Computability complexity and Languages- Fundamentals of Theoretical Computer Science

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Second EditionComputability,Complexity, andLanguagesFundamentals ofTheoretical Computer ScienceMartin D. DavisDepartment of Computer ScienceCourant Institute of Mathematical SciencesNew York UniversityNew York, New YorkRon SigalDepartments of Mathematics and Computer ScienceYale UniversityNew Haven, ConnecticutElaine J. WeyukerDepartment of Computer ScienceCourant Institute of Mathematical SciencesNew York UniversityNew York, New YorkACADEMIC PRESSHarcourt, Brace & CompanyBoston San Diego New YorkLondon Sydney Tokyo Toronto
This book is printed on acid-free paper @)Copyright© 1994, 1983 by Academic Press, Inc.All rights reservedNo part of this publication may be reproduced ortransmitted in any form or by any means, electronicor mechanical, including photocopy, recording, orany information storage and retrieval system, withoutpermission in writing from the publisher.ACADEMIC PRESS, INC.525 B Street, Suite 1900, San Diego, CA 92101-4495United Kingdom Edition published byACADEMIC PRESS LIMITED24-28 Oval Road, London NW1 7DXLibrary of Congress Cataloging-in-Publication DataDavis, Martin 1928-Computability, complexity, and languages: fundamentals oftheoretical computer science/Martin D. Davis, Ron Sigal,Elaine J. Weyuker. --2nd ed.p. em. --(Computer science and applied mathematics)Includes bibliographical references and index.ISBN 0-12-206382-11. Machine theory. 2. Computational complexity. 3. Formallanguages. I. Sigal, Ron. II. Weyuker, Elaine J. III. Title.IV. Series.QA267.D38 1994511.3-dc20 93-26807CIPPrinted in the United States of America94 95 96 97 98 BC 9 8 7 6 5 4 3 2 1
Page 4: To the memory of Helen and Harry Da
Page 7 and 8: viiiContents3 Primitive Recursive F
Page 9 and 10: XContents11 Context-Sensitive Langu
Page 12 and 13: PrefaceTheoretical computer science
Page 14: PrefaceXValso be read immediately f
Page 17 and 18: xviiiAcknowledgmentsAcknowledgments
Page 20 and 21: 1Preliminaries1. Sets and n-tuplesW
Page 22 and 23: 2. Functions 3S 1 X S 2 X ••·
Page 24 and 25: 4. Predicates 5If u E A*, we write
Page 26 and 27: 5. Quantifiers 7Thus, let P(t, x 1
Page 28 and 29: 7. Mathematical Induction 9has no s
Page 30 and 31: 7. Mathematical Induction 11be prov
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Page 36 and 37: 2Programs andComputable Functions1.
Page 38 and 39: 2. Some Examples of Programs 19If t
Page 40 and 41: 2. Some Examples of Programs 21the
Page 42 and 43: 2. Some Examples of Programs 23was
Page 44 and 45: 3. Syntax 25Exercises1. Write a pro
Page 46 and 47: 3. Syntax 27Suppose we have a progr
Page 48 and 49: 4. Computable Functions 29together
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5. More about Macros 33Let f(x 1 ,
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5. More about Macros 35Hence predic
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5. More about Macros37an arbitrary
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40 Chapter 3 Primitive Recursive Fu
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4A Universal Program1. Coding Progr
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1. Coding Programs by Numbers 67The
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2. The Halting Problem 69numbers ca
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3. Universality 71In considering th
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3. UniversalityZ <-Xn+l + 1ns <---
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3. Universality 75Now we can define
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3. Universality 77recursion. Finall
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4. Recursively Enumerable Sets 79We
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4. Recursively Enumerable Sets81[A]
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4. Recursively Enumerable Sets 83Pr
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5. The Parameter Theorem857. Let A,
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5. The Parameter Theorem 87using fi
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6. Dlagonalization and Reducibility
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6. Diagonalization and Reducibility
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6. Diagonalization and Reducibility
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7. Rice's Theorem 95Show that Q(x)
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8. The Recursion Theorem 97thesis.
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8. The Recursion Theorem 99After th
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8. The Recursion Theorem 101know if
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8. The Recursion Theorem 103and con
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9. A Computable Function That Is No
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5Calculations on Strings1. Numerica
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1. Numerical Representation of Stri
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2. A Programming Language for Strin
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3. The Languages .9" and .5';, 127W
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4. Post- Turing Programs 1294. Post
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4. Post-Turing Programs 131Of cours
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4. Post- Turing Programs133[C) RIGH
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5. Simulation of .9';, in fT 1354.
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--------'5. Simulation of Y, In .'T
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5. Simulation of .9';, In !T 139cor
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6. Simulation of .9'" in .'7' 1413.
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6. Simulation of .'T in .'7'143Simi
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6Turing Machines1. Internal StatesN
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1. Internal States147Table 1.1Symbo
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1. Internal States 149and the final
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1. Internal States151[Ad IF s 0 GOT
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3. The Languages Accepted by Turing
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3. The Languages Accepted by Turing
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4. The Halting Problem for Turing M
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5. Nondeterministic Turing Machines
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5. Nondeterministic Turing Machines
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6. Variations on the Turing Machine
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7Processes and Grammars1. Semi-Thue
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2. Simulation of Nondeterministic T
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3. Unsolvable Word Problems 177We w
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3. Unsolvable Word Problems 179Proo
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4. Post's Correspondence Problem 18
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4. Post's Correspondence Problem183
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4. Post's Correspondence Problem185
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5. Grammars187Now, let L accept u E
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5. Grammars 189We now are able to o
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6. Some Unsolvable Problems Concern
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7. Normal Processes 193to indicate
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7. Normal Processes 195Lemma 5. Let
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198 Chapter 8 Classifying Unsolvabl
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9Regular Languages1. Finite Automat
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1. Finite Automata 239is the initia
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1. Finite Automata 241(a) A = {1};
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2. Nondeterministic Finite Automata
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2. Nondeterministic Finite Automata
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3. Additional Examples 2472. For ea
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4. Closure Properties 249aabFigure3
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4. Closure Properties 251(That is,
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5. Kleene's Theorem 253a 1 ••
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5. Kleene's Theorem 255Chapter 4. I
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5. Kleene's Theorem 257For each reg
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5. Kleene's Theorem 259(Note that p
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6. The Pumping Lemma and Its Applic
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7. The Myhill- Nerode Theorem 2637.
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7. The Myhill- Nerode Theorem 265wo
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7. The Myhill- Nerode Theorem 26713
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270 Chapter 10 Context-Free Languag
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272 Chapter 1 0 Context-Free Langua
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286Chapter 1 0 Context-Free Languag
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294 Chapter 1 0 Context-Free lanQua
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308Chapter 1 0 Context-Free Languag
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328 Chapter 11 Context-Sensitive La
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nlwlknlwl342 Chapter 11 Context-Sen
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12Propositional Calculus1. Formulas
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1. Formulas and Assignments 349Tabl
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1. Formulas and Assignments351secon
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3. Normal Forms 353Consider the exa
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3. Normal Forms 355that there are t
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3. Normal Forms 357Use of (II) agai
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3. Normal Forms 359This problem has
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4. The Davis- Putnam Rules 361We wi
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4. The Davis- Putnam Rules363Succes
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4. The Davis- Putnam Rules 365of th
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6. Resolution 367Then it is very ea
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6. Resolution 369j, k < i. Hence, b
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7. The Compactness Theorem 371for n
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7. The Compactness Theorem 373adjac
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376 Chapter 13 Quantification Theor
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Part 4Complexity
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420 Chapter 14 Abstract Complexityi
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422 Chapter 14 Abstract ComplexityO
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424 Chapter 14 Abstract Complexity2
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426 Chapter 14 Abstract ComplexityT
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428 Chapter 14 Abstract Complexity2
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430 Chapter 14 Abstract Complexityr
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432 Chapter 14 Abstract ComplexityX
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434 Chapter 14 Abstract ComplexityH
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436Chapter 14 Abstract ComplexityV+
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438 Chapter 14 Abstract ComplexityF
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440 Chapter 15 Polynomial- Time Com
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442 Chapter 15 Polynomial-Time Comp
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Part 5Semantics
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468 Chapter 16 Approximation Orderi
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17Denotational Semanticsof Recursio
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1. Syntax 507We extend rand p to VA
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1. Syntax 509will remain fixed thro
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2. Semantics of Terms 511(c)Let (W,
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2. Semantics of Terms 5135. For all
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2. Semantics of Terms 515based on :
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2. Semantics of Terms517write Or-;:
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2. Semantics of Terms 519Exercises1
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3. Solutions to W-Programs 521When
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3. Solutions to W-Programs 523Theor
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3. Solutions to W-Programs 525so JL
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3. Solutions to W-Programs 527If i
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3. Solutions to W-Programs 5299. Le
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4. Denotational Semantics of W-Prog
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4. Denotatlonal Semantics of W-Prog
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5. Simple Data Structure Systems 53
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6. lnfinltary Data Structure System
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6. lnfinitary Data Structure System
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18Operational Semanticsof Recursion
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1. Operational Semantics for Simple
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2. Computable Functions5757. Let P
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2. Computable Functions 577Then for
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2. Computable Functions579Let (x 1
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2. Computable Functions 581It is cl
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2. Computable Functions 5833. Let P
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3. lnflnltary Data Structure System
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3. lnflnltary Data Structure System
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Suggestions for Further ReadingC. L
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Notation IndexE 1 (3t), (Vt) 70 1
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Notation Index 597p 444 F(XH ... ,X
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IndexAlgorithms, 3, 68, 79-80, 95,
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Indexnonregular, 270and pushdown au
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IndexHorn clauses, 403Horn programs
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IndexpP,444-446, 448,449,456,461Pai
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IndexRogers, Hartley, 594Rooted sen
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Indexbound, 376-377free, 376-377fun
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Computability complexity and Languages- Fundamentals of Theoretical Computer Science

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