Regularization of Low Error PCPs and an Application to MCSP

Authors Shuichi Hirahara, Dana Moshkovitz

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Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Dana Moshkovitz
  • Department of Computer Science, University of Texas at Austin, TX, USA


We are thankful to Dean Doron for discussions about explicit construction of dispersers.

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Shuichi Hirahara and Dana Moshkovitz. Regularization of Low Error PCPs and an Application to MCSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In a regular PCP the verifier queries each proof symbol in the same number of tests. This number is called the degree of the proof, and it is at least 1/(sq) where s is the soundness error and q is the number of queries. It is incredibly useful to have regularity and reduced degree in PCP. There is an expander-based transformation by Papadimitriou and Yannakakis that transforms any PCP with a constant number of queries and constant soundness error to a regular PCP with constant degree. There are also transformations for low error projection and unique PCPs. Other PCPs are constructed especially to be regular. In this work we show how to regularize and reduce degree of PCPs with a possibly large number of queries and low soundness error. As an application, we prove NP-hardness of an unweighted variant of the collective minimum monotone satisfying assignment problem, which was introduced by Hirahara (FOCS'22) to prove NP-hardness of MCSP^* (the partial function variant of the Minimum Circuit Size Problem) under randomized reductions. We present a simplified proof and sufficient conditions under which MCSP^* is NP-hard under the standard notion of reduction: MCSP^* is NP-hard under deterministic polynomial-time many-one reductions if there exists a function in E that satisfies certain direct sum properties.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • PCP theorem
  • regularization
  • Minimum Circuit Size Problem


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  1. E. Allender and S. Hirahara. New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems. TOCT, 11(4):27:1-27:27, 2019. URL:
  2. J. Alman and L. Chen. Efficient construction of rigid matrices using an NP oracle. In Proc. 60th IEEE Symp. on Foundations of Computer Science, pages 1034-1055, 2019. Google Scholar
  3. A. Beimel. Secret-Sharing Schemes: A Survey. In The Third International Workshop on Coding and Cryptology (IWCC), pages 11-46, 2011. URL:
  4. M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient probabilistically checkable proofs and applications to approximations. In Proc. 25th ACM Symp. on Theory of Computing, pages 294-304, 1993. Google Scholar
  5. E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM Journal on Computing, 36(4):889-974, 2006. Google Scholar
  6. J. C. Benaloh and J. Leichter. Generalized Secret Sharing and Monotone Functions. In Proceedings of the International Cryptology Conference (CRYPTO), pages 27-35, 1988. URL:
  7. M. L. Carmosino, R. Impagliazzo, V. Kabanets, and A. Kolokolova. Learning Algorithms from Natural Proofs. In Proceedings of the Conference on Computational Complexity (CCC), pages 10:1-10:24, 2016. URL:
  8. L. Chen, S. Hirahara, I. C. Oliveira, J. Pich, N. Rajgopal, and R. Santhanam. Beyond Natural Proofs: Hardness Magnification and Locality. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 70:1-70:48, 2020. URL:
  9. J. Cook and D. Moshkovitz. Tighter MA/1 circuit lower bounds from verifier efficient PCPs for PSPACE. Technical Report TR22-014, ECCC, 2022. Google Scholar
  10. I. Dinur, E. Fischer, G. Kindler, R. Raz, and S. Safra. PCP characterizations of NP: Toward a polynomially-small error-probability. Computational Complexity, 20(3):413-504, 2011. Google Scholar
  11. I. Dinur and P. Harsha. Composition of low-error 2-query PCPs using decodable PCPs. In Proc. 50th IEEE Symp. on Foundations of Computer Science, pages 472-481, 2009. Google Scholar
  12. I. Dinur, P. Harsha, and G. Kindler. Polynomially low error pcps with polyloglog n queries via modular composition. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proc. 47th ACM Symp. on Theory of Computing, pages 267-276. ACM, 2015. Google Scholar
  13. I. Dinur and D. Steurer. Analytical approach to parallel repetition. In Proc. 46th ACM Symp. on Theory of Computing, 2014. Google Scholar
  14. U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268-292, 1996. Google Scholar
  15. J. Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001. Google Scholar
  16. A. Healy, S. P. Vadhan, and E. Viola. Using Nondeterminism to Amplify Hardness. SIAM J. Comput., 35(4):903-931, 2006. URL:
  17. S. Hirahara. Non-Black-Box Worst-Case to Average-Case Reductions within NP. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 247-258, 2018. URL:
  18. S. Hirahara. NP-Hardness of Learning Programs and Partial MCSP. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 968-979, 2022. URL:
  19. S. Hirahara and O. Watanabe. Limits of Minimum Circuit Size Problem as Oracle. In Proceedings of the Conference on Computational Complexity (CCC), pages 18:1-18:20, 2016. URL:
  20. J. M. Hitchcock and A. Pavan. On the NP-Completeness of the Minimum Circuit Size Problem. In Proceedings of the Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS), pages 236-245, 2015. URL:
  21. J. Holmerin and S. Khot. A new PCP outer verifier with applications to homogeneous linear equations and max-bisection. In Proc. 36th ACM Symp. on Theory of Computing, pages 11-20, 2004. Google Scholar
  22. R. Ilango. Constant Depth Formula and Partial Function Versions of MCSP are Hard. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 424-433, 2020. URL:
  23. R. Impagliazzo and A. Wigderson. P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the Symposium on the Theory of Computing (STOC), pages 220-229, 1997. URL:
  24. M. Ito, A. Saito, and T. Nishizeki. Multiple Assignment Scheme for Sharing Secret. J. Cryptol., 6(1):15-20, 1993. URL:
  25. Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen. MIP* = RE. Submitted, 2020. Google Scholar
  26. V. Kabanets and J. Cai. Circuit minimization problem. In Proceedings of the Symposium on Theory of Computing (STOC), pages 73-79, 2000. URL:
  27. J. Katz and L. Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proc. 32nd ACM Symp. on Theory of Computing, pages 80-86, 2000. Google Scholar
  28. S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2-epsilon. Journal of Computer and System Sciences, 74(3):335-349, 2008. Google Scholar
  29. S. Khot and N. K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l_1. In Proc. 46th IEEE Symp. on Foundations of Computer Science, pages 53-62, 2005. Google Scholar
  30. J. Kilian. A note on efficient zero-knowledge proofs and arguments (extended abstract. In Proc. 24th ACM Symp. on Theory of Computing, pages 723-732, 1992. Google Scholar
  31. A. R. Klivans and D. van Melkebeek. Graph Nonisomorphism Has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL:
  32. L. A. Levin. Universal sequential search problems. Problemy Peredachi Informatsii, 9(3):115-116, 1973. Google Scholar
  33. Y. Liu and R. Pass. On One-way Functions and Kolmogorov Complexity. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1243-1254, 2020. URL:
  34. D. Moshkovitz and R. Raz. Two query PCP with sub-constant error. Journal of the ACM, 57(5), 2010. Google Scholar
  35. N. Nisan and A. Wigderson. Hardness vs Randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL:
  36. I. C. Oliveira and R. Santhanam. Hardness Magnification for Natural Problems. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 65-76, 2018. Google Scholar
  37. C. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes. Journal of Computer and System Sciences, 43:425-440, 1991. Google Scholar
  38. O. Paradise. Smooth and strong pcps. Comput. Complex., 30(1):1, 2021. Google Scholar
  39. O. Reingold, S. P. Vadhan, and A. Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. Annals of Mathematics, 155(1):157-187, 2002. Google Scholar
  40. H. Ren, R. Santhanam, and Z. Wang. On the Range Avoidance Problem for Circuits. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 640-650, 2022. URL:
  41. M. Saks and R. Santhanam. Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions. In Proceedings of the Computational Complexity Conference (CCC), pages 26:1-26:13, 2020. URL:
  42. R. Santhanam. Pseudorandomness and the Minimum Circuit Size Problem. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 68:1-68:26, 2020. URL:
  43. R. Shaltiel and C. Umans. Pseudorandomness for Approximate Counting and Sampling. Computational Complexity, 15(4):298-341, 2006. URL:
  44. L. Trevisan. Extractors and pseudorandom generators. J. ACM, 48(4):860-879, 2001. URL:
  45. D. Uhlig. On the synthesis of self-correcting schemes from functional elements with a small number of reliable elements. Mathematical Notes of the Academy of Sciences of the USSR, 15(6):558-562, 1974. Google Scholar
  46. D. Uhlig. Networks Computing Boolean Functions for Multiple Input Values. In Poceedings of the London Mathematical Society Symposium on Boolean Function Complexity, pages 165-173, USA, 1992. Cambridge University Press. Google Scholar
  47. S. P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. Google Scholar
  48. I. Wegener. The complexity of Boolean functions. Wiley-Teubner, 1987. URL:
  49. R. Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM Journal on Computing, 42:231-240, 2010. Google Scholar
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