64 research outputs found
The Busy Beaver Competition: a historical survey
Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing
machines to give explicit definitions for some functions that are not
computable and grow faster than any computable function. He put forward the
problem of computing the values of these functions on numbers 1, 2, 3, ... More
and more powerful computers have made possible the computation of lower bounds
for these values. In 1988, Brady extended the definitions to functions on two
variables. We give a historical survey of these works. The successive record
holders in the Busy Beaver Competition are displayed, with their discoverers,
the date they were found, and, for some of them, an analysis of their behavior.Comment: 70 page
Busy beavers gone wild
We show some incompleteness results a la Chaitin using the busy beaver
functions. Then, with the help of ordinal logics, we show how to obtain a
theory in which the values of the busy beaver functions can be provably
established and use this to reveal a structure on the provability of the values
of these functions
Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks
We show that numerical approximations of Kolmogorov complexity (K) applied to
graph adjacency matrices capture some group-theoretic and topological
properties of graphs and empirical networks ranging from metabolic to social
networks. That K and the size of the group of automorphisms of a graph are
correlated opens up interesting connections to problems in computational
geometry, and thus connects several measures and concepts from complexity
science. We show that approximations of K characterise synthetic and natural
networks by their generating mechanisms, assigning lower algorithmic randomness
to complex network models (Watts-Strogatz and Barabasi-Albert networks) and
high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these
results via two different Kolmogorov complexity approximation methods applied
to the adjacency matrices of the graphs and networks. The methods used are the
traditional lossless compression approach to Kolmogorov complexity, and a
normalised version of a Block Decomposition Method (BDM) measure, based on
algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical
Mechanics and its Application
Random semicomputable reals revisited
The aim of this expository paper is to present a nice series of results,
obtained in the papers of Chaitin (1976), Solovay (1975), Calude et al. (1998),
Kucera and Slaman (2001). This joint effort led to a full characterization of
lower semicomputable random reals, both as those that can be expressed as a
"Chaitin Omega" and those that are maximal for the Solovay reducibility. The
original proofs were somewhat involved; in this paper, we present these results
in an elementary way, in particular requiring only basic knowledge of
algorithmic randomness. We add also several simple observations relating lower
semicomputable random reals and busy beaver functions.Comment: 15 page
Algorithmic statistics revisited
The mission of statistics is to provide adequate statistical hypotheses
(models) for observed data. But what is an "adequate" model? To answer this
question, one needs to use the notions of algorithmic information theory. It
turns out that for every data string one can naturally define
"stochasticity profile", a curve that represents a trade-off between complexity
of a model and its adequacy. This curve has four different equivalent
definitions in terms of (1)~randomness deficiency, (2)~minimal description
length, (3)~position in the lists of simple strings and (4)~Kolmogorov
complexity with decompression time bounded by busy beaver function. We present
a survey of the corresponding definitions and results relating them to each
other
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
The Busy Beaver Competition: a historical survey
70 pagesTibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, ... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior
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