Random closed sets

Authors:
Paul Brodhead;Douglas Cenzer;Seyyed Dashti
Affiliations:
Department of Mathematics, University of Florida, Gainesville, Florida;Department of Mathematics, University of Florida, Gainesville, Florida;Department of Mathematics, University of Florida, Gainesville, Florida
Venue:
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Year:
2006

Citing 0
Cited 4

Random Continuous Functions

Electronic Notes in Theoretical Computer Science (ENTCS)
K-Trivial Closed Sets and Continuous Functions

CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Computability of probability measures and Martin-Löf randomness over metric spaces

Information and Computation
Immunity for Closed Sets

CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice

Quantified Score

Hi-index	0.00

Visualization

Abstract

We investigate notions of randomness in the space ${\mathcal {C}}[2^{\mathbb {N}}]$ of nonempty closed subsets of {0,1}ℕ. A probability measure is given and a version of the Martin-Löf Test for randomness is defined. Π02 random closed sets exist but there are no random Π01 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of $2^{{\mathbb N}}$ may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitin-style notion of randomness for closed sets.