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-Löf test for randomness is defined. Π2 random closed sets exist but there are no random Π1 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}, 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}∗.