199 research outputs found
Random semicomputable reals revisited
The aim of this expository paper is to present a nice series of results,
obtained in the papers of Chaitin (1976), Solovay (1975), Calude et al. (1998),
Kucera and Slaman (2001). This joint effort led to a full characterization of
lower semicomputable random reals, both as those that can be expressed as a
"Chaitin Omega" and those that are maximal for the Solovay reducibility. The
original proofs were somewhat involved; in this paper, we present these results
in an elementary way, in particular requiring only basic knowledge of
algorithmic randomness. We add also several simple observations relating lower
semicomputable random reals and busy beaver functions.Comment: 15 page
Kolmogorov Complexity and Solovay Functions
Solovay proved that there exists a computable upper bound f of the
prefix-free Kolmogorov complexity function K such that f (x) = K(x) for
infinitely many x. In this paper, we consider the class of computable functions
f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for
infinitely many x, which we call Solovay functions. We show that Solovay
functions present interesting connections with randomness notions such as
Martin-L\"of randomness and K-triviality
K-trivial, K-low and MLR-low sequences: a tutorial
A remarkable achievement in algorithmic randomness and algorithmic
information theory was the discovery of the notions of K-trivial, K-low and
Martin-Lof-random-low sets: three different definitions turns out to be
equivalent for very non-trivial reasons. This paper, based on the course taught
by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014,
provides an exposition of the proof of this equivalence and some related
results. We assume that the reader is familiar with basic notions of
algorithmic information theory.Comment: 25 page
How powerful are integer-valued martingales?
In the theory of algorithmic randomness, one of the central notions is that
of computable randomness. An infinite binary sequence X is computably random if
no recursive martingale (strategy) can win an infinite amount of money by
betting on the values of the bits of X. In the classical model, the martingales
considered are real-valued, that is, the bets made by the martingale can be
arbitrary real numbers. In this paper, we investigate a more restricted model,
where only integer-valued martingales are considered, and we study the class of
random sequences induced by this model.Comment: Long version of the CiE 2010 paper
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
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