265 research outputs found
On Time-Bounded Incompressibility of Compressible Strings and Sequences
For every total recursive time bound , a constant fraction of all
compressible (low Kolmogorov complexity) strings is -bounded incompressible
(high time-bounded Kolmogorov complexity); there are uncountably many infinite
sequences of which every initial segment of length is compressible to yet -bounded incompressible below ; and there are
countable infinitely many recursive infinite sequence of which every initial
segment is similarly -bounded incompressible. These results are related to,
but different from, Barzdins's lemma.Comment: 9 pages, LaTeX, no figures, submitted to Information Processing
Letters. Changed and added a Barzdins-like lemma for infinite sequences with
different quantification oreder, a fixed constant, and uncountably many
sequence
On Resource-bounded versions of the van Lambalgen theorem
The van Lambalgen theorem is a surprising result in algorithmic information
theory concerning the symmetry of relative randomness. It establishes that for
any pair of infinite sequences and , is Martin-L\"of random and
is Martin-L\"of random relative to if and only if the interleaved sequence
is Martin-L\"of random. This implies that is relative random
to if and only if is random relative to \cite{vanLambalgen},
\cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this
phenomenon for different notions of time-bounded relative randomness.
We prove the classical van Lambalgen theorem using martingales and Kolmogorov
compressibility. We establish the failure of relative randomness in these
settings, for both time-bounded martingales and time-bounded Kolmogorov
complexity. We adapt our classical proofs when applicable to the time-bounded
setting, and construct counterexamples when they fail. The mode of failure of
the theorem may depend on the notion of time-bounded randomness
Around Kolmogorov complexity: basic notions and results
Algorithmic information theory studies description complexity and randomness
and is now a well known field of theoretical computer science and mathematical
logic. There are several textbooks and monographs devoted to this theory where
one can find the detailed exposition of many difficult results as well as
historical references. However, it seems that a short survey of its basic
notions and main results relating these notions to each other, is missing.
This report attempts to fill this gap and covers the basic notions of
algorithmic information theory: Kolmogorov complexity (plain, conditional,
prefix), Solomonoff universal a priori probability, notions of randomness
(Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff
dimension. We prove their basic properties (symmetry of information, connection
between a priori probability and prefix complexity, criterion of randomness in
terms of complexity, complexity characterization for effective dimension) and
show some applications (incompressibility method in computational complexity
theory, incompleteness theorems). It is based on the lecture notes of a course
at Uppsala University given by the author
Anytime Algorithms for Non-Ending Computations
A program which eventually stops but does not halt âtoo quicklyâ halts at a time which is algorithmically compressible. This result â originally proved in [4] â is proved in a more general setting. Following Manin [11] we convert the result into an anytime algorithm for the halting problem and we show that the stopping time (cut-off temporal bound) cannot be significantly improved
Causality - Complexity - Consistency: Can Space-Time Be Based on Logic and Computation?
The difficulty of explaining non-local correlations in a fixed causal
structure sheds new light on the old debate on whether space and time are to be
seen as fundamental. Refraining from assuming space-time as given a priori has
a number of consequences. First, the usual definitions of randomness depend on
a causal structure and turn meaningless. So motivated, we propose an intrinsic,
physically motivated measure for the randomness of a string of bits: its length
minus its normalized work value, a quantity we closely relate to its Kolmogorov
complexity (the length of the shortest program making a universal Turing
machine output this string). We test this alternative concept of randomness for
the example of non-local correlations, and we end up with a reasoning that
leads to similar conclusions as in, but is conceptually more direct than, the
probabilistic view since only the outcomes of measurements that can actually
all be carried out together are put into relation to each other. In the same
context-free spirit, we connect the logical reversibility of an evolution to
the second law of thermodynamics and the arrow of time. Refining this, we end
up with a speculation on the emergence of a space-time structure on bit strings
in terms of data-compressibility relations. Finally, we show that logical
consistency, by which we replace the abandoned causality, it strictly weaker a
constraint than the latter in the multi-party case.Comment: 17 pages, 16 figures, small correction
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