Affiliation:
1. Iowa State University, Osborn Drive, Ames, IA
2. University of Pennsylvania, Philadelphia, PA
Abstract
We formulate the
conditional Kolmogorov complexity
of
x
given
y
at
precision
r
, where
x
and
y
are points in Euclidean spaces and
r
is a natural number. We demonstrate the utility of this notion in two ways;
(1) We prove a
point-to-set principle
that enables one to use the (relativized, constructive) dimension of a
single point
in a set
E
in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of
E
. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2.
(2)We use conditional Kolmogorov complexity in Euclidean spaces to develop the
lower
and
upper conditional dimensions
dim(
x
|
y
) and Dim(
x
|
y
) of
x
given
y
, where
x
and
y
are points in Euclidean spaces. Intuitively, these are the lower and upper asymptotic algorithmic information densities of
x
conditioned on the information in
y
. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(
x
) and Dim(
x
) and the mutual dimensions mdim(
x
:
y
) and Mdim(
x
:
y
).
Funder
National Science Foundation
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Cited by
15 articles.
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