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|Title:||On initial segment complexity and degrees of randomness|
|Citation:||Miller, J.S., Yu, L. (2008-06). On initial segment complexity and degrees of randomness. Transactions of the American Mathematical Society 360 (6) : 3193-3210. ScholarBank@NUS Repository. https://doi.org/10.1090/S0002-9947-08-04395-X|
|Abstract:||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(Xn) ≤ K(Yn) + 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 closed upward in the K-degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the vL-degrees. Unlike the K-degrees, many basic properties of the vL-degrees are easy to prove. We show that X ≤K Y implies X ≤vL Y , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for ≤C, the analogue of ≤K for plain Kolmogorov complexity. Two other interesting results are included. First, we prove that for any Z ∈ 2ω, a 1-random real computable from a 1-Z-random real is automatically 1-Z-random. Second, we give a plain Kolmogorov complexity characterization of 1-randomness. This characterization is related to our proof that X ≤C Y implies X ≤vL Y . Copyright © 2008 American Mathematical Society.|
|Source Title:||Transactions of the American Mathematical Society|
|Appears in Collections:||Staff Publications|
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