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Multiple Permitting and Bounded Turing Reducibilities
We look at various properties of the computably enumerable (c.e.) not totally ω-c.e. Turing degrees. In particular, we are interested in the variant of multiple permitting given by those degrees. We define a property of left-c.e. sets called universal similarity property which can be viewed as a universal or uniform version of the property of array noncomputable c.e. sets of agreeing with any c.e. set on some component of a very strong array. Using a multiple permitting argument, we prove that the Turing degrees of the left-c.e. sets with the universal similarity property coincide with the c.e. not totally ω-c.e. degrees. We further introduce and look at various notions of socalled universal array noncomputability and show that c.e. sets with those properties can be found exactly in the c.e. not totally ω-c.e. Turing degrees and that they guarantee a special type of multiple permitting called uniform multiple permitting. We apply these properties of the c.e. not totally ω-c.e. degrees to give alternative proofs of well-known results on those degrees as well as to prove new results. E.g., we show that a c.e. Turing degree contains a left-c.e. set which is not cl-reducible to any complex left-c.e. set if and only if it is not totally ω-c.e. Furthermore, we prove that the nondistributive finite lattice S7 can be embedded into the c.e. Turing degrees precisely below any c.e. not totally ω-c.e. degree. We further look at the question of join preservation for bounded Turing reducibilities r and r′ such that r is stronger than r′. We say that join preservation holds for two reducibilities r and r′ if every join in the c.e. r-degrees is also a join in the c.e. r′-degrees. We consider the class of monotone admissible (uniformly) bounded Turing reducibilities, i.e., the reflexive and transitive Turing reducibilities with use bounded by a function that is contained in a (uniformly computable) family of strictly increasing computable functions. This class contains for example identity bounded Turing (ibT-) and computable Lipschitz (cl-) reducibility. Our main result of Chapter 3 is that join preservation fails for cl and any strictly weaker monotone admissible uniformly bounded Turing reducibility. We also look at the dual question of meet preservation and show that for all monotone admissible bounded Turing reducibilities r and r′ such that r is stronger than r′, meet preservation holds. Finally, we completely solve the question of join and meet preservation in the classical reducibilities 1, m, tt, wtt and T.
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