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
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
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We investigate notions of randomness in the space C[2N] of non-empty closed subsets of {0,1}N. A probability measure is given and a version of the Martin-Löf test for randomness is defined. Π20 random closed sets exist but there are no random Π10 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}*.