Random Closed Sets

  • Paul Brodhead
  • Douglas Cenzer
  • Seyyed Dashti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

We investigate notions of randomness in the space \({\mathcal {C}}[2^{\mathbb {N}}]\) of nonempty closed subsets of {0,1}. 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.

Keywords

Computability Randomness Π01 Classes 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Paul Brodhead
    • 1
  • Douglas Cenzer
    • 1
  • Seyyed Dashti
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesville

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