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Computability Theory
Rebecca Weber
STUDENT MATHEMAT ICAL L IBRARYVolume 62
American Mathematical SocietyProvidence, Rhode Island
Editorial Board
Gerald B. FollandRobin Forman
Brad G. Osgood (Chair)John Stillwell
2000 Mathematics Subject Classification. Primary 03Dxx;Secondary 68Qxx.
For additional information and updates on this book, visitwww.ams.org/bookpages/stml-62
Library of Congress Cataloging-in-Publication Data
Weber, Rebecca, 1977–Computability theory / Rebecca Weber.
p. cm. — (Student mathematical library ; v. 62)Includes bibliographical references and index.ISBN 978-0-8218-7392-2 (alk. paper)1. Recursion theory. 2. Computable functions. I. Title.
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10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12
Contents
Chapter 1. Introduction 1
§1.1. Approach 1
§1.2. Some History 4
§1.3. Notes on Use of the Text 6
§1.4. Acknowledgements and References 7
Chapter 2. Background 9
§2.1. First-Order Logic 9
§2.2. Sets 15
§2.3. Relations 21
§2.4. Bijection and Isomorphism 29
§2.5. Recursion and Induction 30
§2.6. Some Notes on Proofs and Abstraction 37
Chapter 3. Defining Computability 41
§3.1. Functions, Sets, and Sequences 41
§3.2. Turing Machines 43
§3.3. Partial Recursive Functions 50
§3.4. Coding and Countability 56
§3.5. A Universal Turing Machine 62
v
vi Contents
§3.6. The Church-Turing Thesis 65
§3.7. Other Definitions of Computability 66
Chapter 4. Working with Computable Functions 77
§4.1. The Halting Problem 77
§4.2. The “Three Contradictions” 78
§4.3. Parametrization 79
§4.4. The Recursion Theorem 81
§4.5. Unsolvability 85
Chapter 5. Computing and Enumerating Sets 95
§5.1. Dovetailing 95
§5.2. Computing and Enumerating 96
§5.3. Aside: Enumeration and Incompleteness 102
§5.4. Enumerating Noncomputable Sets 105
Chapter 6. Turing Reduction and Post’s Problem 109
§6.1. Reducibility of Sets 109
§6.2. Finite Injury Priority Arguments 115
§6.3. Notes on Approximation 124
Chapter 7. Two Hierarchies of Sets 127
§7.1. Turing Degrees and Relativization 127
§7.2. The Arithmetical Hierarchy 131
§7.3. Index Sets and Arithmetical Completeness 135
Chapter 8. Further Tools and Results 139
§8.1. The Limit Lemma 139
§8.2. The Arslanov Completeness Criterion 142
§8.3. E Modulo Finite Difference 145
Chapter 9. Areas of Research 149
§9.1. Computably Enumerable Sets and Degrees 149
§9.2. Randomness 155
§9.3. Some Model Theory 169
Contents vii
§9.4. Computable Model Theory 174
§9.5. Reverse Mathematics 177
Appendix A. Mathematical Asides 189
§A.1. The Greek Alphabet 189
§A.2. Summations 190
§A.3. Cantor’s Cardinality Proofs 190
Bibliography 193
Index 199
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Index
∅′, 129, see also halting set
1-randomness, 159
1-reduction, 133
Ackermann function, 53
alphabet, 87
and, logical, 9
antecedent, 10
antisymmetry, 22
arithmetic
Peano, 173, 175, 183
Robinson, 104, 173
second-order, 178
standard model, 104, 173
arithmetic transfinite recursion, 183
arithmetical
completeness, 133
comprehension axiom, 183
hierarchy, 131
arity, 103, 170
Arslanov completeness criterion,143
associativity, 33
automorphism, 30
of a lattice, 147
axiom of a production system, 88
axiomatizable, 163
base case, 31
Berry’s paradox, 163
bijection, 30
binary sequence, 43, 157
binary string, 43, 157
bit-flip, 46
Boolean algebra, 147
bounding, 186
BΠ0n, 186
BΣ0n, 186
c.e., see computably enumerable
capping, 150, 153
cardinality, 20, 25, 191
Cartesian product, 16
Chaitin’s Ω, 165
characteristic function, 43
Church-Turing thesis, 65
closure operator, 33
co-c.e., 102
code, 56
of a Turing machine, 59
codomain, 29
cofinite, 106
coinfinite, 106
complement, 17
completeness theorem, 103, 172
composition, 51
via indices, 80
comprehension, 179
arithmetical, 183
Π11, 184
199
200 Index
recursive, 179
computable
function, 65
set, 96
computable model theory, 174
computably (in)separable sets, 101,136, 175
computably categorical, 176
computably enumerable set, 97
relativized, 129
conjunction, 9
consequent, 10
contradiction, 10
countable, 20
countably infinite, 20
cupping, 150, 153
low, 151
De Morgan’s Laws
for predicate logic, 10
for sets, 19
decidable, 85
definability, 147, 150
degree, see Turing degree
degree invariance, 154
diagonalization, 63, 105, 190
diagonally non-recursive, 185
disjoint, 16
disjoint union, 101
disjunction, 10
domain
of a function, 29
of quantification, 13
domination, 64
dovetailing, 96
E, 146, 153E∗, 146effective, 97
effective countability, 56, 62
embedding, 150
enumeration of Turing machines, 60
equality
for partial functions, 42
for sets, 15, 19
equivalence class, 25
equivalence relation, 25
equivalence, logical, 11
finite difference, 114, 145first-order theory, 173
fixed-point theorem, 81function
Ackermann, 53
characteristic, 43computable, 65
equality, 42pairing, 57
partial, 42
partial recursive, 54primitive recursive, 50
Godel’s incompleteness, 105, 145,163, 178
graphgeneral mathematical, 33
of a function, 29, 100, 113of a relation, 24, 27
greatest lower bound, 28
halting probability, 165halting set, 78, 99, 114, 128, 129,
133
relativized, 129
high set, 154
implication, logical, 10
incompleteness theorem, 105, 145,163, 178
independent sentence, 104index, 60
index set, 82, 86, 135
induction, 179on N, 31
on recursively defined sets, 35inductive step, 31
injection, 29
injury, 116, 121internal state, 44
interpretation, 103, 170intersection, 16, 18
interval, 158
invariance, 147for degrees, 154
isomorphism, 29
join, 101, 127, 128, 130, 146jump, 129
Index 201
omega, 130
K, 78, see also halting set
K(σ), 159
K-trivial, 169KC Theorem, 161
Kolmogorov complexity
plain, 167
prefix-free, 159
Kraft inequality, 160, 166
lambda calculus, 69
language, 103, 170
lattice, 146
complemented, 146distributive, 146
embedding, 150
nondistributive, 152
least upper bound, 28
left-c.e. set, 125, 166
limit lemma, 139limit, discrete, 41, 140
linear order, 27, 176
low basis theorem, 183
low for random, 169
low set, 130, 140
noncomputable, 142lower bound, 28
maximal set, 154
measure, 158meet, 128, 146
model, 103, 170, 180
modular arithmetic, 26
modulus, 140
lemma, 140
μ, 55
negation, 10, 15
nondeterministic Turing machine,68
nonstandard model, 104, 174, 175nonuniformity, see uniformity
not, 10
Ω, 165or, logical, 10
oracle, 109
finite, 111
notation, 111oracle Turing machine, 109orbit, 147overspill, 175
padding lemma, 60pairing function, 57parameter, 180
parametrization, 79partial computable functions, 65partial function, 42
argument for allowing, 55, 62
partial order, 27partition, 26Peano arithmetic, 173, 175
in reverse math, 183permutation, 30ϕe, 60Π1
1 comprehension axiom, 184
pigeonhole principle, 186plain Kolmogorov complexity, 167poset, 27Post correspondence system, 91
Post’s problem, 115, 120Post’s theorem, 134power set, 16, 192predicate, 13
prefix-freeKolmogorov complexity, 159Turing machine, 158
primitive recursion, 51
priority, 116, 117, 121product, Cartesian, 16production, 87
normal, 88semi-Thue, 88
projection, 51
quantifiers, 13alternation of, 131collapsing like, 133
quotient structure, 25, see alsoTuring degree, E∗
randomness, 155, 159and Turing degree, 165
for finite strings, 159relative, 168
range, 29
202 Index
read/write head, 44
recursion theorem, 81
with parameters, 142
recursive, 96
recursive comprehension axiom,179
recursive definition
of a function, 35
of a set, 33
reduction property, 100
reflexivity, 22
relation, 21
relative randomness, 168
relativization, 128
requirement, 106, 117
injury, 116
priority, 116
satisfaction, 107
restriction, 112, 157
reverse mathematics, 177
Rice’s theorem, 82
Robinson arithmetic, 104, 173
nonstandard model, 174
s-m-n theorem, 79, 82, 84
relativized, 128
satisfaction, 107
second-order arithmetic, 178
second-order theory, 173, 178
semantics, 103, 172
semi-Thue system, 88, 92
sentence, 13
sequence, 43, 157
set, 15
arithmetical, 131
bi-immune, 124
computable, 96
computably enumerable, 97
creative, 145
d.c.e., 125
high, 154
left-c.e., 125, 166
low, 130, 140
maximal, 154
productive, 145
simple, 106, see also simple set
simple set, 106, 117, 146, 164
effectively, 145
incomplete, 120, 142
nowhere, 153
promptly, 151
solvable, 85
soundness theorem, 103, 172
splitting theorem, 153
stage bounding
notation, 96, 100, 111
use of, 111, 118, 121, 136
standard model, 104, 173
string, 43, 157
structure, 170
computably categorical, 176
subset, 16
successor, 51
surjection, 30
symmetric difference, 114, 136, 144,145
symmetry, 22
syntax, 103, 172
tape, 44
tautology, 10
theorem of a production system, 88
theory, 170
first-order, 173
second-order, 173, 178
total computable functions, 65
total order, 27
transitive closure, 27
transitivity, 22, 147
truth table, 12
tuple, 16
Turing completeness, 114
Turing degree, 127, 149
and randomness, 165
c.e., 128, 149
Turing equivalence, 112
Turing jump, 129
Turing machine, 44
busy beaver, 78
code of, 59
enumeration of, 60
index of, 60
nondeterministic, 68
nonstandard, 67
oracle, 109
prefix-free, 158
Index 203
universal, 62Turing reduction, 112
unbounded search, 54uncountable, 21undecidable problem, see
unsolvable problemuniformity, 3, 61, 80, 85, 86, 97,
101, 104, 108, 110, 112, 128,130, 177
main discussion of, 3, 61, 86, 97,108, 110
union, 16, 18universal Turing machine, 62universe, 16
of a model, 170unlimited register machine, 73unsolvable problem, 85
halting problem, 78, see alsohalting set
in behavior of Turing machines,77, 78, 86
index sets, 82, 87solutions to Post correspondence
systems, 92word problem for groups, 90word problem for semi-Thue
systems, 89word problem for Thue systems,
90upper bound, 27use, 111
of divergent computations, 111
We, 100weak jump, 114, 143
relativized, 130
weak Konig’s lemma, 181, 182weak weak Konig’s lemma, 185without loss of generality, 39witness, 118word, 87word problem
for groups, 90for semi-Thue systems, 89
Selected Titles in This Series
62 Rebecca Weber, Computability Theory, 2012
61 Anthony Bonato and Richard J. Nowakowski, The Game of Copsand Robbers on Graphs, 2011
60 Richard Evan Schwartz, Mostly Surfaces, 2011
59 Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, AlexSchwendner, Dmitry Vaintrob, and Elena Yudovina, Introduction toRepresentation Theory, 2011
58 Alvaro Lozano-Robledo, Elliptic Curves, Modular Forms, and TheirL-functions, 2011
57 Charles M. Grinstead, William P. Peterson, and J. Laurie Snell,Probability Tales, 2011
56 Julia Garibaldi, Alex Iosevich, and Steven Senger, The ErdosDistance Problem, 2011
55 Gregory F. Lawler, Random Walk and the Heat Equation, 2010
54 Alex Kasman, Glimpses of Soliton Theory, 2010
53 Jirı Matousek, Thirty-three Miniatures, 2010
52 Yakov Pesin and Vaughn Climenhaga, Lectures on Fractal Geometryand Dynamical Systems, 2009
51 Richard S. Palais and Robert A. Palais, Differential Equations,Mechanics, and Computation, 2009
50 Mike Mesterton-Gibbons, A Primer on the Calculus of Variations andOptimal Control Theory, 2009
49 Francis Bonahon, Low-Dimensional Geometry, 2009
48 John Franks, A (Terse) Introduction to Lebesgue Integration, 2009
47 L. D. Faddeev and O. A. Yakubovskii, Lectures on QuantumMechanics for Mathematics Students, 2009
46 Anatole Katok and Vaughn Climenhaga, Lectures on Surfaces, 2008
45 Harold M. Edwards, Higher Arithmetic, 2008
44 Yitzhak Katznelson and Yonatan R. Katznelson, A (Terse)Introduction to Linear Algebra, 2008
43 Ilka Agricola and Thomas Friedrich, Elementary Geometry, 2008
42 C. E. Silva, Invitation to Ergodic Theory, 2008
41 Gary L. Mullen and Carl Mummert, Finite Fields and Applications,2007
40 Deguang Han, Keri Kornelson, David Larson, and Eric Weber,Frames for Undergraduates, 2007
39 Alex Iosevich, A View from the Top, 2007
38 B. Fristedt, N. Jain, and N. Krylov, Filtering and Prediction: APrimer, 2007
37 Svetlana Katok, p-adic Analysis Compared with Real, 2007
36 Mara D. Neusel, Invariant Theory, 2007
SELECTED TITLES IN THIS SERIES
35 Jorg Bewersdorff, Galois Theory for Beginners, 2006
34 Bruce C. Berndt, Number Theory in the Spirit of Ramanujan, 2006
33 Rekha R. Thomas, Lectures in Geometric Combinatorics, 2006
32 Sheldon Katz, Enumerative Geometry and String Theory, 2006
31 John McCleary, A First Course in Topology, 2006
30 Serge Tabachnikov, Geometry and Billiards, 2005
29 Kristopher Tapp, Matrix Groups for Undergraduates, 2005
28 Emmanuel Lesigne, Heads or Tails, 2005
27 Reinhard Illner, C. Sean Bohun, Samantha McCollum, and Theavan Roode, Mathematical Modelling, 2005
26 Steven J. Cox, Robin Forman, Frank Jones, Barbara Lee Keyfitz,Frank Morgan, and Michael WolfSix Themes on Variation, 2004
25 S. V. Duzhin and B. D. Chebotarevsky, Transformation Groups forBeginners, 2004
24 Bruce M. Landman and Aaron Robertson, Ramsey Theory on theIntegers, 2004
23 S. K. Lando, Lectures on Generating Functions, 2003
22 Andreas Arvanitoyeorgos, An Introduction to Lie Groups and theGeometry of Homogeneous Spaces, 2003
21 W. J. Kaczor and M. T. Nowak, Problems in Mathematical AnalysisIII, 2003
20 Klaus Hulek, Elementary Algebraic Geometry, 2003
19 A. Shen and N. K. Vereshchagin, Computable Functions, 2003
18 V. V. Yaschenko, Editor, Cryptography: An Introduction, 2002
17 A. Shen and N. K. Vereshchagin, Basic Set Theory, 2002
16 Wolfgang Kuhnel, Differential Geometry, 2002
15 Gerd Fischer, Plane Algebraic Curves, 2001
14 V. A. Vassiliev, Introduction to Topology, 2001
13 Frederick J. Almgren, Jr., Plateau’s Problem, 2001
12 W. J. Kaczor and M. T. Nowak, Problems in Mathematical AnalysisII, 2001
11 Mike Mesterton-Gibbons, An Introduction to Game-TheoreticModelling, 2001
10 John Oprea, The Mathematics of Soap Films: Explorations with Maple�,2000
9 David E. Blair, Inversion Theory and Conformal Mapping, 2000
For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/.
AMS on the Webwww.ams.orgSTML/62
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What can we compute—even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoreti-cally? These questions are at the heart of computability theory.
The goal of this book is to give the reader a fi rm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recur-sive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, exam-ples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.