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Effective strong nullness and effectively closed sets

Kojiro Higuchi & Takayuki Kihara

In S. Barry Cooper (ed.), How the World Computes. pp. 303--312 (2012)

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Propagation of partial randomness.Kojiro Higuchi, W. M. Phillip Hudelson, Stephen G. Simpson & Keita Yokoyama - 2014 - Annals of Pure and Applied Logic 165 (2):742-758.details Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a (...) Download Export citation Bookmark 5 citations
On effectively closed sets of effective strong measure zero.Kojiro Higuchi & Takayuki Kihara - 2014 - Annals of Pure and Applied Logic 165 (9):1445-1469.details The strong measure zero sets of reals have been widely studied in the context of set theory of the real line. The notion of strong measure zero is straightforwardly effectivized. A set of reals is said to be of effective strong measure zero if for any computable sequence {εn}n∈N{εn}n∈N of positive rationals, a sequence of intervals InIn of diameter εnεn covers the set. We observe that a set is of effective strong measure zero if and only if it is of (...) Download Export citation Bookmark 2 citations

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