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Theory and Applications of Computability Theory and Applications of Computability In cooperation with the association Computability in Europe Series Editors Prof. P. Bonizzoni Università degli Studi di Milano-Bicocca Dipartimento di Informatica Sistemistica e Comunicazione (DISCo) Milan Italy [email protected] Prof. V. Brattka Universität der Bundeswehr München Fakultät für Informatik Neubiberg Germany [email protected] Prof. E. Mayordomo Universidad de Zaragoza Departamento de Informática e Ingeniería de Sistemas Zaragoza Spain [email protected] Prof. P. Panangaden McGill University School of Computer Science Montré al Canada [email protected] Founding Editors: P. Bonizzoni, V. Brattka, S.B. Cooper, E. Mayordomo More information about this series at http://www.springer.com/series/8819 Books published in this series will be of interest to the research community and graduate students, with a unique focus on issues of computability. The perspective of the series is multidisciplinary, recapturing the spirit of Turing by linking theoretical and real-world concerns from computer science, mathematics, biology, physics, and the philosophy of science. The series includes research monographs, advanced and graduate texts, and books that offer an original and informative view of computability and computational paradigms. Series Advisory Board Samson Abramsky, University of Oxford Eric Allender, Rutgers, The State University of New Jersey Klaus Ambos-Spies, Universität Heidelberg Giorgio Ausiello, Università di Roma, “La Sapienza” Jeremy Avigad, Carnegie Mellon University Samuel R. Buss, University of California, San Diego Rodney G. Downey, Victoria University of Wellington Sergei S. Goncharov, Novosibirsk State University Peter Jeavons, University of Oxford Nataša Jonoska, University of South Florida, Tampa Ulrich Kohlenbach, Technische Universität Darmstadt Ming Li, University of Waterloo Wolfgang Maass, Technische Universität Graz Grzegorz Rozenberg, Leiden University and University of Colorado, Boulder Alan Selman, University at Buffalo, The State University of New York Wilfried Sieg, Carnegie Mellon University Jan van Leeuwen, Universiteit Utrecht Klaus Weihrauch, FernUniversität Hagen Philip Welch, University of Bristol Robert I. Soare Turing Computability Theory and Applications Robert I. Soare Department of Mathematics The University of Chicago Chicago, Illinois, USA 03Dxx (Computability and recursion theory). ISSN 2190-619X ISSN 2190-6203 (electronic) Theory and Applications of Computability ISBN 978-3-642-31932-7 ISBN 978-3-642-31933-4 (eBook) DOI 10.1007/978-3-642-31933-4 Library of Congress Control Number: 2016944469 © Springer-Verlag Berlin Heidelberg 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover illustration: Damir Dzhafarov designed the image of the Turing machine used in the book cover. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg I dedicate this book to my wife, Pegeen. Contents I Foundations of Computability 1 1 Defining Computability 3 1.1 Algorithmically Computable Functions .......... 3 1.1.1 Algorithms in Mathematics ............ 3 1.1.2 The Obstacle of Diagonalization and Partial Func- tions ......................... 4 1.1.3 The Quest for a Characterization . 5 1.1.4 Turing's Breakthrough ............... 5 1.2 ?? Turing Defines Effectively Calculable . 6 1.3 ?? Turing's Thesis, Turing's Theorem, TT . 7 1.4 ?? Turing Machines ..................... 7 1.4.1 Exercises on Turing Machines ........... 9 1.5 ?? The Basic Results .................... 10 1.5.1 Numbering Turing Programs Pe . 10 1.5.2 Numbering Turing Computations . 10 1.5.3 The Enumeration Theorem and Universal Machine 11 1.5.4 The Parameter Theorem or s-m-n Theorem . 12 1.6 ?? Unsolvable Problems . 13 1.6.1 Computably Enumerable Sets . 13 1.6.2 Noncomputable C.E. Sets . 13 1.6.3 The Index Set Theorem and Rice's Theorem . 16 1.6.4 Computable Approximations to Computations . 17 vii viii Contents 1.6.5 Exercises ....................... 18 1.7 ? Computable Permutations and Isomorphisms . 19 1.7.1 Myhill Isomorphism Theorem . 20 1.7.2 Acceptable Numberings . 21 1.7.3 Exercises ....................... 21 2 Computably Enumerable Sets 23 2.1 ?? Characterizations of C.E. Sets . 23 0 2.1.1 The Σ1 Normal Form for C.E. Sets . 23 2.1.2 The Uniformization Theorem . 25 2.1.3 The Listing Theorem for C.E. Sets . 26 2.1.4 The C.E. and Computable Sets as Lattices . 26 2.1.5 Exercises ....................... 27 2.2 ? Recursion Theorem (Fixed Point Theorem) . 28 2.2.1 Fixed Points in Mathematics . 28 2.2.2 Operating on Indices . 28 2.2.3 ?? A Direct Proof of the Recursion Theorem . 29 2.2.4 A Diagonal Argument Which Fails . 29 2.2.5 Informal Applications of the Recursion Theorem . 31 2.2.6 Other Properties of the Recursion Theorem . 31 2.2.7 Exercises ....................... 32 2.3 Indexing Finite and Computable Sets . 33 2.3.1 Computable Sets and ∆0 and ∆1 Indices . 33 2.3.2 Canonical Index y for Finite Set Dy and String σy 34 2.3.3 Acceptable Numberings of Partial Computable Func- tions ......................... 35 2.3.4 Exercises ....................... 37 2.4 ? Complete Sets and Creative Sets . 38 2.4.1 Productive Sets ................... 38 2.4.2 ?? Creative Sets Are Complete . 39 2.4.3 Exercises ....................... 40 2.5 ?? Elementary Lachlan Games . 43 2.5.1 The Definition of a Lachlan Game . 43 2.5.2 Playing Partial Computable (P.C.) Functions . 44 2.5.3 Some Easy Examples of Lachlan Games . 44 2.5.4 Practicing Lachlan Games . 45 2.5.5 The Significance of Lachlan Games . 45 2.5.6 Exercises on Lachlan Games . 45 2.6 The Order of Enumeration of C.E. Sets . 46 2.6.1 Uniform Sequences and Simultaneous Enumerations 46 2.6.2 Static and Dynamic Properties of C.E. Sets . 47 2.6.3 Exercises ....................... 48 2.7 The Friedberg Splitting Theorem . 49 2.7.1 The Priority Ordering of Requirements . 49 2.7.2 Exercises ....................... 50 Contents ix 3 Turing Reducibility 51 3.1 The Concept of Relative Computability . 51 3.1.1 Turing Suggests Oracle Machines (o-Machines) . 51 3.1.2 Post Develops Relative Computability . 51 3.2 ?? Turing Computability . 52 3.2.1 An o-Machine Model for Relative Computability 52 3.2.2 Turing Computable Functionals Φe . 53 ? 3.3 Oracle Graphs of Turing Functional Φe . 54 3.3.1 The Prefix-Free Graph Fe of Functional Φe . 54 3.3.2 The Oracle Graph Ge of Functional Φe . 56 3.3.3 The Use Principle for Turing Functionals . 57 3.3.4 Permitting Constructions . 57 3.3.5 Lachlan Notation for Approximation by Stages . 57 3.3.6 Standard Theorems Relativized to A . 58 3.3.7 Exercises ....................... 59 3.4 ? Turing Degrees and the Jump Operator . 60 3.4.1 The Structure of the Turing Degrees . 60 3.4.2 The Jump Theorem . 61 3.4.3 Exercises ....................... 62 3.5 ? Limit Computable Sets and Domination . 63 3.5.1 Domination and Quantifiers (81 x) and (91x) . 64 3.5.2 Uniformly Computable Sequences . 64 3.5.3 Limit Computable Sets . 65 3.5.4 Exercises ....................... 66 3.6 ?? The Limit Lemma .................... 66 3.6.1 The Modulus Lemma for C.E. Sets . 68 3.6.2 The Ovals of Σ1 and ∆2 Degrees . 69 3.6.3 Reaching With the Jump: Low and High Sets . 69 3.6.4 Exercises ....................... 70 3.7 ? Trees and the Low Basis Theorem . 71 3.7.1 Notation for Trees . 71 ? 0 3.7.2 The Low Basis Theorem for Π1 Classes . 71 3.7.3 Exercises ....................... 72 3.8 Bounded Reducibilities and n-C.E. Sets . 73 3.8.1 A Matrix Mx for Bounded Reducibilities . 73 3.8.2 Bounded Turing Reducibility . 73 3.8.3 Truth-Table Reductions . 74 3.8.4 Difference of C.E., n-c.e., and !-c.e. Sets . 75 3.8.5 Exercises ....................... 77 4 The Arithmetical Hierarchy 79 4.1 Levels in the Arithmetical Hierarchy . 79 4.1.1 Quantifier Manipulation . 80 4.1.2 Placing a Set in Σn or Πn . 82 4.1.3 Exercises ....................... 83 x Contents 4.2 ?? Post's Theorem and the Hierarchy Theorem . 83 (n) 4.2.1 Post's Theorem Relating Σn to ; . 84 4.2.2 Exercises ....................... 85 ? 4.3 Σn-Complete Sets and Πn-Complete Sets . 86 4.3.1 Classifying Σ2 and Π2 Sets: Fin, Inf, and Tot . 86 4.3.2 Constructions with Movable Markers . 87 4.3.3 Classifying Cof as Σ3-Complete . 87 4.3.4 Classifying Rec as Σ3-Complete . 88 4.3.5 Σ3-Representation Theorems . 89 4.3.6 Exercises ....................... 91 4.4 Relativized Hierarchy: Lown and Highn Sets . 91 4.4.1 Relativized Post's Theorem . 92 4.4.2 Lown and Highn Sets . 92 4.4.3 Common Jump Classes of Degrees .
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