Robust Pcps of Proximity and Shorter Pcps by Prahladh Harsha Bachelor of Technology (Computer Science and Engineering), Indian Institute of Technology, Madras, India
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Robust PCPs of Proximity and Shorter PCPs by Prahladh Harsha Bachelor of Technology (Computer Science and Engineering), Indian Institute of Technology, Madras, India. (1998) S.M. (Electrical Engineering and Computer Science), Massachusetts Institute of Technology, Cambridge MA. (2000) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY. September 2004 c Massachusetts Institute of Technology 2004. All rights reserved. Author Department of Electrical Engineering and Computer Science September 1, 2004 Certified by Madhu Sudan Professor of Computer Science and Engineering Thesis Supervisor Accepted by Arthur C. Smith Chairman, Departmental Committee on Graduate Students 2 Robust PCPs of Proximity and Shorter PCPs by Prahladh Harsha Submitted to the Department of Electrical Engineering and Computer Science on September 1, 2004 in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Abstract Probabilistically Checkable Proofs (PCPs) provide a format of rewriting and verifying mathematical proofs that allow efficient probabilistic verification based on probing very few bits of the rewritten proof. The celebrated PCP Theorem asserts that probing a constant number of bits suffices (in fact just 3 bits suffice). A natural question that arises in the construction of PCPs is by how much does this encoding blow up the original proof while retaining low query complexity. We continue the study of the trade-off between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size n): 1. We present PCPs of length exp(o(log log n)2) n that can be verified by making o(log log n) · Boolean queries. 2. For every ε > 0, we present PCPs of length exp(logε n) n that can be verified by making a · constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrar- ily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi-polylogarithmic in the first case (of query ε complexity o(log log n)), and 2(log n) , for any ε> 0, in the second case (of constant query complex- ity). Our techniques include the introduction of a new variant of PCPs that we call “Robust PCPs of proximity”. These new PCPs facilitate proof composition, which is a central ingredient in con- struction of PCP systems. Our main technical contribution is a construction of a “length-efficient” Robust PCP of proximity. We also obtain analogous quantitative results for locally testable codes. In addition, we intro- duce a relaxed notion of locally decodable codes, and present such codes mapping k information bits to codewords of length k1+ε, for any ε> 0. Thesis Supervisor: Madhu Sudan Title: Professor of Computer Science and Engineering 4 Credits This thesis is a result of joint work with Eli Ben-Sasson, Oded Goldreich, Madhu Sudan and Salil Vadhan. I thank all my co-authors for this collaboration. A preliminary version of the results in this thesis appeared in [BGH+04a]. A more elaborate version of these results can be found at [BGH+04b]. Some of the results mentioned in Chapter 5 (Propositions 5.3.4 and 5.3.5) are from the work of Eli Ben-Sasson and Madhu Sudan [BS04]. I am thankful to them for letting me include these results in this thesis. I would also like to thank the many people who made useful suggestions to me in my research. I would especially like to thank Eli Ben-Sasson, Irit Dinur, Oded Goldreich, Sofya Raskhodnikova, Alon Rosen, Mike Sipser, Madhu Sudan, Salil Vadhan, and Avi Wigderson. My thesis committee consisted of my advisor Madhu Sudan, Shafi Goldwasser and Mike Sipser. 5 6 Acknowledgments This thesis would not have been possible but for the valuable advice and direction provided by Madhu Sudan, Eli Ben-Sasson and Oded Goldreich. I thank my advisor, Madhu Sudan for his invaluable guidance and support. I have greatly benefited from my collaboration with Eli over the the past four years, who has always been there to discuss and clarify any matter. I thank Oded Goldreich for his advice, direction and encouragement during our interaction over the last one year. I would also like to thank the many people with whom I had useful discussions during my research. I would especially like to thank Sofya Raskhodnikova, Mike Sipser, and Salil Vadhan. I am indebted to the the efforts of my math teachers – Kotteswara Rao, Sri Ramaiah, Sheshayya Choudum and amm¯a, who instilled in me the love for mathematics. I owe a lot to the research atmosphere at CSAIL. I also thank all the members of the theory group for having created a very conducive atmosphere for me to work in. Special mention to Alantha Newman, my office-mate of several years. I also greatly benefited from a summer at the NEC Laboratories in the excellent company of Irit Dinur, Lance Fortnow, Joe Kilian, Kobbi Nissim and Ronnit Rubinfeld. Special Thanks to Mike Sipser both for patiently listening to me during various stages of my research and offering me a teaching assistantship for his excellent course “Theory of Computation”. I thank my apartment-mates over several years – Kripa Varanasi, Sridhar Ramachandran, Ra- machandran Balakrishnan, and Rajappa Tadepalli for having created a home away from home. My friends in Boston – Amit Deshpande, Krishnan Sriram and several others; have been very encour- aging and greatly helped me get over my occasional blues. Finally, no words can suffice to thank amm¯a, app¯a, Pavithra, Ramesh and Ajay for their endless encouragement and who keep me going. 7 8 To ñ Èa Àñ Àa È (amm¯a) and (app¯a) 9 10 Contents 1 Introduction 17 1.1 ComplexityTheoryviaProofs . ............... 18 1.2 ProbabilisticallyCheckableProofs . .................... 20 1.3 ContributionsofThesis . ............... 22 1.3.1 ShortPCPs..................................... ........ 22 1.3.2 WhyShortPCPs? ................................. ........ 23 1.3.3 MainResults................................... ......... 23 1.3.4 Newnotionsandmaintechniques . ............ 25 1.3.5 Applicationstocodingproblems . .............. 26 1.4 StructureofthisThesis . ............... 27 2 PCPs and variants 29 2.1 StandardPCPs.................................... ........... 30 2.2 PCPsofProximity................................. ............ 31 2.2.1 RelatedWork ................................... ........ 34 2.2.2 RelationtoPropertyTesting . .............. 35 2.3 RobustSoundness................................. ............ 36 2.4 Various observations and transformations . .................... 38 2.4.1 Queriesvs.proximity. ............ 38 2.4.2 Expectedrobustness . ........... 39 2.4.3 Non-BooleanPCPs ............................... ......... 39 2.4.4 Robustnessvs.proximity. ............. 42 3 Composition of Robust PCPs of Proximity 45 3.1 Whycomposition?................................. ............ 45 3.1.1 ProofCompositionin earlierPCP constructions . ................... 46 3.2 CompositionTheorem ........... ............. ...... ............ 48 3.3 Buildingblockforcomposition. ................. 50 3.4 Relation to Assignment Testers of Dinur and Reingold . ..................... 52 4 A constant-query, exponential-sized PCP of proximity 55 4.1 ConstantqueryPCPofproximity . ............... 55 11 I Proof of the PCP Theorem 59 5 Proof of thePCP Theorem 61 5.1 Introduction .................................... ............ 61 5.1.1 OverviewofProof............................... .......... 62 5.2 ComposingtheMainConstruct . .............. 63 5.3 RobustPCPPsforTwoProblems. .............. 64 5.3.1 LowDegreeTesting......... ............. ........ .......... 65 5.3.2 ZeroonSubcube................................. ......... 67 5.4 A robust PCPP for CIRCUIT VALUE ................................... 72 5.4.1 ArobustPCPfor CIRCUIT SATISFIABILITY ........................... 72 5.4.2 Augmentingwiththeproximitytest . .............. 78 5.4.3 ConvertingtoBinaryAlphabet . ............. 80 II ShortPCPs 83 6 Introduction 85 6.1 Introduction-MainConstruct . ................ 85 6.1.1 OutlineofthisPart ......... ............. ....... ........... 86 6.2 SavingonRandomness ........... ............. ...... ............ 86 6.3 OverviewofMainConstruct . .............. 91 7 A randomness-efficient PCP 95 7.1 Introduction .................................... ............ 95 7.2 Well-structuredBooleancircuits . ................... 96 7.3 Arithmetization ................................. .............101 7.4 ThePCPverifier ................................... ...........104 7.5 AnalysisofthePCPverifier. ...............108 8 A randomness-efficient PCP of proximity 113 8.1 Introduction .................................... ............113 8.2 ProofofTheorem8.1.1 ........... ............. ..... .............114 9 A randomness-efficient robust PCP of proximity 117 9.1 Introduction .................................... ............117 9.2 Robustnessofindividualtests . .................118 9.3 Bundling ........................................ ..........123 9.4 Robustnessoverthebinaryalphabet . .................130 9.5 Linearityofencoding ............ ............. .... ..............131 10 Putting them together: Very short PCPs with very few queries 137 10.1 MainConstruct-Recalled . ...............137 10.2 ComposingtheMainConstruct . ...............138 10.2.1 ProofofTheorem10.2.1.