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Symmetry of Information from Meta-Complexity

Author Shuichi Hirahara

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Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan

Funding

JST, PRESTO Grant Number JPMJPR2024, Japan.

Acknowledgements

I thank Mikito Nanashima for helpful discussion, Rahul Santhanam for raising the question of finding an equivalent notion of the weak universal heuristic scheme, which inspired Lemma 7.5, and anonymous reviewers for strongly inspiring me to include Appendix A.1.

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Shuichi Hirahara. Symmetry of Information from Meta-Complexity. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 26:1-26:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.26

Abstract

Symmetry of information for time-bounded Kolmogorov complexity is a hypothetical inequality that relates time-bounded Kolmogorov complexity and its conditional analogue. In 1992, Longpré and Watanabe showed that symmetry of information holds if NP is easy in the worst case, which has been the state of the art over the last three decades. In this paper, we significantly improve this result by showing that symmetry of information holds under the weaker assumption that NP is easy on average. In fact, our proof techniques are applicable to any resource-bounded Kolmogorov complexity and enable proving symmetry of information from an efficient algorithm that computes resource-bounded Kolmogorov complexity. We demonstrate the significance of our proof techniques by presenting two applications. First, using that symmetry of information does not hold for Levin’s Kt-complexity, we prove that randomized Kt-complexity cannot be computed in time 2^o(n) on inputs of length n, which improves the previous quasi-polynomial lower bound of Oliveira (ICALP 2019). Our proof implements Kolmogorov’s insightful approach to the P versus NP problem in the case of randomized Kt-complexity. Second, we consider the question of excluding Heuristica, i.e., a world in which NP is easy on average but NP ≠ P, from Impagliazzo’s five worlds: Using symmetry of information, we prove that Heuristica is excluded if the problem of approximating time-bounded conditional Kolmogorov complexity K^t(x∣y) up to some additive error is NP-hard for t ≫ |y|. We complement this result by proving NP-hardness of approximating sublinear-time-bounded conditional Kolmogorov complexity up to a multiplicative factor of |x|^{1/(log log |x|)^O(1)} for t ≪ |y|. Our NP-hardness proof presents a new connection between sublinear-time-bounded conditional Kolmogorov complexity and a secret sharing scheme.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • resource-bounded Kolmogorov complexity
  • average-case complexity
  • pseudorandomness
  • hardness of approximation
  • unconditional lower bound

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References

  1. Dorit Aharonov and Oded Regev. Lattice problems in NP ∩ coNP. J. ACM, 52(5):749-765, 2005. URL: https://doi.org/10.1145/1089023.1089025.
  2. Miklós Ajtai. Generating Hard Instances of Lattice Problems (Extended Abstract). In Proceedings of the Symposium on the Theory of Computing (STOC), pages 99-108, 1996. URL: https://doi.org/10.1145/237814.237838.
  3. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from Random Strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL: https://doi.org/10.1137/050628994.
  4. Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-Way Functions and a Conditional Variant of MKTP. In Proceedings of the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 7:1-7:19, 2021. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2021.7.
  5. Eric Allender, John Gouwar, Shuichi Hirahara, and Caleb Robelle. Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity. In Proceedings of the International Symposium on Algorithms and Computation (ISAAC), pages 54:1-54:17, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.54.
  6. Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan. Minimum Circuit Size, Graph Isomorphism, and Related Problems. SIAM J. Comput., 47(4):1339-1372, 2018. URL: https://doi.org/10.1137/17M1157970.
  7. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing Disjunctive Normal Form Formulas and AC⁰ Circuits Given a Truth Table. SIAM J. Comput., 38(1):63-84, 2008. URL: https://doi.org/10.1137/060664537.
  8. Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. J. Comput. Syst. Sci., 77(1):14-40, 2011. URL: https://doi.org/10.1016/j.jcss.2010.06.004.
  9. Luis Antunes, Lance Fortnow, Dieter van Melkebeek, and N. V. Vinodchandran. Computational depth: Concept and applications. Theor. Comput. Sci., 354(3):391-404, 2006. URL: https://doi.org/10.1016/j.tcs.2005.11.033.
  10. Luis Antunes, Sophie Laplante, Alexandre Pinto, and Liliana C. M. Salvador. Cryptographic Security of Individual Instances. In Proceedings of Information Theoretic Security (ICITS), pages 195-210, 2007. URL: https://doi.org/10.1007/978-3-642-10230-1_17.
  11. Luis Filipe Coelho Antunes and Lance Fortnow. Worst-Case Running Times for Average-Case Algorithms. In Proceedings of the Conference on Computational Complexity (CCC), pages 298-303, 2009. URL: https://doi.org/10.1109/CCC.2009.12.
  12. Amos Beimel. Secret-Sharing Schemes: A Survey. In The Third International Workshop on Coding and Cryptology (IWCC), pages 11-46, 2011. URL: https://doi.org/10.1007/978-3-642-20901-7_2.
  13. Josh Cohen Benaloh and Jerry Leichter. Generalized Secret Sharing and Monotone Functions. In Proceedings of the International Cryptology Conference (CRYPTO), pages 27-35, 1988. URL: https://doi.org/10.1007/0-387-34799-2_3.
  14. George Robert Blakley. Safeguarding cryptographic keys. In International Workshop on Managing Requirements Knowledge (MARK), pages 313-318, 1979. URL: https://doi.org/10.1109/MARK.1979.8817296.
  15. Manuel Blum and Sampath Kannan. Designing Programs that Check Their Work. J. ACM, 42(1):269-291, 1995. URL: https://doi.org/10.1145/200836.200880.
  16. 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.
  17. 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.
  18. Harry Buhrman, Lance Fortnow, and Aduri Pavan. Some Results on Derandomization. Theory Comput. Syst., 38(2):211-227, 2005. URL: https://doi.org/10.1007/s00224-004-1194-y.
  19. Harry Buhrman and Elvira Mayordomo. An Excursion to the Kolmogorov Random Strings. J. Comput. Syst. Sci., 54(3):393-399, 1997. URL: https://doi.org/10.1006/jcss.1997.1484.
  20. Lijie Chen, Shuichi Hirahara, and Neekon Vafa. Average-case Hardness of NP and PH from Worst-case Fine-grained Assumptions. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 67:1-67:17, 2022. Google Scholar
  21. Irit Dinur, Prahladh Harsha, and Guy Kindler. Polynomially Low Error PCPs with polyloglog n Queries via Modular Composition. In Proceedings of the Symposium on Theory of Computing (STOC), pages 267-276, 2015. URL: https://doi.org/10.1145/2746539.2746630.
  22. Irit Dinur and Shmuel Safra. On the hardness of approximating label-cover. Inf. Process. Lett., 89(5):247-254, 2004. URL: https://doi.org/10.1016/j.ipl.2003.11.007.
  23. 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.
  24. Lance Fortnow and Martin Kummer. On Resource-Bounded Instance Complexity. Theor. Comput. Sci., 161(1&2):123-140, 1996. URL: https://doi.org/10.1016/0304-3975(95)00097-6.
  25. Lance Fortnow, Troy Lee, and Nikolai K. Vereshchagin. Kolmogorov Complexity with Error. In STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23-25, 2006, Proceedings, pages 137-148, 2006. URL: https://doi.org/10.1007/11672142_10.
  26. Lance Fortnow and Nick Reingold. PP is Closed Under Truth-Table Reductions. Inf. Comput., 124(1):1-6, 1996. URL: https://doi.org/10.1006/inco.1996.0001.
  27. Eran Gat and Shafi Goldwasser. Probabilistic Search Algorithms with Unique Answers and Their Cryptographic Applications. Electron. Colloquium Comput. Complex., page 136, 2011. URL: https://eccc.weizmann.ac.il/report/2011/136.
  28. Halley Goldberg and Valentine Kabanets. A Simpler Proof of the Worst-Case to Average-Case Reduction for Polynomial Hierarchy via Symmetry of Information. Electronic Colloquium on Computational Complexity (ECCC), 007, 2022. Google Scholar
  29. Oded Goldreich and Shafi Goldwasser. On the Limits of Nonapproximability of Lattice Problems. J. Comput. Syst. Sci., 60(3):540-563, 2000. URL: https://doi.org/10.1006/jcss.1999.1686.
  30. Ishay Haviv and Oded Regev. Tensor-based Hardness of the Shortest Vector Problem to within Almost Polynomial Factors. Theory of Computing, 8(1):513-531, 2012. URL: https://doi.org/10.4086/toc.2012.v008a023.
  31. 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
  32. 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
  33. 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.
  34. 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.
  35. Shuichi Hirahara. Average-case hardness of NP from exponential worst-case hardness assumptions. In Proceedings of the Symposium on Theory of Computing (STOC), pages 292-302, 2021. URL: https://doi.org/10.1145/3406325.3451065.
  36. Shuichi Hirahara and Mikito Nanashima. On Worst-Case Learning in Relativized Heuristica. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), 2021. Google Scholar
  37. Shuichi Hirahara, Igor Carboni Oliveira, and Rahul Santhanam. NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits. In Proceedings of the Computational Complexity Conference (CCC), pages 5:1-5:31, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.5.
  38. Shuichi Hirahara and Rahul Santhanam. Errorless versus Error-prone Average-case Complexity. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), pages 38:1-38:23, 2022. Google Scholar
  39. Shuichi Hirahara and Osamu 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: https://doi.org/10.4230/LIPIcs.CCC.2016.18.
  40. 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.
  41. 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
  42. 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.
  43. 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.
  44. Russell Impagliazzo. Relativized Separations of Worst-Case and Average-Case Complexities for NP. In Proceedings of the Conference on Computational Complexity (CCC), pages 104-114, 2011. URL: https://doi.org/10.1109/CCC.2011.34.
  45. 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.
  46. 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.
  47. 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.
  48. Tarik Kaced. Almost-perfect secret sharing. In Proceedings of the International Symposium on Information Theory (ISIT), pages 1603-1607, 2011. URL: https://doi.org/10.1109/ISIT.2011.6033816.
  49. Mauricio Karchmer and Avi Wigderson. On Span Programs. In Proceedings of the Structure in Complexity Theory Conference, pages 102-111, 1993. URL: https://doi.org/10.1109/SCT.1993.336536.
  50. 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.
  51. Andrei N Kolmogorov. Three approaches to the quantitative definition of information. Problems of information transmission, 1(1):1-7, 1965. Google Scholar
  52. Troy Lee and Andrei E. Romashchenko. Resource bounded symmetry of information revisited. Theor. Comput. Sci., 345(2-3):386-405, 2005. URL: https://doi.org/10.1016/j.tcs.2005.07.017.
  53. Leonid A. Levin. The Tale of One-Way Functions. Probl. Inf. Transm., 39(1):92-103, 2003. URL: https://doi.org/10.1023/A:1023634616182.
  54. Leonid A. Levin. How do we succeed in tasks like proving Fermat’s Theorem or predicting the Higgs boson?, 2021. An invited talk given at STOC 2021. The transcript is available at URL: https://www.cs.bu.edu/fac/lnd/expo/stoc21/txt.pdf.
  55. Ming Li and Paul M. B. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications, Third Edition. Texts in Computer Science. Springer, 2008. URL: https://doi.org/10.1007/978-0-387-49820-1.
  56. 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.
  57. Luc Longpré and Sarah Mocas. Symmetry of Information and One-Way Functions. Inf. Process. Lett., 46(2):95-100, 1993. URL: https://doi.org/10.1016/0020-0190(93)90204-M.
  58. Luc Longpré and Osamu Watanabe. On Symmetry of Information and Polynomial Time Invertibility. Inf. Comput., 121(1):14-22, 1995. URL: https://doi.org/10.1006/inco.1995.1120.
  59. Zhenjian Lu and Igor Carboni Oliveira. An Efficient Coding Theorem via Probabilistic Representations and Its Applications. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 94:1-94:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.94.
  60. Daniele Micciancio and Oded Regev. Worst-Case to Average-Case Reductions Based on Gaussian Measures. SIAM J. Comput., 37(1):267-302, 2007. URL: https://doi.org/10.1137/S0097539705447360.
  61. 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.
  62. Igor Carboni Oliveira. Randomness and Intractability in Kolmogorov Complexity. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 32:1-32:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.32.
  63. Igor Carboni Oliveira and Rahul Santhanam. Pseudodeterministic constructions in subexponential time. In Proceedings of the Symposium on Theory of Computing (STOC), pages 665-677, 2017. URL: https://doi.org/10.1145/3055399.3055500.
  64. Ran Raz, Omer Reingold, and Salil P. Vadhan. Extracting all the Randomness and Reducing the Error in Trevisan’s Extractors. J. Comput. Syst. Sci., 65(1):97-128, 2002. URL: https://doi.org/10.1006/jcss.2002.1824.
  65. Hanlin Ren and Rahul Santhanam. Hardness of KT Characterizes Parallel Cryptography. In Proceedings of the Computational Complexity Conference (CCC), pages 35:1-35:58, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.35.
  66. Detlef Ronneburger. Kolmogorov Complexity and Derandomization. PhD thesis, Rutgers UniversityDept. of Computer Science Laboratory for Computer Sci. Research Hill Center, NJ, USA, 2004. Google Scholar
  67. Adi Shamir. How to Share a Secret. Commun. ACM, 22(11):612-613, 1979. URL: https://doi.org/10.1145/359168.359176.
  68. Amnon Ta-Shma, Christopher Umans, and David Zuckerman. Lossless Condensers, Unbalanced Expanders, And Extractors. Combinatorica, 27(2):213-240, 2007. URL: https://doi.org/10.1007/s00493-007-0053-2.
  69. Boris A. Trakhtenbrot. A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms. IEEE Annals of the History of Computing, 6(4):384-400, 1984. URL: https://doi.org/10.1109/MAHC.1984.10036.
  70. Luca Trevisan. Extractors and pseudorandom generators. J. ACM, 48(4):860-879, 2001. URL: https://doi.org/10.1145/502090.502099.
  71. 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.
  72. 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.
  73. 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.
  74. AK Zvonkin and LA Levin. The complexity of finite objects and the algorithmic concepts of randomness and information. UMN (Russian Math. Surveys), 25(6):83-124, 1970. Google Scholar
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