We investigate notions of randomness in the space C½2 N of non-empty closed subsets of f0, 1g N . A probability measure is given and a version of the Martin-Lo¨f test for randomness is defined. Å 0 2 random closed sets exist but there are no random Å 0 1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log 2 ð4=3Þ. A random closed set has no n-c.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If T n ¼ T \ f0, 1g n , then for any random closed set ½T where T has no dead ends, KðT n Þ ! n À Oð1Þ but for any k, KðT n Þ 2 nÀk þ Oð1Þ, where K() is the prefix-free complexity of 2 f0, 1g à .

No abstract

No abstract

No abstract

Electronic Notes in Theoretical Computer Science 167 (2007) 275-287. doi:10.1016/j.entcs.2006.08.016Received by publisher: 0000-01-01Harvest Date: 2016-01-04 12:23:44DOI: 10.1016/j.entcs.2006.08.016Page Range: 275-28