David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Bulletin of Symbolic Logic 11 (1):1-27 (2005)
A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices P w and P s of weak and strong degrees of mass problems given by nonempty Π 1 0 subsets of 2 ω . Using an abstract Gödel/Rosser incompleteness property, we characterize the Π 1 0 subsets of 2 ω whose associated mass problems are of top degree in P w and P s , respectively. Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within P w . Namely, we characterize r as the unique largest weak degree of a Π 1 0 subset of 2 ω of positive measure. Within P w we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect Π 1 0 subsets of 2 ω . In addition, we present other natural examples of intermediate degrees in P w . We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory
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K. Higuchi & T. Kihara (2014). Inside the Muchnik Degrees I: Discontinuity, Learnability and Constructivism. Annals of Pure and Applied Logic 165 (5):1058-1114.
K. Higuchi & T. Kihara (2014). Inside the Muchnik Degrees II: The Degree Structures Induced by the Arithmetical Hierarchy of Countably Continuous Functions. Annals of Pure and Applied Logic 165 (6):1201-1241.
Kojiro Higuchi (2012). Effectively Closed Mass Problems and Intuitionism. Annals of Pure and Applied Logic 163 (6):693-697.
Kojiro Higuchi, W. M. Phillip Hudelson, Stephen G. Simpson & Keita Yokoyama (2014). Propagation of Partial Randomness. Annals of Pure and Applied Logic 165 (2):742-758.
Andrea Sorbi & Sebastiaan A. Terwijn (2008). Intermediate Logics and Factors of the Medvedev Lattice. Annals of Pure and Applied Logic 155 (2):69-85.
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