Book contents
- Frontmatter
- Contents
- Preface
- Resource bounded genericity
- On isolating r.e. and isolated d-r.e. degrees
- A characterisation of the jumps of minimal degrees below 0′
- Array nonrecursive degrees and genericity
- Dynamic properties of computably enumerable sets
- Axioms for subrecursion theories
- On the ∀ ∃ - theory of the factor lattice by the major subset relation
- Degrees of generic sets
- Embeddings into the recursively enumerable degrees
- On a question of Brown and Simpson
- Relativization of structures arising from computability theory
- A Hierarchy of domains with totality, but without density
- Inductive inference of total functions
- The Medvedev lattice of degrees of difficulty
- Extension of embeddings on the recursively enumerable degrees modulo the cappable degrees
- APPENDIX: Questions in Recursion Theory
Dynamic properties of computably enumerable sets
Published online by Cambridge University Press: 23 February 2010
- Frontmatter
- Contents
- Preface
- Resource bounded genericity
- On isolating r.e. and isolated d-r.e. degrees
- A characterisation of the jumps of minimal degrees below 0′
- Array nonrecursive degrees and genericity
- Dynamic properties of computably enumerable sets
- Axioms for subrecursion theories
- On the ∀ ∃ - theory of the factor lattice by the major subset relation
- Degrees of generic sets
- Embeddings into the recursively enumerable degrees
- On a question of Brown and Simpson
- Relativization of structures arising from computability theory
- A Hierarchy of domains with totality, but without density
- Inductive inference of total functions
- The Medvedev lattice of degrees of difficulty
- Extension of embeddings on the recursively enumerable degrees modulo the cappable degrees
- APPENDIX: Questions in Recursion Theory
Summary
Abstract
A set A ⊆ ω is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let ε denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating the definable (especially ε-definable) properties of a c.e. set A to its “information content”, namely its Turing degree, deg(A), under ≤T, the usual Turing reducibility. [Turing 1939]. Recently, Harrington and Soare answered a question arising from Post's program by constructing a nonemptly ε-definable property Q(A) which guarantees that A is incomplete (A <TK). The property Q(A) is of the form (∃C)[A ⊂mC & Q−(A, C)], where A ⊂mC abbreviates that “A is a major subset of C”, and Q−(A,C) contains the main ingredient for incompleteness.
A dynamic property P(A), such as prompt simplicity, is one which is defined by considering how fast elements elements enter A relative to some simultaneous enumeration of all c.e. sets. If some set in deg(A) is promptly simple then A is prompt and otherwise tardy. We introduce here two new tardiness notions, small-tardy (A, C) and Q-tardy(A, C). We begin by proving that small-tardy(A, C) holds iff A is small in C (A ⊂sC) as defined by Lachlan [1968]. Our main result is that Q-tardy(A, C) holds iff Q−(A,C). Therefore, the dynamic property, Q-tardy(A, C), which is more intuitive and easier to work with than the ε-definable counterpart, Q−(A,C), is exactly equivalent and captures the same incompleteness phenomenon.
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- Computability, Enumerability, UnsolvabilityDirections in Recursion Theory, pp. 105 - 122Publisher: Cambridge University PressPrint publication year: 1996
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