930 research outputs found
Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view.Comment: 24 page
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
We introduce the zeta number, natural halting probability and natural
complexity of a Turing machine and we relate them to Chaitin's Omega number,
halting probability, and program-size complexity. A classification of Turing
machines according to their zeta numbers is proposed: divergent, convergent and
tuatara. We prove the existence of universal convergent and tuatara machines.
Various results on (algorithmic) randomness and partial randomness are proved.
For example, we show that the zeta number of a universal tuatara machine is
c.e. and random. A new type of partial randomness, asymptotic randomness, is
introduced. Finally we show that in contrast to classical (algorithmic)
randomness--which cannot be naturally characterised in terms of plain
complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin
On zeros of Martin-L\"of random Brownian motion
We investigate the sample path properties of Martin-L\"of random Brownian
motion. We show (1) that many classical results which are known to hold almost
surely hold for every Martin-L\"of random Brownian path, (2) that the effective
dimension of zeroes of a Martin-L\"of random Brownian path must be at least
1/2, and conversely that every real with effective dimension greater than 1/2
must be a zero of some Martin-L\"of random Brownian path, and (3) we will
demonstrate a new proof that the solution to the Dirichlet problem in the plane
is computable
Primordial RNA Replication and Applications in PCR Technology
The emergence of self-replication and information transmission in life's
origin remains unexplained despite extensive research on the topic. A
hypothesis explaining the transition from a simple organic world to a complex
RNA world is offered here based on physical factors in hydrothermal vent
systems. An interdisciplinary approach is taken using techniques from
thermodynamics, fluid dynamics, oceanography, statistical mechanics, and
stochastic processes to examine nucleic acid dynamics and kinetics in a
hydrothermal vent from first principles. Analyses are carried out using both
analytic and computational methods and confirm the plausibility of a reaction
involving the PCR-like assembly of ribonucleotides. The proposal is put into
perspective with established theories on the origin of life and more generally
the onset of order and information transmission in prebiotic systems. A
biomimicry application of this hypothetical process to PCR technology is
suggested and its viability is evaluated in a rigorous logical analysis.
Optimal temperature curves begin to be established using Monte Carlo
simulation, variational calculus, and Fourier analysis. The converse argument
is also made but qualitatively, asserting that the success of such a
modification to PCR would in turn reconfirm the biological theory.Comment: 20 pages, 7 figure
The rapid points of a complex oscillation
By considering a counting-type argument on Brownian sample paths, we prove a
result similar to that of Orey and Taylor on the exact Hausdorff dimension of
the rapid points of Brownian motion. Because of the nature of the proof we can
then apply the concepts to so-called complex oscillations (or 'algorithmically
random Brownian motion'), showing that their rapid points have the same
dimension.Comment: 11 page
Kolmogorov complexity and probability measures
summary:Classes of strings (infinite sequences resp.) with a specific flow of Kolmogorov complexity are introduced. Namely, lower bounds of Kolmogorov complexity are prescribed to strings (initial segments of infinite sequences resp.) of specified lengths. Dependence of probabilities of the classes on lower bounds of Kolmogorov complexity is the main theme of the paper. Conditions are found under which the probabilities of the classes of the strings are close to one. Similarly, conditions are derived under which the probabilities of the classes of the sequences equal one. It is shown that there are lower bounds of Kolmogorov complexity such that the studied classes of the strings are of probability close to one, classes of the sequences are of probability one, both with respect to almost all probability measures used in practice. A variant of theorem on infinite oscillations is derived
The Dimensions of Individual Strings and Sequences
A constructive version of Hausdorff dimension is developed using constructive
supergales, which are betting strategies that generalize the constructive
supermartingales used in the theory of individual random sequences. This
constructive dimension is used to assign every individual (infinite, binary)
sequence S a dimension, which is a real number dim(S) in the interval [0,1].
Sequences that are random (in the sense of Martin-Lof) have dimension 1, while
sequences that are decidable, \Sigma^0_1, or \Pi^0_1 have dimension 0. It is
shown that for every \Delta^0_2-computable real number \alpha in [0,1] there is
a \Delta^0_2 sequence S such that \dim(S) = \alpha.
A discrete version of constructive dimension is also developed using
termgales, which are supergale-like functions that bet on the terminations of
(finite, binary) strings as well as on their successive bits. This discrete
dimension is used to assign each individual string w a dimension, which is a
nonnegative real number dim(w). The dimension of a sequence is shown to be the
limit infimum of the dimensions of its prefixes.
The Kolmogorov complexity of a string is proven to be the product of its
length and its dimension. This gives a new characterization of algorithmic
information and a new proof of Mayordomo's recent theorem stating that the
dimension of a sequence is the limit infimum of the average Kolmogorov
complexity of its first n bits.
Every sequence that is random relative to any computable sequence of
coin-toss biases that converge to a real number \beta in (0,1) is shown to have
dimension \H(\beta), the binary entropy of \beta.Comment: 31 page
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