4,944 research outputs found
Busy Beaver Scores and Alphabet Size
We investigate the Busy Beaver Game introduced by Rado (1962) generalized to
non-binary alphabets. Harland (2016) conjectured that activity (number of
steps) and productivity (number of non-blank symbols) of candidate machines
grow as the alphabet size increases. We prove this conjecture for any alphabet
size under the condition that the number of states is sufficiently large. For
the measure activity we show that increasing the alphabet size from two to
three allows an increase. By a classical construction it is even possible to
obtain a two-state machine increasing activity and productivity of any machine
if we allow an alphabet size depending on the number of states of the original
machine. We also show that an increase of the alphabet by a factor of three
admits an increase of activity
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
Problems in number theory from busy beaver competition
By introducing the busy beaver competition of Turing machines, in 1962, Rado
defined noncomputable functions on positive integers. The study of these
functions and variants leads to many mathematical challenges. This article
takes up the following one: How can a small Turing machine manage to produce
very big numbers? It provides the following answer: mostly by simulating
Collatz-like functions, that are generalizations of the famous 3x+1 function.
These functions, like the 3x+1 function, lead to new unsolved problems in
number theory.Comment: 35 page
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
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