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Parameter Definability in the Recursively Enumerable Degrees
"... 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 defin ..."
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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...
Effectively dense Boolean algebras and their applications
"... A computably enumerable boolean algebra B is effectively dense if for each x 2 B we can effectively determine an F (x) x such that x 6= 0 implies 0 ! F (x) ! x. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a boolean algebra. A ..."
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A computably enumerable boolean algebra B is effectively dense if for each x 2 B we can effectively determine an F (x) x such that x 6= 0 implies 0 ! F (x) ! x. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of E (the lattice of computably enumerable sets under inclusion) which are not boolean algebras. We derive a similar result for theories of certain initial segments "low down" of subrecursive degree structures. 1 Introduction We describe a uniform method to interpret Th(N; +; \Theta) in the theories of a wide variety of seemingly wellbehaved structures. These structures stem from formal logic, complexity theory and computability theory. In many cases, they are closely related to dense distributive lattices. The results can be summarized by saying that, in spite of the structure's apparen...
Model Theory of the Computably Enumerable ManyOne Degrees
"... We investigate model theoretic properties of Rm , the partial order of computably enumerable manyone degrees. We prove that all nontrivial final segments and the set of minimal degrees are automorphism bases, and that some proper half open initial segment is an elementary substructure of Rm \Gamma f ..."
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We investigate model theoretic properties of Rm , the partial order of computably enumerable manyone degrees. We prove that all nontrivial final segments and the set of minimal degrees are automorphism bases, and that some proper half open initial segment is an elementary substructure of Rm \Gamma f1g. This shows that Rm is not a minimal model. In an appendix, we show that the manyone degree of an rmaximal set is join irreducible. Keywords: Model theory, manyone degrees This article is dedicated to Mari Santos. 1 Introduction Manyone reducibility, introduced by Post [9], is a rather fine way to measure the relative complexity of subsets of !: X is manyone reducible to Y , written X m Y , if X = f \Gamma1 (Y ) for a computable function f (we assume that f can also assume the values TRUE and FALSE to avoid trivialities). However, it appears naturally in a wide variety of contexts, for instance interpretability of theories and word problems of subgroups. Let Dm and Rm denote t...
Parameter Definable Subsets of the Computably Enumerable Degrees
"... 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 ..."
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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, BIconjecture) states that there is a parameter defined copy M of (N; +; \Theta) and a parameter definable 11 map f : M 7! A. This h...
CONTENTS
"... 1 Introduction to computability theory 1 1.1 The basic concepts 2 1.1.1 Partial computable functions 2 1.1.2 Computably enumerable sets 5 ..."
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1 Introduction to computability theory 1 1.1 The basic concepts 2 1.1.1 Partial computable functions 2 1.1.2 Computably enumerable sets 5
Definability in the c.e. degrees: Questions and results
"... . 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 h ..."
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. 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, BIconjecture) states that there is a parameter defined copy M of (N; +; \Theta) and a parameter definable 11 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 BIconjecture holds, we ca...
Interpreting N in the computably enumerable weak truth table degrees
"... We give a firstorder 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 im ..."
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We give a firstorder 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 manyone 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...