786 research outputs found
Uniform distribution and algorithmic randomness
A seminal theorem due to Weyl states that if (a_n) is any sequence of
distinct integers, then, for almost every real number x, the sequence (a_n x)
is uniformly distributed modulo one. In particular, for almost every x in the
unit interval, the sequence (a_n x) is uniformly distributed modulo one for
every computable sequence (a_n) of distinct integers. Call such an x "UD
random". Here it is shown that every Schnorr random real is UD random, but
there are Kurtz random reals that are not UD random. On the other hand, Weyl's
theorem still holds relative to a particular effectively closed null set, so
there are UD random reals that are not Kurtz random
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
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
Generic algorithms for halting problem and optimal machines revisited
The halting problem is undecidable --- but can it be solved for "most"
inputs? This natural question was considered in a number of papers, in
different settings. We revisit their results and show that most of them can be
easily proven in a natural framework of optimal machines (considered in
algorithmic information theory) using the notion of Kolmogorov complexity. We
also consider some related questions about this framework and about asymptotic
properties of the halting problem. In particular, we show that the fraction of
terminating programs cannot have a limit, and all limit points are Martin-L\"of
random reals. We then consider mass problems of finding an approximate solution
of halting problem and probabilistic algorithms for them, proving both positive
and negative results. We consider the fraction of terminating programs that
require a long time for termination, and describe this fraction using the busy
beaver function. We also consider approximate versions of separation problems,
and revisit Schnorr's results about optimal numberings showing how they can be
generalized.Comment: a preliminary version was presented at the ICALP 2015 conferenc
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
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts in the same volume. Part I is dedicated to information theory and
the mathematical formalization of randomness based on Kolmogorov complexity.
This last application goes back to the 60's and 70's with the work of
Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last
years.Comment: 40 page
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