Sound 3-Query PCPPs Are Long

We initiate the study of the trade-off between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3-query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot obtain the same soundness as that obtained by a verifier querying a long proof. Moreover, we quantify the soundness deficiency as a function of the proof-length and show that any verifier obtaining “best possible” soundness must query an exponentially long proof. In terms of techniques, we focus on the special class of inspective verifiers that read at most 2 proof-bits per invocation. For such verifiers, we prove exponential length-soundness trade-offs that are later on used to imply our main results for the case of general (i.e., not necessarily inspective) verifiers. To prove the exponential trade-off for inspective verifiers, we show a connection between PCPP proof length and property-testing query complexity that may be of independent interest. The connection is that any linear property that can be verified with proofs of length ℓ by linear inspective verifiers must be testable with query complexity ≈ logℓ.

[1]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[2]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[3]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[4]  Ronitt Rubinfeld,et al.  Fast approximate probabilistically checkable proofs , 2004, Inf. Comput..

[5]  Ran Raz,et al.  Two Query PCP with Sub-Constant Error , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Ran Raz,et al.  Sub-constant error low degree test of almost-linear size , 2006, STOC '06.

[7]  Luca Trevisan,et al.  Approximation algorithms for unique games , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[8]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[9]  Eli Ben-Sasson,et al.  Sound 3-Query PCPPs Are Long , 2008, ICALP.

[10]  Eldar Fischer,et al.  The Art of Uninformed Decisions , 2001, Bull. EATCS.

[11]  Eli Ben-Sasson,et al.  Some 3CNF properties are hard to test , 2003, STOC '03.

[12]  Uriel Feige,et al.  Impossibility results for recycling random bits in two-prover proof systems , 1995, STOC '95.

[13]  Michael Horstein,et al.  Review of 'Low-Density Parity-Check Codes' (Gallager, R. G.; 1963) , 1964, IEEE Transactions on Information Theory.

[14]  Silvio Micali,et al.  Computationally Sound Proofs , 2000, SIAM J. Comput..

[15]  David P. Woodruff.  Corruption and Recovery-Efficient Locally Decodable Codes , 2008, APPROX-RANDOM.

[16]  Anupam Gupta,et al.  Approximating unique games , 2006, SODA '06.

[17]  Eli Ben-Sasson,et al.  Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding , 2004, SIAM J. Comput..

[18]  Oded Goldreich,et al.  Locally testable codes and PCPs of almost-linear length , 2006, JACM.

[19]  E. Fischer.  THE ART OF UNINFORMED DECISIONS: A PRIMER TO PROPERTY TESTING , 2004 .

[20]  Daniel A. Spielman,et al.  Nearly-linear size holographic proofs , 1994, STOC '94.

[21]  Subhash Khot,et al.  Query efficient PCPs with perfect completeness , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[22]  Luca Trevisan,et al.  Gowers uniformity, influence of variables, and PCPs , 2005, STOC '06.

[23]  Madhu Sudan,et al.  Small PCPs with low query complexity , 2000, computational complexity.

[24]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[25]  Venkatesan Guruswami,et al.  Tolerant Locally Testable Codes , 2005, APPROX-RANDOM.

[26]  Lance Fortnow,et al.  Tolerant Versus Intolerant Testing for Boolean Properties , 2005, Computational Complexity Conference.

[27]  Moses Charikar,et al.  Near-optimal algorithms for unique games , 2006, STOC '06.

[28]  Luca Trevisan,et al.  A PCP characterization of NP with optimal amortized query complexity , 2000, STOC '00.

[29]  Mario Szegedy,et al.  Many-Valued Logics and Holographic Proofs , 1999, ICALP.

[30]  Uri Zwick,et al.  Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint , 1998, SODA '98.

[31]  Mihir Bellare,et al.  Linearity testing in characteristic two , 1996, IEEE Trans. Inf. Theory.

[32]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[33]  E. Ben-Sasson,et al.  CNF Properties are Hard to Test , 2002 .

[34]  Eli Ben-Sasson,et al.  Robust pcps of proximity, shorter pcps and applications to coding , 2004, STOC '04.

[35]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[36]  Sanjeev Arora,et al.  Probabilistic checking of proofs; a new characterization of NP , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[37]  Eli Ben-Sasson,et al.  Randomness-efficient low degree tests and short PCPs via epsilon-biased sets , 2003, STOC '03.

[38]  Subhash Khot,et al.  A 3-query non-adaptive PCP with perfect completeness , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[39]  Joe Kilian,et al.  A note on efficient zero-knowledge proofs and arguments (extended abstract) , 1992, STOC '92.

[40]  Eldar Fischer,et al.  A Review of Graph Grammars and Preview of ICGT 2002: The First International Conference on Graph Transformation. , 2001 .

[41]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[42]  Eli Ben-Sasson,et al.  Bounds on 2-Query Codeword Testing , 2003, RANDOM-APPROX.

[43]  Eli Ben-Sasson,et al.  Short PCPs with Polylog Query Complexity , 2008, SIAM J. Comput..

[44]  Ronitt Rubinfeld,et al.  Tolerant property testing and distance approximation , 2006, J. Comput. Syst. Sci..

[45]  Irit Dinur,et al.  The PCP theorem by gap amplification , 2006, STOC.

[46]  Venkatesan Guruswami.  On 2-Query Codeword Testing with Near-Perfect Completeness , 2006, ISAAC.

[47]  Jonas Holmerin,et al.  More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP , 2008, Random Struct. Algorithms.

[48]  Ran Raz,et al.  Sub-Constant Error Probabilistically Checkable Proof of Almost-Linear Size , 2010, computational complexity.

[49]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[50]  Venkatesan Guruswami,et al.  A tight characterization of NP with 3 query PCPs , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[51]  Eli Ben-Sasson,et al.  Simple PCPs with poly-log rate and query complexity , 2005, STOC '05.

[52]  Omer Reingold,et al.  Assignment testers: towards a combinatorial proof of the PCP-theorem , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.