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LOGCOM

2007

2007

We investigate notions of randomness in the space C[2N ] of nonempty closed subsets of {0, 1}N . A probability measure is given and a version of the Martin-L¨of test for randomness is deﬁned. Π0 2 random closed sets exist but there are no random Π0 1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log2 4 3 . A random closed set has no n-c.e. elements. A closed subset of 2N may be deﬁned as the set of inﬁnite paths through a tree and so the problem of compressibility of trees is explored. If Tn = T ∩ {0, 1}n , then for any random closed set [T] where T has no dead ends, K(Tn) ≥ n − O(1) but for any k, K(Tn) ≤ 2n−k +O(1), where K(σ) is the preﬁx-free complexity of σ ∈ {0, 1}∗ . .

Related Content

Added |
16 Dec 2010 |

Updated |
16 Dec 2010 |

Type |
Journal |

Year |
2007 |

Where |
LOGCOM |

Authors |
George Barmpalias, Paul Brodhead, Douglas Cenzer, Seyyed Dashti, Rebecca Weber |

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