4,022 research outputs found
Algorithmic Randomness and Capacity of Closed Sets
We investigate the connection between measure, capacity and algorithmic
randomness for the space of closed sets. For any computable measure m, a
computable capacity T may be defined by letting T(Q) be the measure of the
family of closed sets K which have nonempty intersection with Q. We prove an
effective version of Choquet's capacity theorem by showing that every
computable capacity may be obtained from a computable measure in this way. We
establish conditions on the measure m that characterize when the capacity of an
m-random closed set equals zero. This includes new results in classical
probability theory as well as results for algorithmic randomness. For certain
computable measures, we construct effectively closed sets with positive
capacity and with Lebesgue measure zero. We show that for computable measures,
a real q is upper semi-computable if and only if there is an effectively closed
set with capacity q
The interplay of classes of algorithmically random objects
We study algorithmically random closed subsets of , algorithmically
random continuous functions from to , and algorithmically
random Borel probability measures on , especially the interplay
between these three classes of objects. Our main tools are preservation of
randomness and its converse, the no randomness ex nihilo principle, which say
together that given an almost-everywhere defined computable map between an
effectively compact probability space and an effective Polish space, a real is
Martin-L\"of random for the pushforward measure if and only if its preimage is
random with respect to the measure on the domain. These tools allow us to prove
new facts, some of which answer previously open questions, and reprove some
known results more simply.
Our main results are the following. First we answer an open question of
Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that
is a random closed set if and only if it is the
set of zeros of a random continuous function on . As a corollary we
obtain the result that the collection of random continuous functions on
is not closed under composition. Next, we construct a computable
measure on the space of measures on such that
is a random closed set if and only if
is the support of a -random measure. We also establish a
correspondence between random closed sets and the random measures studied by
Culver in previous work. Lastly, we study the ranges of random continuous
functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
Uniform test of algorithmic randomness over a general space
The algorithmic theory of randomness is well developed when the underlying
space is the set of finite or infinite sequences and the underlying probability
distribution is the uniform distribution or a computable distribution. These
restrictions seem artificial. Some progress has been made to extend the theory
to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary
distributions (by Levin). We recall the main ideas and problems of Levin's
theory, and report further progress in the same framework.
- We allow non-compact spaces (like the space of continuous functions,
underlying the Brownian motion).
- The uniform test (deficiency of randomness) d_P(x) (depending both on the
outcome x and the measure P should be defined in a general and natural way.
- We see which of the old results survive: existence of universal tests,
conservation of randomness, expression of tests in terms of description
complexity, existence of a universal measure, expression of mutual information
as "deficiency of independence.
- The negative of the new randomness test is shown to be a generalization of
complexity in continuous spaces; we show that the addition theorem survives.
The paper's main contribution is introducing an appropriate framework for
studying these questions and related ones (like statistics for a general family
of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of
Theorem 7 adde
Effective Capacity and Randomness of Closed Sets
We investigate the connection between measure and capacity for the space of
nonempty closed subsets of {0,1}*. For any computable measure, a computable
capacity T may be defined by letting T(Q) be the measure of the family of
closed sets which have nonempty intersection with Q. We prove an effective
version of Choquet's capacity theorem by showing that every computable capacity
may be obtained from a computable measure in this way. We establish conditions
that characterize when the capacity of a random closed set equals zero or is
>0. We construct for certain measures an effectively closed set with positive
capacity and with Lebesgue measure zero
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