Algorithmic Randomness of Closed Sets

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}∗ . .

• Type: Journal
• Year: 2007
• Where: LOGCOM
• Authors: George Barmpalias, Paul Brodhead, Douglas Cenzer, Seyyed Dashti, Rebecca Weber
• Keywords: LOGCOM 2007, Probability Measure, Random, Randomness