STUDENT MATHEMAT ICAL L IBRARYVolume 62

ComputabilityTheory

Rebecca Weber

Computability Theory

http://dx.doi.org/10.1090/stml/062

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.

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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 set1-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

Gödel’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, 60Π11 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 König’s lemma, 181, 182weak weak König’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 Álvaro 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 ErdősDistance Problem, 2011

55 Gregory F. Lawler, Random Walk and the Heat Equation, 2010

54 Alex Kasman, Glimpses of Soliton Theory, 2010

53 Jǐŕı Matoušek, 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. Yakubovskĭi, 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 Jörg 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 Kühnel, 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.