13 research outputs found
Lines Missing Every Random Point
We prove that there is, in every direction in Euclidean space, a line that
misses every computably random point. We also prove that there exist, in every
direction in Euclidean space, arbitrarily long line segments missing every
double exponential time random point.Comment: Added a section: "Betting in Doubly Exponential Time.
Mutual Dimension
We define the lower and upper mutual dimensions and
between any two points and in Euclidean space. Intuitively these are
the lower and upper densities of the algorithmic information shared by and
. We show that these quantities satisfy the main desiderata for a
satisfactory measure of mutual algorithmic information. Our main theorem, the
data processing inequality for mutual dimension, says that, if is computable and Lipschitz, then the inequalities
and hold for all and . We use this inequality and related
inequalities that we prove in like fashion to establish conditions under which
various classes of computable functions on Euclidean space preserve or
otherwise transform mutual dimensions between points.Comment: This article is 29 pages and has been submitted to ACM Transactions
on Computation Theory. A preliminary version of part of this material was
reported at the 2013 Symposium on Theoretical Aspects of Computer Science in
Kiel, German
Curves That Must Be Retraced
We exhibit a polynomial time computable plane curve that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization of and every positive integer , there is some positive-length subcurve of that retraces at least times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem
Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension
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)
The power of backtracking and the confinement of length
We show that there is a point on a computable arc that does not belong to any computable rectifiable curve. We also show that there is a point on a computable rectifiable curve with computable length that does not belong to any computable arc
Point-Separable Classes of Simple Computable Planar Curves
In mathematics curves are typically defined as the images of continuous real
functions (parametrizations) defined on a closed interval. They can also be
defined as connected one-dimensional compact subsets of points. For simple
curves of finite lengths, parametrizations can be further required to be
injective or even length-normalized. All of these four approaches to curves are
classically equivalent. In this paper we investigate four different versions of
computable curves based on these four approaches. It turns out that they are
all different, and hence, we get four different classes of computable curves.
More interestingly, these four classes are even point-separable in the sense
that the sets of points covered by computable curves of different versions are
also different. However, if we consider only computable curves of computable
lengths, then all four versions of computable curves become equivalent. This
shows that the definition of computable curves is robust, at least for those of
computable lengths. In addition, we show that the class of computable curves of
computable lengths is point-separable from the other four classes of computable
curves