Baire reductions and good Borel reducibilities

In reference [8] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [8] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class $\xi$ functions).Comment: 21 page

A bounded jump for the bounded Turing degrees

We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.Comment: 22 pages. Minor changes for publicatio

Lipschitz and uniformly continuous reducibilities on ultrametric Polish spaces

We analyze the reducibilities induced by, respectively, uniformly continuous, Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces, and determine whether under suitable set-theoretical assumptions the induced degree-structures are well-behaved.Comment: 37 pages, 2 figures, revised version, accepted for publication in the Festschrift that will be published on the occasion of Victor Selivanov's 60th birthday by Ontos-Verlag. A mistake has been corrected in Section

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

Total Representations

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.Comment: 30 page

P-Selectivity, Immunity, and the Power of One Bit

We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is not EXP/1-immune. That is, we prove that some infinite P-selective set has no infinite EXP-time subset, but we also prove that every infinite P-selective set has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so fragile that it is pierced by a single bit of information. The above claims follow from broader results that we obtain about the immunity of the P-selective sets. In particular, we prove that for every recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is not \Pi_2^p/1-immune