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تسجيل مستخدم جديدOn the interplay between effective notions of randomness and genericity
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In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence (which answers an open question posed by Barmpalias, Day, and Lewis) and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager $mathcal{G}subseteq 2^omega$, there is some weakly 2-random sequence $X$ that computes some $Yinmathcal{G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.
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