Randomness for computable measures and initial segment complexity
نویسندگان
چکیده
منابع مشابه
Randomness for Non-computable Measures
Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gács, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent. Having clarified what it means to be random for a non-computable probabilit...
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One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤K Y to mean that (∀n) K(X n) ≤ K(Y n) + O(1). The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ Y is 1-random, then X and Y have no upper bound in the K-degrees (hence, no join). We also prove that n-randomness is clos...
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Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with Martin-Löf randomness relative to the halting problem K; that is, a set A is Martin-Löf random relative to K iff there is no function f such that for all m an...
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ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 2017
ISSN: 0168-0072
DOI: 10.1016/j.apal.2016.10.014