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Excluding PH Pessiland

Authors Shuichi Hirahara, Rahul Santhanam

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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • average-case complexity
  • pseudorandomness
  • meta-complexity

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Abstract

Heuristica and Pessiland are "worlds" of average-case complexity [Impagliazzo95] that are considered unlikely but that current techniques are unable to rule out. Recently, [Hirahara20] considered a PH (Polynomial Hierarchy) analogue of Heuristica, and showed that to rule it out, it would be sufficient to prove the NP-completeness of the problem GapMINKT^PH of estimating the PH-oracle time-bounded Kolmogorov complexity of a string.
In this work, we analogously define "PH Pessiland" to be a world where PH is hard on average but PH-computable pseudo-random generators do not exist. We unconditionally rule out PH-Pessiland in both non-uniform and uniform settings, by showing that the distributional problem of computing PH-oracle time-bounded Kolmogorov complexity of a string over the uniform distribution is complete for an (error-prone) average-case analogue of PH. Moreover, we show the equivalence between error-prone average-case hardness of PH and the existence of PH-computable pseudorandom generators.

Cite As Get BibTex

Shuichi Hirahara and Rahul Santhanam. Excluding PH Pessiland. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 85:1-85:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.85

Author Details

Shuichi Hirahara
  • Principle of Informatics Research Division, National Institute of Informatics, Tokyo, Japan
Rahul Santhanam
  • Department of Computer Science, University of Oxford, UK

Funding

  • Hirahara, Shuichi: Supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Santhanam, Rahul: Supported by ESPRC New Horizons Grant No. EP/V048201/1 on "Structure vs Randomness in Algorithms and Computation", UK.

References

  1. Adi Akavia, Oded Goldreich, Shafi Goldwasser, and Dana Moshkovitz. On basing one-way functions on NP-hardness. In Proceedings of the Symposium on Theory of Computing (STOC), pages 701-710, 2006. URL: https://doi.org/10.1145/1132516.1132614.
  2. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP Has Subexponential Time Simulations Unless EXPTIME has Publishable Proofs. Computational Complexity, 3:307-318, 1993. URL: https://doi.org/10.1007/BF01275486.
  3. Shai Ben-David, Benny Chor, Oded Goldreich, and Michael Luby. On the Theory of Average Case Complexity. J. Comput. Syst. Sci., 44(2):193-219, 1992. URL: https://doi.org/10.1016/0022-0000(92)90019-F.
  4. Manuel Blum and Silvio Micali. How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. SIAM J. Comput., 13(4):850-864, 1984. URL: https://doi.org/10.1137/0213053.
  5. Andrej Bogdanov and Christina Brzuska. On Basing Size-Verifiable One-Way Functions on NP-Hardness. In Proceedings of the Theory of Cryptography Conference (TCC), pages 1-6, 2015. URL: https://doi.org/10.1007/978-3-662-46494-6_1.
  6. Andrej Bogdanov and Luca Trevisan. Average-Case Complexity. Foundations and Trends in Theoretical Computer Science, 2(1), 2006. URL: https://doi.org/10.1561/0400000004.
  7. Andrej Bogdanov and Luca Trevisan. On Worst-Case to Average-Case Reductions for NP Problems. SIAM J. Comput., 36(4):1119-1159, 2006. URL: https://doi.org/10.1137/S0097539705446974.
  8. Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning Algorithms from Natural Proofs. In Proceedings of the Conference on Computational Complexity (CCC), pages 10:1-10:24, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.10.
  9. Larry Carter and Mark N. Wegman. Universal Classes of Hash Functions. J. Comput. Syst. Sci., 18(2):143-154, 1979. URL: https://doi.org/10.1016/0022-0000(79)90044-8.
  10. Joan Feigenbaum and Lance Fortnow. Random-Self-Reducibility of Complete Sets. SIAM J. Comput., 22(5):994-1005, 1993. URL: https://doi.org/10.1137/0222061.
  11. Oded Goldreich, Noam Nisan, and Avi Wigderson. On Yao’s XOR-Lemma. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 273-301. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_23.
  12. Johan Håstad, Russell Impagliazzo, Leonid A. Levin, and Michael Luby. A Pseudorandom Generator from any One-way Function. SIAM J. Comput., 28(4):1364-1396, 1999. URL: https://doi.org/10.1137/S0097539793244708.
  13. Shuichi Hirahara. Identifying an Honest EXP^NP Oracle Among Many. In Proceedings of the Conference on Computational Complexity (CCC), pages 244-263, 2015. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.244.
  14. Shuichi 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. Google Scholar
  15. Shuichi Hirahara. Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 50-60, 2020. Google Scholar
  16. Shuichi Hirahara. Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions. In Proceedings of the Computational Complexity Conference (CCC), pages 20:1-20:47, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.20.
  17. Shuichi Hirahara. Unexpected hardness results for Kolmogorov complexity under uniform reductions. In Proceedings of the Symposium on Theory of Computing (STOC), pages 1038-1051, 2020. URL: https://doi.org/10.1145/3357713.3384251.
  18. Shuichi Hirahara. Average-Case Hardness of NP from Exponential Worst-Case Hardness Assumptions. Electron. Colloquium Comput. Complex., 28:58, 2021. To appear in STOC 2021. URL: https://eccc.weizmann.ac.il/report/2021/058.
  19. Shuichi Hirahara and Rahul Santhanam. On the Average-Case Complexity of MCSP and Its Variants. In Proceedings of the Computational Complexity Conference (CCC), pages 7:1-7:20, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.7.
  20. Rahul Ilango. Approaching MCSP from Above and Below: Hardness for a Conditional Variant and AC⁰[p]. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 34:1-34:26, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.34.
  21. Rahul 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. Google Scholar
  22. Rahul Ilango, Bruno Loff, and Igor Carboni Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In Proceedings of the Computational Complexity Conference (CCC), pages 22:1-22:36, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.22.
  23. Russell Impagliazzo. A Personal View of Average-Case Complexity. In Proceedings of the Structure in Complexity Theory Conference, pages 134-147, 1995. URL: https://doi.org/10.1109/SCT.1995.514853.
  24. Russell Impagliazzo. Hard-Core Distributions for Somewhat Hard Problems. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 538-545, 1995. URL: https://doi.org/10.1109/SFCS.1995.492584.
  25. Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets, and Avi Wigderson. Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized. SIAM J. Comput., 39(4):1637-1665, 2010. URL: https://doi.org/10.1137/080734030.
  26. Russell Impagliazzo and Leonid A. Levin. No Better Ways to Generate Hard NP Instances than Picking Uniformly at Random. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 812-821, 1990. URL: https://doi.org/10.1109/FSCS.1990.89604.
  27. Russell Impagliazzo and Michael Luby. One-way Functions are Essential for Complexity Based Cryptography (Extended Abstract). In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 230-235, 1989. URL: https://doi.org/10.1109/SFCS.1989.63483.
  28. Russell Impagliazzo and Avi 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: https://doi.org/10.1145/258533.258590.
  29. Russell Impagliazzo and Avi Wigderson. Randomness vs Time: Derandomization under a Uniform Assumption. J. Comput. Syst. Sci., 63(4):672-688, 2001. URL: https://doi.org/10.1006/jcss.2001.1780.
  30. Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In Proceedings of the Symposium on Theory of Computing (STOC), pages 73-79, 2000. URL: https://doi.org/10.1145/335305.335314.
  31. Richard M. Karp and Richard J. Lipton. Turing machines that take advice. L'Enseignement Mathématique, 28:191-209, 1982. Google Scholar
  32. Ker-I Ko. On the Complexity of Learning Minimum Time-Bounded Turing Machines. SIAM J. Comput., 20(5):962-986, 1991. URL: https://doi.org/10.1137/0220059.
  33. Yanyi Liu and Rafael Pass. On One-way Functions and Kolmogorov Complexity. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1243-1254, 2020. Google Scholar
  34. Yanyi Liu and Rafael Pass. On One-way Functions from NP-Complete Problems. Electron. Colloquium Comput. Complex., 28:59, 2021. URL: https://eccc.weizmann.ac.il/report/2021/059.
  35. Andrei A. Muchnik and Nikolai K. Vereshchagin. Shannon Entropy vs. Kolmogorov Complexity. In Computer Science - Theory and Applications, First International Computer Science Symposium in Russia, CSR 2006, St. Petersburg, Russia, June 8-12, 2006, Proceedings, pages 281-291, 2006. URL: https://doi.org/10.1007/11753728_29.
  36. Noam Nisan and Avi Wigderson. Hardness vs Randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  37. Aduri Pavan, Rahul Santhanam, and N. V. Vinodchandran. Some Results on Average-Case Hardness Within the Polynomial Hierarchy. In Proceedings of the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 188-199, 2006. URL: https://doi.org/10.1007/11944836_19.
  38. Rahul 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: https://doi.org/10.4230/LIPIcs.ITCS.2020.68.
  39. Rainer Schuler and Osamu Watanabe. Towards Average-Case Complexity Analysis of NP Optimization Problems. In Proceedings of the Structure in Complexity Theory Conference, pages 148-159, 1995. URL: https://doi.org/10.1109/SCT.1995.514854.
  40. Larry J. Stockmeyer. The Complexity of Approximate Counting (Preliminary Version). In Proceedings of the Symposium on Theory of Computing (STOC), pages 118-126, 1983. URL: https://doi.org/10.1145/800061.808740.
  41. Luca Trevisan and Salil P. Vadhan. Pseudorandomness and Average-Case Complexity Via Uniform Reductions. Computational Complexity, 16(4):331-364, 2007. URL: https://doi.org/10.1007/s00037-007-0233-x.
  42. Salil P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  43. Leslie G. Valiant and Vijay V. Vazirani. NP is as Easy as Detecting Unique Solutions. Theor. Comput. Sci., 47(3):85-93, 1986. URL: https://doi.org/10.1016/0304-3975(86)90135-0.
  44. Emanuele Viola. On Constructing Parallel Pseudorandom Generators from One-Way Functions. In Proceedings of the Conference on Computational Complexity (CCC), pages 183-197, 2005. URL: https://doi.org/10.1109/CCC.2005.16.
  45. Emanuele Viola. The complexity of constructing pseudorandom generators from hard functions. Computational Complexity, 13(3-4):147-188, 2005. URL: https://doi.org/10.1007/s00037-004-0187-1.
  46. Emanuele Viola. The Sum of D Small-Bias Generators Fools Polynomials of Degree D. Computational Complexity, 18(2):209-217, 2009. URL: https://doi.org/10.1007/s00037-009-0273-5.
  47. Andrew Chi-Chih Yao. Theory and Applications of Trapdoor Functions (Extended Abstract). In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 80-91, 1982. URL: https://doi.org/10.1109/SFCS.1982.45.
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