261,856 research outputs found
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
Effective complexity of stationary process realizations
The concept of effective complexity of an object as the minimal description
length of its regularities has been initiated by Gell-Mann and Lloyd. The
regularities are modeled by means of ensembles, that is probability
distributions on finite binary strings. In our previous paper we propose a
definition of effective complexity in precise terms of algorithmic information
theory. Here we investigate the effective complexity of binary strings
generated by stationary, in general not computable, processes. We show that
under not too strong conditions long typical process realizations are
effectively simple. Our results become most transparent in the context of
coarse effective complexity which is a modification of the original notion of
effective complexity that uses less parameters in its definition. A similar
modification of the related concept of sophistication has been suggested by
Antunes and Fortnow.Comment: 14 pages, no figure
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Training-free Measures Based on Algorithmic Probability Identify High Nucleosome Occupancy in DNA Sequences
We introduce and study a set of training-free methods of
information-theoretic and algorithmic complexity nature applied to DNA
sequences to identify their potential capabilities to determine nucleosomal
binding sites. We test our measures on well-studied genomic sequences of
different sizes drawn from different sources. The measures reveal the known in
vivo versus in vitro predictive discrepancies and uncover their potential to
pinpoint (high) nucleosome occupancy. We explore different possible signals
within and beyond the nucleosome length and find that complexity indices are
informative of nucleosome occupancy. We compare against the gold standard
(Kaplan model) and find similar and complementary results with the main
difference that our sequence complexity approach. For example, for high
occupancy, complexity-based scores outperform the Kaplan model for predicting
binding representing a significant advancement in predicting the highest
nucleosome occupancy following a training-free approach.Comment: 8 pages main text (4 figures), 12 total with Supplementary (1 figure
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