Short Locally Testable Codes and Proofs: A Survey in Two Parts

We survey known results regarding locally testable codes and locally testable proofs (known as PCPs), with emphasis on the length of these constructs. Local testability refers to approximately testing large objects based on a very small number of probes, each retrieving a single bit in the representation of the object. This yields super-fast approximatetesting of the corresponding property (i.e., be a codeword or a valid proof). We also review the related concept of local decodable codes. The survey consists of two independent (i.e., self-contained) parts that cover the same material at different levels of rigor and detail. Still, in spite of the repetitions, there may be a benefit in reading both parts.

[1]  Yuval Ishai,et al.  Breaking the O(n/sup 1/(2k-1)/) barrier for information-theoretic Private Information Retrieval , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

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

[3]  Mihir Bellare,et al.  Free Bits, PCPs, and Nonapproximability-Towards Tight Results , 1998, SIAM J. Comput..

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

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

[6]  D. Spielman,et al.  Computationally efficient error-correcting codes and holographic proofs , 1995 .

[7]  Lance Fortnow,et al.  On the Power of Multi-Prover Interactive Protocols , 1994, Theor. Comput. Sci..

[8]  Simon Litsyn,et al.  Breaking the Epsilon-Soundness Bound of the Linearity Test over GF(2) , 2008, SIAM J. Comput..

[9]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

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

[11]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

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

[13]  Sorin C. Popescu,et al.  Lidar Remote Sensing , 2011 .

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

[15]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[16]  Sergey Yekhanin,et al.  Towards 3-query locally decodable codes of subexponential length , 2008, JACM.

[17]  J. Davenport.  Editor , 1960 .

[18]  Ronitt Rubinfeld,et al.  Fast approximate PCPs , 1999, STOC '99.

[19]  Ronitt Rubinfeld,et al.  Self-testing/correcting for polynomials and for approximate functions , 1991, STOC '91.

[20]  Adi Shamir,et al.  Fully Parallelized Multi-Prover Protocols for NEXP-Time , 1997, J. Comput. Syst. Sci..

[21]  Adi Shamir,et al.  Fully parallelized multi prover protocols for NEXP-time , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[22]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[23]  Madhu Sudan,et al.  Some improvements to total degree tests , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[24]  Luca Trevisan,et al.  Lower bounds for linear locally decodable codes and private information retrieval , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

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

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

[27]  Simon Litsyn,et al.  Breaking the ε-Soundness Bound of the Linearity Test over GF(2) , 2007, Electron. Colloquium Comput. Complex..

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

[29]  Sanjeev Arora.  Probabilistic checking of proofs and hardness of approximation problems , 1995 .

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

[31]  Oded Goldreich,et al.  Computational complexity: a conceptual perspective , 2008, SIGA.

[32]  Dana Ron,et al.  On proximity oblivious testing , 2009, STOC '09.

[33]  Or Meir.  Combinatorial Construction of Locally Testable Codes , 2009, SIAM J. Comput..

[34]  J. Håstad.  Clique is hard to approximate within n 1-C , 1996 .

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

[36]  Or Meir,et al.  Combinatorial PCPs with Efficient Verifiers , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

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

[38]  Boaz Barak,et al.  How to go beyond the black-box simulation barrier , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[40]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[41]  Venkatesan Guruswami,et al.  Locally Testable Codes Require Redundant Testers , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[42]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[43]  Oded Goldreich,et al.  Short Locally Testable Codes and Proofs (Survey) , 2005, Electron. Colloquium Comput. Complex..

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

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

[46]  Eyal Kushilevitz,et al.  Private information retrieval , 1998, JACM.

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

[48]  Irit Dinur,et al.  Composition of Low-Error 2-Query PCPs Using Decodable PCPs , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[49]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[50]  G. David Forney,et al.  Concatenated codes , 2009, Scholarpedia.

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

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

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

[54]  N. Meyers,et al.  H = W. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[55]  Johan Håstad,et al.  Clique is hard to approximate within n/sup 1-/spl epsiv// , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[56]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[57]  Eli Ben-Sasson,et al.  Short PCPs verifiable in polylogarithmic time , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[58]  Ran Canetti,et al.  The random oracle methodology, revisited , 2000, JACM.

[59]  Eli Ben-Sasson,et al.  Robust locally testable codes and products of codes , 2004, Random Struct. Algorithms.

[60]  Ran Canetti,et al.  The random oracle methodology, revisited , 2000, JACM.

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

[62]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[63]  Noga Alon,et al.  Testing Low-Degree Polynomials over GF(2( , 2003, RANDOM-APPROX.

[64]  Mihir Bellare,et al.  Improved non-approximability results , 1994, STOC '94.

[65]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

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

[67]  Andris Ambainis,et al.  On Lower Bounds for the Communication Complexity of Private Information Retrieval ∗ , 2000 .

[68]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

[69]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[70]  Dana Ron,et al.  Property Testing , 2000 .

[71]  M. Bellare,et al.  Efficient probabilistic checkable proofs and applications to approximation , 1994, STOC '94.