Abstract
We investigate notions of randomness in the space \({\mathcal {C}}[2^{\mathbb {N}}]\) 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. Π0 2 random closed sets exist but there are no random Π0 1 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of \(2^{{\mathbb N}}\) may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitin-style notion of randomness for closed sets.
Research was partially supported by the National Science Foundation.
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Brodhead, P., Cenzer, D., Dashti, S. (2006). Random Closed Sets. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_6
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DOI: https://doi.org/10.1007/11780342_6
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