Computability, Enumerability, Unsolvability: Directions in Recursion Theory

doi:10.1017/CBO9780511629167

The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will make this volume an invaluable resource.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-viii
- By S. Barry Cooper, Theodore A. Slaman, Stanley S. Wainer
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Resource bounded genericity
pp 1-60
- By Klaus Ambos-Spies, Mathematisches Institut, Universitat Heidelberg, D-69120 Heidelberg, Germany, ambos@math.uni-heidelberg.de
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On isolating r.e. and isolated d-r.e. degrees
pp 61-80
- By Marat M. Arslanov, Kazan University, Kazan, Russia, Steffen Lempp, University of Wisconsin, Madison WI 53706-1388 USA, Richard A. Shore, Cornell University, Ithaca NY 14853 USA
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A characterisation of the jumps of minimal degrees below 0′
pp 81-92
- By S. Barry Cooper, School of Mathematics, University of Leeds, Leeds LS2 9JT, England
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Array nonrecursive degrees and genericity
pp 93-104
- By Rod Downey, Carl G. Jockusch, Michael Stob
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Dynamic properties of computably enumerable sets
pp 105-122
- By Leo Harrington, Robert I. Soare
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Axioms for subrecursion theories
pp 123-138
- By Andrew J. Heaton, Stanley S. Wainer
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On the ∀ ∃ - theory of the factor lattice by the major subset relation
pp 139-166
- By Eberhard Herrmann
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Degrees of generic sets
pp 167-184
- By Masahiro Kumabe, University of the air, 2-11, Wakaba, Mihama-ku, Chiba 261, Japan
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Embeddings into the recursively enumerable degrees
pp 185-204
- By Manuel Lerman, Department of Mathematics, University of Connecticut Storrs, CT 06269-3009, e-mail: mlerman@math.uconn.edu
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On a question of Brown and Simpson
pp 205-218
- By Michael E. Mytilinaios, Theodore A. Slaman
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Relativization of structures arising from computability theory
pp 219-232
- By André Nies, Cornell University
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A Hierarchy of domains with totality, but without density
pp 233-258
- By Dag Normann, University of Oslo
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Inductive inference of total functions
pp 259-288
- By Piergiorgio Odifreddi, University of Torino, Italy
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The Medvedev lattice of degrees of difficulty
pp 289-312
- By Andrea Sorbi, Department of Mathematics, Via del Capitano 15, 53100 Siena, Italy
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Extension of embeddings on the recursively enumerable degrees modulo the cappable degrees
pp 313-332
- By Xiaoding Yi
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APPENDIX: Questions in Recursion Theory
pp 333-347
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Computability, Enumerability, Unsolvability

Directions in Recursion Theory

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Resource bounded genericity
pp 1-60

On isolating r.e. and isolated d-r.e. degrees
pp 61-80

A characterisation of the jumps of minimal degrees below 0′
pp 81-92

Array nonrecursive degrees and genericity
pp 93-104

Dynamic properties of computably enumerable sets
pp 105-122

Axioms for subrecursion theories
pp 123-138

On the ∀ ∃ - theory of the factor lattice by the major subset relation
pp 139-166

Degrees of generic sets
pp 167-184

Embeddings into the recursively enumerable degrees
pp 185-204

On a question of Brown and Simpson
pp 205-218

Relativization of structures arising from computability theory
pp 219-232

A Hierarchy of domains with totality, but without density
pp 233-258

Inductive inference of total functions
pp 259-288

The Medvedev lattice of degrees of difficulty
pp 289-312

Extension of embeddings on the recursively enumerable degrees modulo the cappable degrees
pp 313-332

APPENDIX: Questions in Recursion Theory
pp 333-347

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Computability, Enumerability, Unsolvability

Directions in Recursion Theory

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