36,015 research outputs found
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem
In classical information theory, entropy rate and Kolmogorov complexity per
symbol are related by a theorem of Brudno. In this paper, we prove a quantum
version of this theorem, connecting the von Neumann entropy rate and two
notions of quantum Kolmogorov complexity, both based on the shortest qubit
descriptions of qubit strings that, run by a universal quantum Turing machine,
reproduce them as outputs.Comment: 26 pages, no figures. Reference to publication added: published in
the Communications in Mathematical Physics
(http://www.springerlink.com/content/1432-0916/
Effective Complexity and its Relation to Logical Depth
Effective complexity measures the information content of the regularities of
an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of
the disadvantages of Kolmogorov complexity, also known as algorithmic
information content. In this paper, we give a precise formal definition of
effective complexity and rigorous proofs of its basic properties. In
particular, we show that incompressible binary strings are effectively simple,
and we prove the existence of strings that have effective complexity close to
their lengths. Furthermore, we show that effective complexity is related to
Bennett's logical depth: If the effective complexity of a string exceeds a
certain explicit threshold then that string must have astronomically large
depth; otherwise, the depth can be arbitrarily small.Comment: 14 pages, 2 figure
Kolmogorov complexity in perspective
We survey the diverse approaches to the notion of information content: from
Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov
complexity are presented namely, the mathematical notion of randomness (which
goes back to the 60's with the work of Martin-Lof, Schnorr, Chaitin, Levin),
and classification, which is a recent idea with provocative implementation by
Vitanyi and Cilibrasi.Comment: 37 page
Algorithmic Clustering of Music
We present a fully automatic method for music classification, based only on
compression of strings that represent the music pieces. The method uses no
background knowledge about music whatsoever: it is completely general and can,
without change, be used in different areas like linguistic classification and
genomics. It is based on an ideal theory of the information content in
individual objects (Kolmogorov complexity), information distance, and a
universal similarity metric. Experiments show that the method distinguishes
reasonably well between various musical genres and can even cluster pieces by
composer.Comment: 17 pages, 11 figure
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
Algorithmic Statistics
While Kolmogorov complexity is the accepted absolute measure of information
content of an individual finite object, a similarly absolute notion is needed
for the relation between an individual data sample and an individual model
summarizing the information in the data, for example, a finite set (or
probability distribution) where the data sample typically came from. The
statistical theory based on such relations between individual objects can be
called algorithmic statistics, in contrast to classical statistical theory that
deals with relations between probabilistic ensembles. We develop the
algorithmic theory of statistic, sufficient statistic, and minimal sufficient
statistic. This theory is based on two-part codes consisting of the code for
the statistic (the model summarizing the regularity, the meaningful
information, in the data) and the model-to-data code. In contrast to the
situation in probabilistic statistical theory, the algorithmic relation of
(minimal) sufficiency is an absolute relation between the individual model and
the individual data sample. We distinguish implicit and explicit descriptions
of the models. We give characterizations of algorithmic (Kolmogorov) minimal
sufficient statistic for all data samples for both description modes--in the
explicit mode under some constraints. We also strengthen and elaborate earlier
results on the ``Kolmogorov structure function'' and ``absolutely
non-stochastic objects''--those rare objects for which the simplest models that
summarize their relevant information (minimal sufficient statistics) are at
least as complex as the objects themselves. We demonstrate a close relation
between the probabilistic notions and the algorithmic ones.Comment: LaTeX, 22 pages, 1 figure, with correction to the published journal
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Global Software Development: Measuring, Approximating and Reducing the Complexity of Global Software Development Using Extended Axiomatic Design Theory
This paper considers GSD projects as designed artefacts, and proposes the application of an Extended Axiomatic Design theory to reduce their complexity in order to increase the probability of project success. Using an upper bound estimation of the Kolmogorov complexity of the so-called ‘design matrix’ (as a proxy of Information Content as a complexity measure) we demonstrate on two hypothetical examples how good and bad designs of GSD planning compare in terms of complexity. We also demonstrate how to measure and calculate the ‘structural’ complexity of GSD projects and show that by satisfying all design axioms this ‘structural’ complexity could be minimised
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