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STUDENT MATHEMATICAL Volume 62

Computability

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

Computability Theory

STUDENT MATHEMATICAL LIBRARY Volume 62

Computability Theory

Rebecca Weber

American Mathematical Society Providence, Rhode Island Editorial Board Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell

2000 Subject Classification. Primary 03Dxx; Secondary 68Qxx.

For additional information and updates on this book, visit www.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 and index. ISBN 978-0-8218-7392-2 (alk. paper) 1. theory. 2. Computable functions. I. Title.

QA9.6.W43 2012 511.352—dc23 2011050912

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 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 9 §2.2. Sets 15 §2.3. Relations 21 §2.4. and 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 41 §3.2. Turing Machines 43 §3.3. Partial Recursive Functions 50 §3.4. Coding and Countability 56 §3.5. A Universal 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 77 §4.2. The “Three ” 78 §4.3. Parametrization 79 §4.4. The Recursion 81 §4.5. Unsolvability 85

Chapter 5. and Enumerating Sets 95 §5.1. Dovetailing 95 §5.2. Computing and Enumerating 96 §5.3. Aside: and Incompleteness 102 §5.4. Enumerating Noncomputable Sets 105

Chapter 6. Turing and Post’s Problem 109 §6.1. Reducibility of Sets 109 §6.2. Finite Injury Priority 115 §6.3. Notes on Approximation 124

Chapter 7. Two Hierarchies of Sets 127 §7.1. Turing Degrees and Relativization 127 §7.2. The 131 §7.3. Index Sets and Arithmetical 135

Chapter 8. Further Tools and Results 139 §8.1. The Lemma 139 §8.2. The Arslanov Completeness Criterion 142 §8.3. E Modulo Finite Difference 145

Chapter 9. Areas of Research 149 §9.1. Sets and Degrees 149 §9.2. Randomness 155 §9.3. Some 169 Contents vii

§9.4. Computable Model Theory 174 §9.5. 177

Appendix A. Mathematical Asides 189 §A.1. The Greek Alphabet 189 §A.2. Summations 190 §A.3. Cantor’s Proofs 190

Bibliography 193

Index 199

Bibliography

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Index

∅, 129, see also halting set bijection, 30 1-randomness, 159 binary sequence, 43, 157 1-reduction, 133 binary string, 43, 157 bit-flip, 46 , 53 , 147 alphabet, 87 bounding, 186 0 and, logical, 9 BΠn, 186 0 antecedent, 10 BΣn, 186 antisymmetry, 22 arithmetic c.e., see computably enumerable Peano, 173, 175, 183 capping, 150, 153 Robinson, 104, 173 cardinality, 20, 25, 191 second-order, 178 , 16 standard model, 104, 173 Chaitin’s Ω, 165 arithmetic transfinite recursion, 183 characteristic function, 43 arithmetical Church-Turing thesis, 65 completeness, 133 , 33 comprehension , 183 co-c.e., 102 hierarchy, 131 code, 56 arity, 103, 170 of a Turing machine, 59 Arslanov completeness criterion, , 29 143 cofinite, 106 associativity, 33 coinfinite, 106 automorphism, 30 , 17 of a lattice, 147 completeness theorem, 103, 172 axiom of a production system, 88 composition, 51 axiomatizable, 163 via indices, 80 comprehension, 179 base case, 31 arithmetical, 183 1 Berry’s paradox, 163 Π1, 184

199 200 Index

recursive, 179 finite difference, 114, 145 computable first-, 173 function, 65 fixed-point theorem, 81 set, 96 function computable model theory, 174 Ackermann, 53 computably (in)separable sets, 101, characteristic, 43 136, 175 computable, 65 computably categorical, 176 , 42 computably enumerable set, 97 pairing, 57 relativized, 129 partial, 42 conjunction, 9 partial recursive, 54 consequent, 10 primitive recursive, 50 , 10 countable, 20 G¨odel’s incompleteness, 105, 145, countably infinite, 20 163, 178 cupping, 150, 153 graph low, 151 general mathematical, 33 of a function, 29, 100, 113 De Morgan’s Laws of a relation, 24, 27 for predicate logic, 10 greatest lower bound, 28 for sets, 19 halting probability, 165 decidable, 85 halting set, 78, 99, 114, 128, 129, definability, 147, 150 133 degree, see relativized, 129 degree invariance, 154 high set, 154 diagonalization, 63, 105, 190 diagonally non-recursive, 185 implication, logical, 10 disjoint, 16 incompleteness theorem, 105, 145, disjoint union, 101 163, 178 disjunction, 10 independent sentence, 104 domain index, 60 of a function, 29 index set, 82, 86, 135 of quantification, 13 induction, 179 domination, 64 on N,31 dovetailing, 96 on recursively defined sets, 35 inductive step, 31 E , 146, 153 injection, 29 E∗ , 146 injury, 116, 121 effective, 97 internal state, 44 effective countability, 56, 62 , 103, 170 embedding, 150 intersection, 16, 18 enumeration of Turing machines, 60 interval, 158 equality invariance, 147 for partial functions, 42 for degrees, 154 for sets, 15, 19 isomorphism, 29 equivalence class, 25 , 25 join, 101, 127, 128, 130, 146 equivalence, logical, 11 jump, 129 Index 201

omega, 130 notation, 111 oracle Turing machine, 109 K, 78, see also halting set orbit, 147 K(σ), 159 overspill, 175 K-trivial, 169 KC Theorem, 161 padding lemma, 60 Kolmogorov complexity pairing function, 57 plain, 167 parameter, 180 prefix-free, 159 parametrization, 79 Kraft inequality, 160, 166 partial computable functions, 65 , 42 , 69 for allowing, 55, 62 , 103, 170 partial order, 27 lattice, 146 partition, 26 complemented, 146 Peano arithmetic, 173, 175 distributive, 146 in reverse math, 183 embedding, 150 permutation, 30 nondistributive, 152 ϕe,60 1 least upper bound, 28 Π1 comprehension axiom, 184 left-c.e. set, 125, 166 , 186 limit lemma, 139 plain Kolmogorov complexity, 167 limit, discrete, 41, 140 poset, 27 linear order, 27, 176 Post correspondence system, 91 low basis theorem, 183 Post’s problem, 115, 120 low for random, 169 Post’s theorem, 134 low set, 130, 140 , 16, 192 noncomputable, 142 predicate, 13 lower bound, 28 prefix-free Kolmogorov complexity, 159 maximal set, 154 Turing machine, 158 measure, 158 primitive recursion, 51 meet, 128, 146 priority, 116, 117, 121 model, 103, 170, 180 product, Cartesian, 16 modular arithmetic, 26 production, 87 modulus, 140 normal, 88 lemma, 140 semi-Thue, 88 μ,55 projection, 51 negation, 10, 15 quantifiers, 13 nondeterministic Turing machine, alternation of, 131 68 collapsing like, 133 nonstandard model, 104, 174, 175 quotient structure, 25, see also nonuniformity, see uniformity Turing degree, E∗ not, 10 randomness, 155, 159 Ω, 165 and Turing degree, 165 or, logical, 10 for finite strings, 159 oracle, 109 relative, 168 finite, 111 range, 29 202 Index read/write head, 44 incomplete, 120, 142 recursion theorem, 81 nowhere, 153 with parameters, 142 promptly, 151 recursive, 96 solvable, 85 recursive comprehension axiom, theorem, 103, 172 179 splitting theorem, 153 recursive definition stage bounding of a function, 35 notation, 96, 100, 111 of a set, 33 use of, 111, 118, 121, 136 reduction property, 100 standard model, 104, 173 reflexivity, 22 string, 43, 157 relation, 21 structure, 170 relative randomness, 168 computably categorical, 176 relativization, 128 , 16 requirement, 106, 117 successor, 51 injury, 116 surjection, 30 priority, 116 symmetric difference, 114, 136, 144, satisfaction, 107 145 restriction, 112, 157 symmetry, 22 reverse mathematics, 177 , 103, 172 Rice’s theorem, 82 Robinson arithmetic, 104, 173 tape, 44 nonstandard model, 174 , 10 theorem of a production system, 88 s-m-n theorem, 79, 82, 84 theory, 170 relativized, 128 first-order, 173 satisfaction, 107 second-order, 173, 178 second-order arithmetic, 178 total computable functions, 65 second-order theory, 173, 178 total order, 27 , 103, 172 transitive closure, 27 semi-Thue system, 88, 92 transitivity, 22, 147 sentence, 13 table, 12 sequence, 43, 157 , 16 set, 15 , 114 arithmetical, 131 Turing degree, 127, 149 bi-immune, 124 and randomness, 165 computable, 96 c.e., 128, 149 computably enumerable, 97 Turing equivalence, 112 creative, 145 , 129 d.c.e., 125 Turing machine, 44 high, 154 , 78 left-c.e., 125, 166 code of, 59 low, 130, 140 enumeration of, 60 maximal, 154 index of, 60 productive, 145 nondeterministic, 68 simple, 106, see also simple set nonstandard, 67 simple set, 106, 117, 146, 164 oracle, 109 effectively, 145 prefix-free, 158 Index 203

universal, 62 , 112 unbounded search, 54 uncountable, 21 , see unsolvable problem uniformity, 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, 18 , 62 universe, 16 of a model, 170 unlimited , 73 unsolvable problem, 85 halting problem, 78, see also halting set in behavior of Turing machines, 77, 78, 86 index sets, 82, 87 solutions to Post correspondence systems, 92 , 90 word problem for semi-Thue systems, 89 word problem for Thue systems, 90 upper bound, 27 use, 111 of divergent , 111

We, 100 weak jump, 114, 143 relativized, 130 weak K¨onig’s lemma, 181, 182 weak weak K¨onig’s lemma, 185 without loss of generality, 39 witness, 118 word, 87 word problem for groups, 90 for semi-Thue systems, 89

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35 J¨org 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 , 2006 30 Serge Tabachnikov, Geometry and Billiards, 2005 29 Kristopher Tapp, Groups for Undergraduates, 2005 28 Emmanuel Lesigne, Heads or Tails, 2005 27 Reinhard Illner, C. Sean Bohun, Samantha McCollum, and Thea van 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 for Beginners, 2004 24 Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, 2004 23 S. K. Lando, Lectures on Generating Functions, 2003 22 Andreas Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, 2003 21 W. J. Kaczor and M. T. Nowak, Problems in III, 2003 20 Klaus Hulek, Elementary , 2003 19 A. Shen and N. K. Vereshchagin, Computable Functions, 2003 18 V. V. Yaschenko, Editor, : An Introduction, 2002 17 A. Shen and N. K. Vereshchagin, Basic , 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 Analysis II, 2001 11 Mike Mesterton-Gibbons, An Introduction to Game-Theoretic Modelling, 2001 10 John Oprea, The Mathematics of Soap Films: Explorations with Maple, 2000 9 David E. Blair, Inversion Theory and Conformal Mapping, 2000

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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.

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