10 found
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  1.  22
    Π 1 0 Classes, L R Degrees and Turing Degrees.George Barmpalias, Andrew E. M. Lewis & Frank Stephan - 2008 - Annals of Pure and Applied Logic 156 (1):21-38.
    We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.
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  2.  3
    The ibT Degrees of Computably Enumerable Sets Are Not Dense.George Barmpalias & Andrew E. M. Lewis - 2006 - Annals of Pure and Applied Logic 141 (1-2):51-60.
    We show that the identity bounded Turing degrees of computably enumerable sets are not dense.
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  3.  14
    Randomness, Lowness and Degrees.George Barmpalias, Andrew E. M. Lewis & Mariya Soskova - 2008 - Journal of Symbolic Logic 73 (2):559 - 577.
    We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ. In other words. B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumberable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α in not GL₂ the LR degree of (...)
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  4.  4
    Diagonally Non-Computable Functions and Bi-Immunity.Carl G. Jockusch & Andrew E. M. Lewis - 2013 - Journal of Symbolic Logic 78 (3):977-988.
  5.  33
    The Importance of Π1 0 Classes in Effective Randomness.George Barmpalias, Andrew E. M. Lewis & Keng Meng Ng - 2010 - Journal of Symbolic Logic 75 (1):387-400.
    We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.
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  6.  34
    A C.E. Real That Cannot Be SW-Computed by Any Ω Number.George Barmpalias & Andrew E. M. Lewis - 2006 - Notre Dame Journal of Formal Logic 47 (2):197-209.
    The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.
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  7.  19
    Properly Σ2 Minimal Degrees and 0″ Complementation.S. Barry Cooper, Andrew E. M. Lewis & Yue Yang - 2005 - Mathematical Logic Quarterly 51 (3):274-276.
    We show that there exists a properly Σ2 minimal degree b, and moreover that b can be chosen to join with 0′ to 0″ – so that b is a 0″ complement for every degree a such that 0′ ≤ a < 0″.
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  8.  26
    The Minimal Complementation Property Above 0′.Andrew E. M. Lewis - 2005 - Mathematical Logic Quarterly 51 (5):470-492.
    Let us say that any degree d > 0satisfies the minimal complementation property if for every degree 0 < a < d there exists a minimal degree b < d such that a ∨ b = d . We show that every degree d ≥ 0′ satisfies MCP.
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  9.  28
    The Hypersimple-Free C.E. WTT Degrees Are Dense in the C.E. WTT Degrees.George Barmpalias & Andrew E. M. Lewis - 2006 - Notre Dame Journal of Formal Logic 47 (3):361-370.
    We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand, we know that there are intervals which contain no hypersimple set.
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  10. The Importance of $\Pi _1^0$ Classes in Effective Randomness. The Journal of Symbolic Logic, Vol. 75.George Barmpalias, Andrew E. M. Lewis, Keng Meng Ng & Frank Stephan - 2012 - Bulletin of Symbolic Logic 18 (3):409-412.