Computability Theory

420 Pages·2004·2.65 MB·English

by  S. Barry Cooper

Computability
Theory
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COMPUTABILITY
THEORY
S. BARRY COOPER
CRC Press
Taylor & Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Croup, an informa business
A PRODUCTIVITY PRESS BOOK
C2379 disclaimer Page 1 Monday, October 13, 2003 11:49 AM
Library of Congress Cataloging-in-Publication Data
Cooper, S. B. (S. Barry)
Computability theory / by S. Barry Cooper.
p. cm. — (Chapman & Hall/CRC mathematics)
Includes bibliographical references and index.
ISBN 1-58488-237-9 (alk. paper)
1. Computable functions. I. Title. II. Series.
QA9.59.C68 2003
511.3—dc22 2003055823
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Preface
This book may be your first contact with computability theory, although cer-
tainly not with computability. On the other hand, you may already have a
good working knowledge of the subject. Either way, there should be plenty
here to interest and inform everyone, from the beginner to the expert. The
treatment is unashamedly contemporary, and many topics were previously
available only through articles in academic journals or weighty specialist ref-
erence works.
As you will probably know already, the history of the computer is unusual,
in that the theory — the Universal Turing Machine — preceded its physical
embodiment by around ten years. I have tried to write Computability Theory
very much in the spirit of Alan Turing, with a keen curiosity about patterns
andunderlyingtheory,buttheoryfirmlytiedtoareal-worldcontext. Newde-
velopments are strongly featured, without sacrificing readability — although
you the reader will be the best judge of that! Anyway, I do believe that there
is no other book like this currently. There are admirable specialist volumes
you will be able to take in hand later.
Butthisisprimarilyabooktoberead,andnotjusttobereferredto. Ihave
triedtogivesomehistoricalandintuitivebackgroundforallthemoretechnical
partsofthebook. Theexercises,bothroutineandmorechallenging,havebeen
carefully chosen to complement the main text, and have been positioned so
thatyoucanseesomepoint inyourdoingthem. Ifyougethalfasmuchmental
satisfaction and stimulation to further explorations as I got in selecting the
topics,youwillbemorethanadequatelyrewardedforyourhardwork. Andof
course, what cannot be hidden from you — computability theory sometimes
has a reputation for being a difficult subject understood by a few people. I
hope you will soon be joining those for whom it is also exciting and relevant,
like no other subject, to the world we live in.
I should say that what you have here is a very personal view of an in-
creasingly large subject. Although I have tried to choose a wide range of
topics relevant to current research, there are inevitably many gaps — I will
not advertise them, but the experts will notice! Also, this is not an academic
article,sonotallresultshavetheirdiscoverers’namesattached. Ihavechosen
to credit people with the reader’s interests in mind, not those of the expert
looking for his or her name in the index. Hopefully I have mentioned just
enough names to make the reader fairly well informed about the history of
the subject and to give an impression of a living breathing subject full of real
people.
v
vi Preface
If you get through to the Further Reading selection at the end of the book,
you will find lots of ways of filling out the picture given here. Please send me
any comments or corrections. You never know, I may one day be preparing
a second edition. Many thanks to all at CRC, especially Jasmin Naim and
Mimi Williams, for their ever-present help with what you now have in your
hands.
Finally, many thanks to everyone who has helped me and encouraged me
with the writing of this book, without whom — as is often said, and never
more truly — this book would not have been written. Too many to mention,
but you know who you are. Yes, and that includes other writers of books,
friends and colleagues, and — more than anyone! — my family who have
bravely put up with me over the last three years.
Barry Cooper
Leeds
Contents
Part I Computability and Unsolvable Problems 1
1 Hilbert and the Origins of Computability Theory 3
1.1 Algorithms and Algorithmic Content . . . . . . . . . . . . . 3
1.2 Hilbert’s Programme . . . . . . . . . . . . . . . . . . . . . . 5
1.3 G¨odel and the Discovery of Incomputability . . . . . . . . . 7
1.4 Computability and Unsolvability in the Real World . . . . . 8
2 Models of Computability and the Church–Turing Thesis 11
2.1 The Recursive Functions . . . . . . . . . . . . . . . . . . . . 12
2.2 Church’s Thesis and the Computability of Sets and Relations 19
2.3 Unlimited Register Machines . . . . . . . . . . . . . . . . . . 25
2.4 Turing’s Machines . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Church, Turing, and the Equivalence of Models . . . . . . . . 42
3 Language, Proof and Computable Functions 45
3.1 Peano Arithmetic and Its Models . . . . . . . . . . . . . . . 45
3.2 What Functions Can We Describe in a Theory? . . . . . . . 56
4 Coding, Self-Reference and the Universal Turing Machine 61
4.1 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 G¨odel Numberings . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 A Universal Turing Machine . . . . . . . . . . . . . . . . . . 65
4.4 The Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . 66
4.5 Computable Approximations . . . . . . . . . . . . . . . . . . 67
5 Enumerability and Computability 69
5.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 The Normal Form Theorem . . . . . . . . . . . . . . . . . . . 73
5.3 Incomputable Sets and the Unsolvability of the Halting Prob-
lem for Turing Machines . . . . . . . . . . . . . . . . . . . . 78
5.4 The Busy Beaver Function . . . . . . . . . . . . . . . . . . . 81
6 The Search for Natural Examples of Incomputable Sets 87
6.1 The Ubiquitous Creative Sets . . . . . . . . . . . . . . . . . . 88
6.2 Some Less Natural Examples of Incomputable Sets . . . . . 90
vii
viii Contents
6.3 Hilbert’s Tenth Problem and the Search for Really Natural
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7 ComparingComputabilityandtheUbiquityofCreativeSets 101
7.1 Many–One Reducibility . . . . . . . . . . . . . . . . . . . . . 101
7.2 The Non-Computable Universe and Many–One Degrees . . . 106
7.3 Creative Sets Revisited . . . . . . . . . . . . . . . . . . . . . 112
8 G¨odel’s Incompleteness Theorem 117
8.1 Semi-Representability and C.E. Sets . . . . . . . . . . . . . . 117
8.2 Incomputability and G¨odel’s Theorem . . . . . . . . . . . . . 122
9 Decidable and Undecidable Theories 127
9.1 PA is Undecidable . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Other Undecidable Theories and Their Many–One Equivalence 128
9.3 Some Decidable Theories . . . . . . . . . . . . . . . . . . . . 132
Part II Incomputability and Information Content 137
10 Computing with Oracles 139
10.1 Oracle Turing Machines . . . . . . . . . . . . . . . . . . . . . 139
10.2 Relativising, and Listing the Partial Computable Functionals 142
10.3 Introducing the Turing Universe . . . . . . . . . . . . . . . . 144
10.4 Enumerating with Oracles, and the Jump Operator . . . . . 147
10.5 The Arithmetical Hierarchy and Post’s Theorem . . . . . . . 154
10.6 The Structure of the Turing Universe . . . . . . . . . . . . . 161
11 Nondeterminism, Enumerations and Polynomial Bounds 173
11.1 Oracles versus Enumerations of Data . . . . . . . . . . . . . 173
11.2 Enumeration Reducibility and the Scott Model for Lambda
Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11.3 The Enumeration Degrees and the Natural Embedding of the
Turing Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.4 The Structure ofDDD and the Arithmetical Hierarchy . . . . . 199
e
11.5 The Medvedev Lattice . . . . . . . . . . . . . . . . . . . . . . 202
11.6 Polynomial Bounds and P=?NP . . . . . . . . . . . . . . . 205
Part III More Advanced Topics 217
12 Post’s Problem: Immunity and Priority 219
12.1 Information Content and Structure . . . . . . . . . . . . . . 219
12.2 Immunity Properties . . . . . . . . . . . . . . . . . . . . . . 226
12.3 Approximation and Priority . . . . . . . . . . . . . . . . . . 237
Contents ix
12.4 Sacks’ Splitting Theorem and Cone Avoidance . . . . . . . . 246
12.5 Minimal Pairs and Extensions of Embeddings . . . . . . . . 252
12.6 The Π Theory — Information Content Regained . . . . . . 260
3
12.7 Higher Priority and Maximal Sets . . . . . . . . . . . . . . . 266
13 Forcing and Category 273
13.1 Forcing in Computability Theory . . . . . . . . . . . . . . . 273
13.2 Baire Space, Category and Measure . . . . . . . . . . . . . . 278
13.3 n-Genericity and Applications . . . . . . . . . . . . . . . . . 287
13.4 Forcing with Trees and Minimal Degrees . . . . . . . . . . . 299
14 Applications of Determinacy 311
14.1 Gale–Stewart Games . . . . . . . . . . . . . . . . . . . . . . 311
14.2 An Upper Cone of Minimal Covers . . . . . . . . . . . . . . . 315
14.3 Borel and Projective Determinacy, and the Global Theory ofDDD 318
15 The Computability of Theories 321
15.1 Feferman’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 321
15.2 Truth versus Provability . . . . . . . . . . . . . . . . . . . . 323
15.3 Complete Extensions of Peano Arithmetic and Π0-Classes . . 324
1
15.4 The Low Basis Theorem . . . . . . . . . . . . . . . . . . . . 329
15.5 Arslanov’s Completeness Criterion . . . . . . . . . . . . . . . 331
15.6 A Priority-Free Solution to Post’s Problem . . . . . . . . . . 333
15.7 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
16 Computability and Structure 343
16.1 Computable Models . . . . . . . . . . . . . . . . . . . . . . . 343
16.2 Computability and Mathematical Structures . . . . . . . . . 348
16.3 Effective Ramsey Theory . . . . . . . . . . . . . . . . . . . . 362
16.4 Computability in Analysis . . . . . . . . . . . . . . . . . . . 371
16.5 Computability and Incomputability in Science . . . . . . . . 379
Further Reading 383
Index 389