. Bennett identified logical depth as the
appropriate measure of organized complexity, and hence as being better suited
to the evaluation of the complexity of objects in the physical world. Its use
results in a different, and in some sense a finer characterization than is
obtained through the application of the concept of Kolmogorov complexity alone.
We use this measure to classify images by their information content. The method
provides a means for classifying and evaluating the complexity of objects by
way of their visual representations. To the authors' knowledge, the method and
application inspired by the concept of logical depth presented herein are being
proposed and implemented for the first time.Comment: 30 pages, 21 figure
Estimating the Algorithmic Complexity of Stock Markets
Randomness and regularities in Finance are usually treated in probabilistic
terms. In this paper, we develop a completely different approach in using a
non-probabilistic framework based on the algorithmic information theory
initially developed by Kolmogorov (1965). We present some elements of this
theory and show why it is particularly relevant to Finance, and potentially to
other sub-fields of Economics as well. We develop a generic method to estimate
the Kolmogorov complexity of numeric series. This approach is based on an
iterative "regularity erasing procedure" implemented to use lossless
compression algorithms on financial data. Examples are provided with both
simulated and real-world financial time series. The contributions of this
article are twofold. The first one is methodological : we show that some
structural regularities, invisible with classical statistical tests, can be
detected by this algorithmic method. The second one consists in illustrations
on the daily Dow-Jones Index suggesting that beyond several well-known
regularities, hidden structure may in this index remain to be identified
A short note on number preference
Subjects usually show a preference for number 7 above any other in the bracket 0-9, at least in Europe and the USA. Two explanations (a cognitive one, and a cultural one) have been put forward as a bases for this "seven phenomenon". Here we advocate for a "reconciliation" between these two theories
Algorithmic Complexity of Financial Motions
We survey the main applications of algorithmic (Kolmogorov) complexity to the problem of price dynamics in financial markets. We stress the differences between these works and put forward a general algorithmic framework in order to highlight its potential for financial data analysis. This framework is “general" in the sense that it is not constructed on the common assumption that price variations are predominantly stochastic in nature.algorithmic information theory; Kolmogorov complexity; financial returns; market efficiency; compression algorithms; information theory; randomness; price movements; algorithmic probability
Automated Proofs in Geometry : Computing Upper Bounds for the Heilbronn Problem for Triangles
We propose a method for computing upper bounds for the Heilbronn problem for
triangles
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