The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...

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Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppersemilattice. The local theory concerns

We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, and ask whether A \Gamma S is still hard. It turns out that for most of the reductions considered and for an arbitrary given sparseness condition, there is a single subexponential time computable set S that meets this condition, such that A \Gamma S is not hard for any A. Not only is this set S subexponential time computable, but a slight modification of the construction can make the complexity of S meet any reasonable superpolynomial function. On the other hand we show that for any polynomial-time computable sparse set S, the set A \Gamma S remains hard. There are other properties than time complexity that make a set `almost' polynomial-time computable. For sparse p-selective sets...

Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a subset A of the natural numbers #, i.e. an external procedure that answers questions of the form "is n in A", they define the basic notion of relative computability or Turing reducibility (from Turing (1939)). We say that A is computable from (or recursive in) B if there is a Turing machine which, when equipped with an oracle for B, computes (the characteristic function of) A, i.e. for some e, # B e = A. We denote this relation by A # T<F10

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The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1

We prove that, for any D, A and U with D > T A # U and r.e. in A# U , there are pairs X 0 , X 1 and Y 0 , Y 1 such that D # T X 0 #X 1 ; D # T Y 0 # Y 1 ; and, for any i and j from {0, 1} and any set B, if X i #A # T B and Y j # A # T B then A # T B. We then deduce that for any degrees d, a, and b such that a and b are recursive in d, a ## T b, and d is n-REA in to a, d can be split over a avoiding b. This shows that the Main Theorem of Cooper [1990] and [1993] is false.

We extend the Shoenfield jump inversion theorem to the members of any Π0 1 class P⊆2ω with nonzero measure; i.e., for every Σ0 2 set S ≥T ∅ ′, there is a ∆0 2 real A ∈Psuch that A ′ ≡T S. In particular, we get jump inversion for ∆0 2 1-random reals. This paper is part of an ongoing program to study the relationship between two fundamental notions of complexity for real numbers. The first is the computational complexity of a real as captured, for example, by its Turing degree. The second is the intrinsic randomness of a real. In particular, we are interested in the 1random reals, which were introduced by Martin-Löf [13] and represent the most widely studied randomness class. For the purposes of this introduction, we will assume that the reader is somewhat familiar with basic algorithmic randomness, as per Li-Vitányi [12], and with computability theory [18]. A review of notation and terminology will be given in Section 1. Intuitively, a 1-random real is very complex. This complexity can be captured formally in terms of unpredictability or incompressibility, but is it reflected in the