Lowness properties and randomness
نویسنده
چکیده
The set A is low for (Martin-Löf) randomness if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely ∀n K(A!n)"K(n) + O(1). We show that these classes coincide. This answers a question of Ambos-Spies and Kučera in: P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000: each low for Martin-Löf random set is !2. Our class induces a natural intermediate "3 ideal in the r.e. Turing degrees, which generates the whole class under downward closure. Answering a further question in P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000, we prove that each low for computably random set is computable. © 2004 Elsevier Inc. All rights reserved.
منابع مشابه
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تاریخ انتشار 2003