11 research outputs found
Large Alphabets and Incompressibility
We briefly survey some concepts related to empirical entropy -- normal
numbers, de Bruijn sequences and Markov processes -- and investigate how well
it approximates Kolmogorov complexity. Our results suggest th-order
empirical entropy stops being a reasonable complexity metric for almost all
strings of length over alphabets of size about when surpasses
Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer
Since human randomness production has been studied and widely used to assess
executive functions (especially inhibition), many measures have been suggested
to assess the degree to which a sequence is random-like. However, each of them
focuses on one feature of randomness, leading authors to have to use multiple
measures. Here we describe and advocate for the use of the accepted universal
measure for randomness based on algorithmic complexity, by means of a novel
previously presented technique using the the definition of algorithmic
probability. A re-analysis of the classical Radio Zenith data in the light of
the proposed measure and methodology is provided as a study case of an
application.Comment: To appear in Behavior Research Method
Words and Transcendence
Is it possible to distinguish algebraic from transcendental real numbers by
considering the -ary expansion in some base ? In 1950, \'E. Borel
suggested that the answer is no and that for any real irrational algebraic
number and for any base , the -ary expansion of should
satisfy some of the laws that are shared by almost all numbers. There is no
explicitly known example of a triple , where is an integer,
a digit in and a real irrational algebraic number, for
which one can claim that the digit occurs infinitely often in the -ary
expansion of . However, some progress has been made recently, thanks mainly
to clever use of Schmidt's subspace theorem. We review some of these results
A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties
In the general context of computable metric spaces and computable measures we
prove a kind of constructive Borel-Cantelli lemma: given a sequence
(constructive in some way) of sets with effectively summable measures,
there are computable points which are not contained in infinitely many .
As a consequence of this we obtain the existence of computable points which
follow the \emph{typical statistical behavior} of a dynamical system (they
satisfy the Birkhoff theorem) for a large class of systems, having computable
invariant measure and a certain ``logarithmic'' speed of convergence of
Birkhoff averages over Lipshitz observables. This is applied to uniformly
hyperbolic systems, piecewise expanding maps, systems on the interval with an
indifferent fixed point and it directly implies the existence of computable
numbers which are normal with respect to any base.Comment: Revised version. Several results are generalize