STUDENT MATHEMATICAL LIBRARY Volume 62 Computability Theory 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 Mathematics 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 references and index. ISBN 978-0-8218-7392-2 (alk. paper) 1. Recursion 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 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 Bibliography [1] Wilhelm Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), no. 1, 118–133 (German). [2] Klaus Ambos-Spies, Carl G. Jockusch Jr., Richard A. Shore, and Robert I. Soare, An algebraic decomposition of the recursively enumerable degrees and the coin- cidence of several degree classes with the promptly simple degrees, Trans. Amer. Math. Soc. 281 (1984), no. 1, 109–128. [3] Klaus Ambos-Spies, Bjørn Kjos-Hanssen, Steffen Lempp, and Theodore A. Sla- man, Comparing DNR and WWKL, J. Symbolic Logic 69 (2004), no. 4, 1089– 1104. [4] Klaus Ambos-Spies and Anton´ın Kuˇcera, Randomness in computability theory, Computability Theory and its Applications (Boulder, CO, 1999), 2000, pp. 1–14. [5] M. M. Arslanov, Some generalizations of a fixed-point theorem, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (1981), 9–16 (Russian). [6] M.M.Arslanov,R.F.Nadyrov,andV.D.Solovev, A criterion for the complete- ness of recursively enumerable sets, and some generalizations of a fixed point theorem,Izv.Vysˇs. Uˇcebn. Zaved. Matematika 4 (179) (1977), 3–7 (Russian). [7] C. J. Ash and J. Knight, Computable Structures and the Hyperarithmetical Hi- erarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North- Holland Publishing Co., Amsterdam, 2000. [8] Jon Barwise (ed.), Handbook of mathematical logic, with the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra, Studies in Logic and the Foundations of Mathematics, Vol. 90, North-Holland Publishing Co., Amsterdam, 1977. [9] Charles H. Bennett and Martin Gardner, The random number Omega bids fair to hold the mysteries of the universe, Scientific American 241 (1979), no. 5, 20–34. [10] George S. Boolos, John P. Burgess, and Richard C. Jeffrey, Computability and Logic, 4th ed., Cambridge University Press, Cambridge, 2002. [11] Gregory J. Chaitin, Information-theoretic characterizations of recursive infinite strings,Theoret.Comput.Sci.2 (1976), no. 1, 45–48. [12] G. J. Chaitin, Incompleteness theorems for random reals, Adv. in Appl. Math. 8 (1987), no. 2, 119–146. 193 194 Bibliography [13] Peter A. Cholak, Carl G. Jockusch, and Theodore A. Slaman, On the strength of Ramsey’s theorem for pairs,J.SymbolicLogic66 (2001), no. 1, 1–55. [14] Alonzo Church, An Unsolvable Problem of Elementary Number Theory,Amer. J. Math. 58 (1936), no. 2, 345–363. Reprinted in [18]. [15] , A note on the Entscheidungsproblem, J. Symbolic Logic 1 (1936), no. 1, 40–41. Correction in J. Symbolic Logic 1 No. 3 (1936), 101–102. Reprinted in [18]. [16] Robert L. Constable, Harry B. Hunt, and Sartaj Sahni, On the computational complexity of scheme equivalence, Technical Report 74–201, Department of Com- puter Science, Cornell University, Ithaca, NY, 1974. [17] Nigel Cutland, Computability: An introduction to recursive function theory, Cambridge University Press, Cambridge, 1980. [18] Martin Davis (ed.), The Undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, N.Y., 1965. [19] Martin Davis, Hilbert’s tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233–269. [20] , Why G¨odel didn’t have Church’s thesis, Inform. and Control 54 (1982), no. 1-2, 3–24. [21] , American logic in the 1920s, Bull. Symbolic Logic 1 (1995), no. 3, 273– 278. [22] Rodney G. Downey and Denis R. Hirschfeldt, Algorithmic Randomness and Com- plexity, Theory and Applications of Computability, Springer, New York, 2010. [23] Bruno Durand and Alexander Zvonkin, Kolmogorov complexity, Kolmogorov’s Heritage in Mathematics, 2007, pp. 281–299. [24] A. Ehrenfeucht, J. Karhum¨aki, and G. Rozenberg, The (generalized) Post cor- respondence problem with lists consisting of two words is decidable,Theoret. Comput. Sci. 21 (1982), no. 2, 119–144. [25] Herbert B. Enderton, A Mathematical Introduction to Logic,2nded.,Har- court/Academic Press, Burlington, MA, 2001. [26] Yu. L. Ershov, S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek (eds.), Handbook of Recursive Mathematics. Vol. 1: Recursive Model Theory, Studies in Logic and the Foundations of Mathematics, vol. 138, North-Holland, Amsterdam, 1998. [27] Richard M. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post’s problem, 1944), Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 236–238. [28] , Three theorems on recursive enumeration. I. Decomposition. II. Max- imal set. III. Enumeration without duplication, J. Symb. Logic 23 (1958), 309– 316. [29] P´eter G´acs, Every sequence is reducible to a random one, Inform. and Control 70 (1986), no. 2-3, 186–192. [30] Kurt G¨odel, Uber¨ formal unentscheidbare S¨atze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38 (1931), no. 1, 173–198 (German). An English translation appears in [18]. [31] S. S. Gonˇcarov, The number of nonautoequivalent constructivizations,Algebra Logika 16 (1977), no. 3, 257–282, 377 (Russian); English transl., Algebra Logic 16 (1977), no. 3, 169–185. [32] , The problem of the number of nonautoequivalent constructivizations, Algebra Logika 19 (1980), no. 6, 621–639, 745 (Russian); English transl., Algebra Logic 19 (1980), no. 6, 404–414. [33] S. S. Gonˇcarov and V. D. Dzgoev, Autostability of models, Algebra Logika 19 (1980), no. 1, 45–58, 132 (Russian); English transl., Algebra Logic 19 (1980), no. 1, 28–37. Bibliography 195 [34] Eitan Gurari, An Introduction to the Theory of Computation, Computer Sci- ence Press, 1989. Available at http://www.cse.ohio-state.edu/~gurari/theory-bk/ theory-bk.html. [35] Petr H´ajek and Pavel Pudl´ak, Metamathematics of First-Order Arithmetic,Per- spectives in Mathematical Logic, Springer-Verlag, Berlin, 1993. [36] Valentina S. Harizanov, Computability-theoretic complexity of countable struc- tures, Bull. Symbolic Logic 8 (2002), no. 4, 457–477. [37] David Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), no. 10, 437–479. [38] Denis R. Hirschfeldt, Andr´e Nies, and Frank Stephan, Using random sets as or- acles,J.Lond.Math.Soc.(2)75 (2007), no. 3, 610–622.