Computational Complexity Theory This page intentionally left blank https://doi.org/10.1090//pcms/010 IAS/PARK CIT Y MATHEMATICS SERIES Volume 1 0 Computational Complexity Theory Steven Rudic h Avi Wigderson Editors American Mathematical Societ y Institute for Advanced Stud y IAS/Park Cit y Mathematics Institute runs mathematics educatio n programs that brin g together hig h schoo l mathematic s teachers , researcher s i n mathematic s an d mathematic s education, undergraduat e mathematic s faculty , graduat e students , an d undergraduate s t o participate i n distinc t bu t overlappin g program s o f researc h an d education . Thi s volum e contains th e lectur e note s fro m th e Graduat e Summe r Schoo l progra m o n Computationa l Complexity Theor y hel d i n Princeto n i n the summe r o f 2000 . 2000 Mathematics Subject Classification. Primar y 68Qxx ; Secondar y 03D15 . Library o f Congress Cataloging-in-Publicatio n Dat a Computational complexit y theor y / Steve n Rudich , Av i Wigderson, editors , p. cm . — (IAS/Park Cit y mathematic s series , ISS N 1079-563 4 ; v. 10) "Volume contain s the lecture note s fro m th e Graduate Summe r Schoo l progra m o n Computa - tional Complexit y Theor y hel d i n Princeton i n the summer o f 2000"—T.p. verso . Includes bibliographica l references . ISBN 0-8218-2872- X (hardcove r : acid-fre e paper ) 1. Computational complexity . I . Rudich, Steven . II . Wigderson, Avi . III . Series. QA267.7.C685 200 4 511.3'52—dc22 2004049026 Copying an d reprinting. Materia l i n this boo k ma y be reproduced b y any means fo r edu - cational an d scientific purpose s withou t fe e or permission wit h th e exception o f reproduction b y services that collec t fee s fo r delivery o f documents an d provided tha t th e customary acknowledg - ment o f the source i s given. 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Printed i n the United State s o f America. @ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guidelines established t o ensure permanenc e an d durability. Visit th e AMS home pag e at http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 9 08 07 06 05 0 4 Contents Preface xii i Introduction 1 Week One : COMPLEXITY THEORY : FRO M GODE L T O FEYNMA N 3 Steven Rudich , Complexit y Theory : Fro m Gode l t o Feynma n 5 Lecture 1 . History an d Basi c Concept s 7 1.1. Histor y 7 1.2. Th e Turin g Machin e 8 1.3. Som e Basic Definition s 11 1.4. Th e Church-Turin g Thesi s 12 1.5. Computationa l Resource s 12 1.6. Godel' s Lette r 13 1.7. Th e Moder n Da y Versio n o f P = N P 13 1.8. Appendi x 17 Lecture 2 . Resources, Reduction s an d P vs . N P 19 2.1. Tim e an d Spac e 19 2.2. Polynomia l Tim e 19 2.3. Non-Deterministi c Turin g Machine s 20 2.4. Consequence s o f P = N P 22 2.5. Reducibilit y 23 2.6. Completenes s 24 2.7. Cook-Levi n Theore m 25 2.8. Othe r NP-complet e Problem s 27 2.9. Wha t Gode l Misse d 27 Lecture 3 . Probabilistic an d Quantu m Computatio n 29 3.1. Schwartz-Zippe l Theore m 29 3.2. Verifyin g Arithmeti c 30 3.3. Probabilisti c Complexit y Classe s 30 3.4. Quantu m Computatio n 32 3.5. Conclusio n 33 V vi CONTENT S Lecture 4 . Complexity Classe s 3 5 4.1. Simulatio n 3 5 4.2. Hierarch y Theorem s 3 7 4.3. Ladner' s Theore m 3 8 4.4. Relativizatio n 3 9 4.5. Relation s Betwee n Som e Complexit y Classe s 4 0 4.6. Co-classe s 4 1 Lecture 5 . Space Complexit y an d Circui t Complexit y 4 5 5.1. Savitch' s Theore m 4 5 5.2. Th e Immerman-Szelepcseny i Theore m 4 6 5.3. PSPACE-Completenes s 4 8 5.4. Boolea n Circuit s 4 9 5.5. Circui t Complexit y Classe s 5 1 5.6. Non-Unifor m Circuit s an d Advic e Turing Machine s 5 2 Lecture 6 . Oracle s an d the Polynomia l Tim e Hierarch y 5 5 6.1. Complexit y Classe s Relativ e to a n Oracl e 5 5 6.2. Polynomia l Hierarch y 5 6 6.3. Placin g BPP i n the Worl d Pictur e 6 0 6.4. Karp-Lipto n Theore m 6 2 Lecture 7 . Circuit Lowe r Bound s 6 5 7.1. Circui t Complexit y an d Lo w Degree Polynomials 6 5 7.2. Approximatio n Metho d 6 6 Lecture 8 . "Natural " Proof s o f Lowe r Bound s 7 5 8.1. Ho w to Reaso n Tha t a Problem i s Hard 7 5 8.2. A n Ol d "Stumblin g Block" : Relativizatio n 7 5 8.3. A New Direction: Non-Unifor m Lowe r Bound s 7 6 8.4. A New "Stumblin g Block" : Natura l Proof s 7 6 8.5. Natura l Proof s o f Lowe r Bound s fo r AC o 7 7 8.6. Generalizin g Ou r Definition s 7 9 8.7. "Naturalizing " Smolensky' s Proo f 7 9 8.8. What' s "Bad " Abou t a Natural Proof ? 8 1 8.9. Wh y D o Natural Proof s Arise ? 8 3 8.10. Unnatura l Circui t Lowe r Bound s 8 3 8.11. Th e Bi g Picture 8 3 Bibliography 8 5 Avi Wigderson , Averag e Cas e Complexit y 8 9 Lecture 1 . Average Cas e Complexit y 9 1 1.1. Introductio n 9 1 1.2. Levin' s Theory o f Average-Case Complexit y 9 1 1.3. A "Generic " Dist-NP Complet e Problem 9 4 1.4. Convertin g Worst-Cas e Hardnes s int o Average-Cas e Hardnes s 9 5 1.5. Fiv e Possible World s 9 6 Bibliography 99 CONTENTS vi i Sanjeev Arora , Explorin g Complexit y throug h Reduction s 10 1 Introduction 10 3 Lecture 1 . PCP Theore m an d Hardnes s o f Computing Approximat e Solution s 10 5 1. Approximatio n Algorithm s 10 5 2. Probabilisticall y Checkabl e Proof s 10 6 3. Hastad' s PC P an d Inapproximabilit y o f MAX-3SAT 10 8 4. Inapproximabilit y o f MAX-3SAT(13) 10 8 5. Inapproximabilit y o f MAX-INDEP-SET 10 9 6. Inapproximabilit y o f Other Problem s 11 0 7. Histor y 11 1 Lecture 2 . Which Problem s Hav e Strongly Exponentia l Complexity ? 11 3 1. SERF-Reduction s 11 3 2. Th e Mai n Theore m 11 4 Lecture 3 . Toda's Theorem : PH C P* p 11 9 1. Classe s #P an d 0P 11 9 2. Th e Mai n Lemm a 12 1 3. Proo f o f Theorem 2 1 12 2 4. Ope n Problem s 12 3 Bibliography 12 5 Ran Raz , Quantu m Computatio n 12 7 Lecture 1 . Introduction 12 9 1.1. Classica l Deterministi c Machine s 13 0 1.2. Classica l Probabilisti c Machine s 13 1 1.3. Quantu m System s 13 2 1.4. Dirac' s Ke t Notatio n 13 3 1.5. Quantu m Measuremen t 13 4 1.6. Transitio n Matri x an d Interferenc e 13 4 1.7. Measuremen t Accordin g to a Differen t Bas e 13 5 1.8. Th e Polarizer s Experimen t 13 6 1.9. Historica l Backgroun d 13 7 Lecture 2 . Bipartite Quantu m System s 13 9 2.1.