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...

We prove definability results for the structure R T of computably enumerable Turing degrees. Some of the results can be viewed as approximations to an affirmative answer for the biinterpretability conjecture in parameters for. For instance, all uniformly computably enumerable sets of nonzero c.e. Turing degrees can be defined from parameters by a fixed formula. This implies that the finite subsets are uniformly definable. As a consequence we obtain a new ;-definable ideal, and all arithmetical ideals are parameter definable. 1 Introduction Let R T denote the upper semilattice of computably enumerable (c.e.) Turing degrees. We are concerned with definability results in R T which can be viewed as approximations to the biinterpretability conjecture for R T . The biinterpretability conjecture in parameters for an arithmetical structure A (in brief, BI-conjecture) states that there is a parameter defined copy M of (N; +; \Theta) and a parameter definable 1-1 map f : M 7! A. This h...

We investigate degree structures induced by many-one reducibility and Turing reducibility on the computably enumerable (c.e.), the arithmetical all all subsets of N. We study which subsets of the degree structure automorphism bases: for instance the minimal degrees form an automorphism base for the c.e. many one degrees, but not for the other degree structures based on m . We develop a method to show a subset is an automorphism base and apply it to the c.e. m-degrees and to give a modified proof of a result of Ambos-Spies that initial intervals of the c.e. degrees are automorphism bases. Also, we show that the arithmetical m-degrees form a prime model. A central topic of computability theory is the study of sets of natural numbers under a notion of relative computability. Specifying a reducibility and an appropriate class of sets gives rise to a degree structure which may have interesting "global" properties. This is the case for the degree structures induced by many-one redu...

. We ask questions and state results about definability in the partial order R T of computably enumerable Turing degrees. The main open question is whether R T is biinterpretable with N in parameters. Some of the results can be viewed as approximations to an affirmative answer. For instance, we have proved that all uniformly computably enumerable sets of nonzero c.e. Turing degrees can be defined from parameters by a fixed formula. As a consequence we obtain a new ;-definable ideal. 1. Introduction The biinterpretability conjecture in parameters for an arithmetical structure A (in brief, BI-conjecture) states that there is a parameter defined copy M of (N; +; \Theta) and a parameter definable 1-1 map f : A !M . This has far reaching consequences for A, for instance that all automorphisms are arithmetical (and therefore there are only countable many), and that each orbit in A n is ;-definable (in other words, A is a prime model of its theory). If the BI-conjecture holds, we ca...

We give a first-order coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an important further problem is the question of the actual complexity of the theory. If the structure is arithmetical, then its theory can be interpreted in true arithmetic, i.e. Th(N; +; \Theta). Thus an upper bound is ; (!) , the complexity of Th(N; +; \Theta). Here an interpretation of theories is a many--one reduction based on a computable map defined on sentences in some natural way. An example of an arithmetical structure is D T ( ; 0 ), the Turing-- degrees of \Delta 0 2 --sets. Shore [16] proved that true arithmetic can be interpreted in Th(D T ( ; 0 )). A stronger result is interpretability without parameters of a copy of (N; +; \Theta) in the structure (interpretability of struc...