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 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

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© 200 4 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l rights except thos e grante d t o the United State s Government . 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. Th e Quantu m Registe r 13 9 2.2. Bipartit e Quantu m System s 14 0 2.3. Tenso r Produc t o f Vectors 14 0 2.4. Tenso r Produc t o f Matrices 14 1 2.5. Partia l Measurement s 14 2 2.6. Th e EPR Parado x 14 2 2.7. Entanglemen t an d Interferenc e 14 3 2.8. Th e No-Clonin g Theore m 14 4

Lecture 3 . Quantum Circuit s an d Shor' s Factorin g Algorith m 14 7 3.1. Quantu m Circui t Complexit y 14 7 3.2. Quantu m Simulatio n o f Classica l Computation s 14 9 viii CONTENT S

3.3. Quantu m Parallelizatio n 15 0 3.4. Quantu m Fourie r Transfor m 15 0 3.5. Findin g th e Perio d o f a Vector 15 1 3.6. Findin g Orde r modul o N 15 2

Bibliography 15 5

Week Two : LOWER BOUND S 15 7

Ran Raz , Circui t an d Communicatio n Complexit y 15 9

Lecture 1 . Communication Complexit y 16 1 1.1. Basi c Mode l an d Som e Examples 16 1 1.2. Deterministi c versu s Probabilisti c Complexit y 16 2 1.3. Communicatio n Complexit y o f Equality 16 3 1.4. Inpu t Matri x an d Monochromati c Cover s 16 4 1.5. Th e Ran k Lowe r Bound 16 5

Lecture 2 . Lower Bound s fo r Probabilisti c Communicatio n Complexit y 16 7 2.1. Probabilisti c Protocol s an d Unbalance d Rectangle s 16 7 2.2. Lowe r Boun d fo r Inner-Produc t 16 8 2.3. Lowe r Boun d fo r Se t Disjointnes s 17 1 2.4. Th e 3-Distinctnes s Proble m 17 2

Lecture 3 . Communication Complexit y an d Circui t Dept h 17 5 3.1. Karchme r - Wigderson Game s 17 5 3.2. Monoton e Complexit y 17 8 3.3. Monoton e Karchmer-Wigderso n Game s 18 0 3.4. Lowe r Boun d fo r Matchin g 18 2

Lecture 4 . Lowe r Boun d fo r Directe d st-Connectivit y 18 5 4.1. Th e FORK Gam e 18 5 4.2. Restricte d FORK Protocol s 18 7 4.3. Trivia l Lowe r Boun d o f ft(log(w)) 18 8

Lecture 5 . Lower Boun d fo r FORK (Continued ) 19 1 5.1. Increasin g th e Densit y 19 1 5.2. Combinatoria l Clai m 19 2 5.3. Proo f o f The Mai n Lemm a 19 2

Bibliography 19 7

Paul Beame , Proo f Complexit y 19 9

Lecture 1 . An Introduction t o Proo f Complexit y 20 1 1.1. Proo f System s 20 1 1.2. Example s o f Propositional Proo f System s 20 3 1.3. Polynomia l Calculu s wit h Resolutio n - PCR 21 0 1.4. Proo f Syste m Hierarch y 21 3 CONTENTS ix

Lecture 2 . Lower Bound s i n Proo f Complexit y 21 5 2.1. Th e Pigeonhol e Principl e 21 5 2.2. Widt h vs . Siz e o f Resolution Proof s 21 8 2.3. Resolutio n Proof s Base d o n the Width-Siz e Relationshi p 22 0 2.4. Nullstellensat z an d Polynomia l Calculu s Lowe r Bound s 22 3

Lecture 3 . Automatizability an d Interpolatio n 22 7 3.1. Automatizabilit y 22 7 3.2. Interpolatio n 22 8 3.3. Lowe r Bound s Usin g Interpolation 22 9 3.4. Limitation s 23 1

Lecture 4 . The Restrictio n Metho d 23 3 4.1. Decisio n Tree s 23 4 4.2. Restrictio n Metho d i n Circui t Complexit y 23 5 4.3. Restrictio n Metho d i n Proo f Complexit y 23 6

Lecture 5 . Other Researc h an d Ope n Problem s 24 1

Bibliography 24 3

Week Three : RANDOMNESS I N COMPUTATIO N 24 7

Preface t o "Wee k Three: RANDOMNES S I N COMPUTATION" 24 9

Oded Goldreich , Pseudorandomnes s - Par t I 25 3

Preface 25 5

Lecture 1 . Computational Indistinguishabilit y 25 7 1.1. Introductio n 25 7 1.2. Th e Notio n o f Pseudorandom Generator s 25 8 1.3. Th e Definitio n o f Computational Indistinguishabilit y 25 9 1.4. Relatio n t o Statistica l Closenes s 25 9 1.5. Indistinguishabilit y b y Repeated Experiment s 26 1 Lecture 2 . Pseudorandom Generator s 26 5 2.1. Basi c Definitio n an d Initia l Discussio n 26 5 2.2. Amplifyin g th e Stretc h Functio n 26 6 2.3. Ho w to Construc t Pseudorando m Generator s 26 7

Lecture 3 . Pseudorandom Function s an d Concludin g Remark s 27 3 3.1. Definitio n an d Constructio n o f Pseudorandom Function s 27 3 3.2. Application s o f Pseudorandom Function s 27 4 3.3. Concludin g Remark s 27 6

Appendix 27 9 Proof o f Theorem 2. 7 27 9

Bibliography 283 x CONTENT S

Luca Trevisan , Pseudorandomnes s — Part I I 28 7

Introduction 28 9

Lecture 1 . Deterministic Simulatio n o f Randomized Algorithm s 29 1 1. Probabilisti c Algorithm s versu s Deterministi c Algorithm s 29 1 2. De-randomizatio n Unde r Complexit y Assumption s 29 3 Lecture 2 . The Nisan-Wigderso n Generato r 29 7 1. Pseudorando m Generator s 29 7 2. Th e Tw o Main Theorem s 29 8 3. Error-Correctin g Code s an d Worst-Cas e t o Average-Cas e Reduction s 29 9 4. Th e Nisan-Wigderso n Constructio n 30 0

Lecture 3 . Analysis o f the Nisan-Wigderso n Generato r 30 5

Lecture 4 . Randomness Extractor s 30 9 1. Us e o f Weak Rando m Source s 30 9 2. Extractor s 31 0 3. Application s 31 0 4. A n Extractor fro m Nisan-Wigderso n 31 1

Bibliography 31 3

Salil Vadhan , Probabilisti c Proo f System s — Par t I 31 5

Lecture 1 . Interactive Proof s 31 7 1.1. Definition s 31 8 1.2. Grap h Nonisomorphis m 31 9 1.3. co-N P an d Mor e 32 1 1.4. Additiona l Topic s 32 5 1.5. Exercise s 32 7

Lecture 2 . Zero-Knowledge Proof s 33 1 2.1. Definitio n 33 1 2.2. Zero-Knowledg e Proof s fo r NP 33 2 2.3. Additiona l Topic s 33 8 2.4. Exercise s 34 1

Suggestions fo r Furthe r Readin g 34 3

Bibliography 34 5

Madhu Sudan , Probabilisticall y Checkabl e Proof s 34 9

Lecture 1 . Introduction t o PCPs 35 1 1. Overvie w 35 1 2. Definition s an d Forma l Statemen t o f Results 35 2 3. Broa d Skeleto n o f the Proo f 35 6 4. Ga p Problem s an d Polynomia l Constrain t Satisfactio n 35 6 5. Low-Degre e Testin g 35 8 6. Self-Correctio n o f Polynomials 35 9 CONTENTS x i

7. Obtainin g a Non-trivial PCP 35 9 Lecture 2 . NP-Hardness o f PCS 36 1 1. Multivariat e Polynomial s 36 1 2. Hardnes s o f Gap-PCS 36 3 3. Low-Degre e Testin g 36 7 4. Self-Correctio n 36 7

Lecture 3 . A Couple o f Digressions 36 9 1. A 3-Prover MI P fo r N P 37 0 2. NPCPCP[poly,0(l) ] 37 2 Lecture 4 . Proof Compositio n an d th e PCP Theore m 37 9 1. Wher e Ar e We? 37 9 2. Composin g the Verifier s 37 9 3. Th e PCP Theore m 38 2 4. Toward s Optima l PCPs 38 3 5. Roadma p t o the Optima l PCP 38 4

Bibliography 387 This page intentionally left blank Preface

The IAS/Park Cit y Mathematic s Institut e (PCMI ) wa s founded i n 199 1 as part o f the "Regiona l Geometr y Institute " initiativ e o f the Nationa l Scienc e Foundation . In mid 199 3 the program foun d a n institutional hom e at the Institute fo r Advance d Study (IAS ) i n Princeton, Ne w Jersey. Th e PCMI no w holds its summer program s either i n Park Cit y o r i n Princeton . The IAS/Par k Cit y Mathematic s Institut e encourage s bot h researc h an d ed - ucation i n mathematic s an d foster s interactio n betwee n th e two . Th e three-wee k summer institut e offer s program s fo r researcher s an d postdoctoral scholars , gradu - ate students, undergraduate students , hig h school teachers, mathematics educatio n researchers, and undergraduate faculty . On e o f PCMFs main goals is to make all of the participants awar e o f the total spectrum o f activities that occu r i n mathematic s education an d research : w e wish to involv e professiona l mathematician s i n educa - tion an d t o brin g moder n concept s i n mathematic s t o th e attentio n o f educators . To that en d th e summe r institut e feature s genera l session s designe d t o encourag e interaction amon g the various groups. In-yea r activities at site s around the countr y form a n integra l part o f the Hig h Schoo l Teacher Program . Each summe r a differen t topi c i s chosen a s the focu s o f the Researc h Progra m and Graduat e Summe r School . Activitie s i n the Undergraduate Progra m dea l with this topi c a s well . Lectur e note s fro m th e Graduat e Summe r Schoo l ar e bein g published eac h yea r i n this series . Th e firs t te n volume s are :

Volume 1 : Geometry and Quantum Field Theory (1991 ) Volume 2: Nonlinear Partial Differential Equations in Differential Geometry (1992 ) Volume 3 : Complex Algebraic Geometry (1993 ) Volume 4 : Gauge Theory and Four-Manifolds (1994 ) Volume 5 : Hyperbolic Equations and Frequency Interactions (1995 ) Volume 6 : Probability Theory and Applications (1996 ) Volume 7 : Symplectic Geometry and Topology (1997 ) Volume 8 : Representation Theory of Lie Groups (1998 ) Volume 9 : Arithmetic Algebraic Geometry (1999 ) Volume 10 : Computational Complexity Theory (2000 )

Future volume s fro m th e 200 1 Summer Schoo l o n Quantum Field Theory, Super- symmetry and Enumerative Geometry an d fro m the 200 2 Summer Schoo l on Auto- morphic Forms and Applications ar e i n preparation. Th e 200 3 Research Progra m

xiii XIV PREFACE and Graduat e Summe r Schoo l topi c i s Harmonic Analysis and Partial Differential Equations. Some material from the Undergraduate Program i s published as part o f the Stu- dent Mathematica l Librar y serie s o f the America n Mathematical Society . W e hope to publis h materia l fro m othe r part s o f the IAS/Par k Cit y Mathematic s Institut e in th e future . Thi s wil l includ e materia l fro m th e Hig h Schoo l Teache r Progra m and publication s documentin g th e interactiv e activitie s whic h ar e a primary focu s of the PCMI . A t th e summe r institut e lat e afternoon s ar e devote d t o seminar s o f common interes t t o al l participants . Man y dea l wit h curren t issue s i n education ; others trea t mathematica l topic s a t a leve l whic h encourage s broa d participation . The PCM I ha s als o spawned interaction s betwee n universitie s an d hig h school s a t a loca l level . W e hop e t o shar e thes e activitie s wit h a wide r audienc e i n futur e volumes. David R . Morrison , Serie s Edito r March, 200 3 This page intentionally left blank Titles i n Thi s Serie s

10 Steve n Rudic h an d Av i Wigderson , Editors , Computationa l Complexit y Theory , 200 4 9 Bria n Conra d an d Kar l Rubin , Editors , Arithmeti c Algebrai c Geometry , 200 1 8 Jeffre y Adam s an d Davi d Vogan , Editors , Representatio n Theor y o f Li e Groups , 200 0 7 Yako v Eliashber g an d Lis a Traynor , Editors , Symplecti c Geometr y an d Topology , 1999 6 Elto n P . Hs u an d S . R . S . Varadhan , Editors , Probabilit y Theor y an d Applications , 1999 5 Lui s Caffarell i an d Weina n E , Editors , Hyperboli c Equation s an d Frequenc y Interactions, 199 9 4 Rober t Friedma n an d Joh n W . Morgan , Editors , Gaug e Theor y an d th e Topolog y o f Four-Manifolds, 199 8 3 Jano s Kollar , Editor , Comple x Algebrai c Geometry , 199 7 2 Rober t Hard t an d Michae l Wolf , Editors , Nonlinea r Partia l Differentia l Equation s i n Differential Geometry , 199 6 1 Danie l S . Free d an d Kare n K . Uhlenbeck , Editors , Geometr y an d Quantu m Fiel d Theory, 199 5