Results 1  10
of
22
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 ..."
Abstract

Cited by 40 (15 self)
 Add to MetaCart
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...
Defining the Turing Jump
 MATHEMATICAL RESEARCH LETTERS
, 1999
"... 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&q ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
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
Natural Definability in Degree Structures
"... . A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
. A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the issues of getting denitions of interesting, apparently external, relations on degrees that are ordertheoretically natural in the structures D and R of all the Turing degrees and of the r.e. Turing degrees, respectively. Of course, we have no formal denition of natural but we oer some guidelines, examples and suggestions for further research. 1. Introduction A major focus of research in computability theory in recent years has involved denability issues in degree structures. The basic question is, which interesting apparently external relations on degrees can actually be dened in the structures themselves, that is, in the rst order language with the single fundamental relation...
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
"... ..."
Conjectures and Questions from Gerald Sacks’s Degrees of Unsolvability
 Archive for Mathematical Logic
, 1993
"... We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, parti ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals
Biinterpretability up to double jump in the degrees below 0´
, 2011
"... We prove that, for every z 0´ with z´´ > 0´´ (i.e. z 2 L2), the structure D ( z) of the Turing degrees below x is biinterpretable with first first order arithmetic up to double jump. As a corollary, every relation on D ( z) which is invariant under double jump is definable in D ( z) if and only i ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We prove that, for every z 0´ with z´´ > 0´´ (i.e. z 2 L2), the structure D ( z) of the Turing degrees below x is biinterpretable with first first order arithmetic up to double jump. As a corollary, every relation on D ( z) which is invariant under double jump is definable in D ( z) if and only if it is definable in arithmetic.
The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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...