Randomness for computable measures and initial segment complexity

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• 作者：Rupert Hö ; lzl^a ; ^r@hoelzl.fr ; Christopher P. Porter^b ; ^{cp@cpporter.com}
• 关键词：03D32
• 刊名：Annals of Pure and Applied Logic
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：168
• 期：4
• 页码：860-886
• 全文大小：634 K
• 卷排序：168

文摘

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2ω2ω, the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure μ, some sequence in this family fails to be random with respect to μ  . (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2ω2ω, namely diminutive measures and trivial measures.