64 research outputs found
m-sophistication
The m-sophistication of a finite binary string x is introduced as a
generalization of some parameter in the proof that complexity of complexity is
rare. A probabilistic near sufficient statistic of x is given which length is
upper bounded by the m-sophistication of x within small additive terms. This
shows that m-sophistication is lower bounded by coarse sophistication and upper
bounded by sophistication within small additive terms. It is also shown that
m-sophistication and coarse sophistication can not be approximated by an upper
or lower semicomputable function, not even within very large error.Comment: 13 pages, draf
Asymmetry of the Kolmogorov complexity of online predicting odd and even bits
Symmetry of information states that .
We show that a similar relation for online Kolmogorov complexity does not hold.
Let the even (online Kolmogorov) complexity of an n-bitstring
be the length of a shortest program that computes on input ,
computes on input , etc; and similar for odd complexity. We
show that for all n there exist an n-bit x such that both odd and even
complexity are almost as large as the Kolmogorov complexity of the whole
string. Moreover, flipping odd and even bits to obtain a sequence
, decreases the sum of odd and even complexity to .Comment: 20 pages, 7 figure
Influence tests I: ideal composite hypothesis tests, and causal semimeasures
Ratios of universal enumerable semimeasures corresponding to hypotheses are
investigated as a solution for statistical composite hypotheses testing if an
unbounded amount of computation time can be assumed.
Influence testing for discrete time series is defined using generalized
structural equations. Several ideal tests are introduced, and it is argued that
when Halting information is transmitted, in some cases, instantaneous cause and
consequence can be inferred where this is not possible classically.
The approach is contrasted with Bayesian definitions of influence, where it
is left open whether all Bayesian causal associations of universal semimeasures
are equal within a constant. Finally the approach is also contrasted with
existing engineering procedures for influence and theoretical definitions of
causation.Comment: 29 pages, 3 figures, draf
Relating and contrasting plain and prefix Kolmogorov complexity
In [3] a short proof is given that some strings have maximal plain Kolmogorov
complexity but not maximal prefix-free complexity. The proof uses Levin's
symmetry of information, Levin's formula relating plain and prefix complexity
and Gacs' theorem that complexity of complexity given the string can be high.
We argue that the proof technique and results mentioned above are useful to
simplify existing proofs and to solve open questions.
We present a short proof of Solovay's result [21] relating plain and prefix
complexity: and , (here denotes , etc.).
We show that there exist such that is infinite and is
finite, i.e. the infinitely often C-trivial reals are not the same as the
infinitely often K-trivial reals (i.e. [1,Question 1]).
Solovay showed that for infinitely many we have
and , (here
denotes the length of and , etc.). We show that this
result holds for prefixes of some 2-random sequences.
Finally, we generalize our proof technique and show that no monotone relation
exists between expectation and probability bounded randomness deficiency (i.e.
[6, Question 1]).Comment: 20 pages, 1 figur
Linear list-approximation for short programs (or the power of a few random bits)
A -short program for a string is a description of of length at
most , where is the Kolmogorov complexity of . We show that
there exists a randomized algorithm that constructs a list of elements that
contains a -short program for . We also show a polynomial-time
randomized construction that achieves the same list size for -short programs. These results beat the lower bounds shown by Bauwens et al.
\cite{bmvz:c:shortlist} for deterministic constructions of such lists. We also
prove tight lower bounds for the main parameters of our result. The
constructions use only ( for the polynomial-time
result) random bits . Thus using only few random bits it is possible to do
tasks that cannot be done by any deterministic algorithm regardless of its
running time
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