Document Open Access Logo

A Relativization Perspective on Meta-Complexity

Authors Hanlin Ren , Rahul Santhanam



PDF
Thumbnail PDF

File

LIPIcs.STACS.2022.54.pdf
  • Filesize: 0.8 MB
  • 13 pages

Document Identifiers

Author Details

Hanlin Ren
  • University of Oxford, UK
Rahul Santhanam
  • University of Oxford, UK

Cite AsGet BibTex

Hanlin Ren and Rahul Santhanam. A Relativization Perspective on Meta-Complexity. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 54:1-54:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.54

Abstract

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where: 1) MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC'16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof. 2) The complexities of MCSP[2^{n/2}] and MCSP[2^{n/4}] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not "robust" to the choice of the size function. 3) Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2+ε) in polynomial time, for any ε > 0. 4) Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS'20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets. 5) DistNP does not reduce to GapMINKT by a family of "robust" reductions. This presents a technical barrier for solving a question of Hirahara [FOCS'20].

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Circuit complexity
  • Theory of computation → Problems, reductions and completeness
Keywords
  • meta-complexity
  • relativization
  • minimum circuit size problem

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Scott Aaronson and Avi Wigderson. Algebrization: A new barrier in complexity theory. ACM Transactions on Computation Theory, 1(1):2:1-2:54, 2009. URL: https://doi.org/10.1145/1490270.1490272.
  2. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM Journal of Computing, 35(6):1467-1493, 2006. URL: https://doi.org/10.1137/050628994.
  3. Eric Allender and Shuichi Hirahara. New insights on the (non-)hardness of circuit minimization and related problems. ACM Transactions on Computation Theory, 11(4):27:1-27:27, 2019. URL: https://doi.org/10.1145/3349616.
  4. Eric Allender, Rahul Ilango, and Neekon Vafa. The non-hardness of approximating circuit size. In Proc. 14th International Computer Science Symposium in Russia (CSR) , volume 11532 of Lecture Notes in Computer Science, pages 13-24, 2019. URL: https://doi.org/10.1007/978-3-030-19955-5_2.
  5. Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences, 77(1):14-40, 2011. URL: https://doi.org/10.1016/j.jcss.2010.06.004.
  6. Barış Aydınlıoğlu and Eric Bach. Affine relativization: Unifying the algebrization and relativization barriers. ACM Transactions on Computation Theory, 10(1):1:1-1:67, 2018. URL: https://doi.org/10.1145/3170704.
  7. László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. URL: https://doi.org/10.1007/BF01200056.
  8. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computatioanl Complexity, 3:307-318, 1993. URL: https://doi.org/10.1007/BF01275486.
  9. Theodore P. Baker, John Gill, and Robert Solovay. Relativizations of the P = ? NP question. SIAM Journal of Computing, 4(4):431-442, 1975. URL: https://doi.org/10.1137/0204037.
  10. Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In Proc. 31st Computational Complexity Conference (CCC), volume 50 of LIPIcs, pages 10:1-10:24, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.10.
  11. Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis. Circuit lower bounds for MCSP from local pseudorandom generators. ACM Transactions on Computation Theory, 12(3):21:1-21:27, 2020. URL: https://doi.org/10.1145/3404860.
  12. Bin Fu. Hardness of sparse sets and minimal circuit size problem. In Proc. 26th International Computing and Combinatorics Conference (COCOON) , volume 12273 of Lecture Notes in Computer Science, pages 484-495, 2020. URL: https://doi.org/10.1007/978-3-030-58150-3_39.
  13. Rosario Gennaro, Yael Gertner, Jonathan Katz, and Luca Trevisan. Bounds on the efficiency of generic cryptographic constructions. SIAM Journal of Computing, 35(1):217-246, 2005. URL: https://doi.org/10.1137/S0097539704443276.
  14. Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM, 38(3):691-729, 1991. URL: https://doi.org/10.1145/116825.116852.
  15. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within NP. In Proc. 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 247-258, 2018. URL: https://doi.org/10.1109/FOCS.2018.00032.
  16. Shuichi Hirahara. Characterizing average-case complexity of PH by worst-case meta-complexity. In Proc. 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 50-60, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00014.
  17. Shuichi Hirahara. Unexpected hardness results for Kolmogorov complexity under uniform reductions. In Proc. 52nd Annual ACM 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. In Proc. 53rd Annual ACM Symposium on Theory of Computing (STOC), pages 292-302, 2021. URL: https://doi.org/10.1145/3406325.3451065.
  19. Shuichi Hirahara and Rahul Santhanam. On the average-case complexity of MCSP and its variants. In Proc. 32nd Computational Complexity Conference (CCC), volume 79 of LIPIcs, pages 7:1-7:20, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.7.
  20. Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In Proc. 31st Computational Complexity Conference (CCC), volume 50 of LIPIcs, pages 18:1-18:20, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.18.
  21. John M. Hitchcock and Aduri Pavan. On the NP-completeness of the minimum circuit size problem. In Proc. 35th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 45 of LIPIcs, pages 236-245, 2015. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2015.236.
  22. Rahul Ilango. Approaching CSP from above and below: Hardness for a conditional variant and AC⁰[p]. In Proc. 11th Conference on Innovations in Theoretical Computer Science (ITCS), volume 151 of LIPIcs, pages 34:1-34:26, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.34.
  23. Rahul Ilango. Connecting perebor conjectures: Towards a search to decision reduction for minimizing formulas. In Proc. 35th Computational Complexity Conference (CCC), volume 169 of LIPIcs, pages 31:1-31:35, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.31.
  24. Rahul Ilango. Constant depth formula and partial function versions of MCSP are hard. In Proc. 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 424-433, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00047.
  25. Rahul Ilango, Bruno Loff, and Igor Carboni Oliveira. NP-hardness of circuit minimization for multi-output functions. In Proc. 35th Computational Complexity Conference (CCC), volume 169 of LIPIcs, pages 22:1-22:36, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.22.
  26. Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. An axiomatic approach to algebrization. In Proc. 41st Annual ACM Symposium on Theory of Computing (STOC), pages 695-704, 2009. URL: https://doi.org/10.1145/1536414.1536509.
  27. Russell Impagliazzo and Avi Wigderson. Randomness vs time: Derandomization under a uniform assumption. Journal of Computer and System Sciences, 63(4):672-688, 2001. URL: https://doi.org/10.1006/jcss.2001.1780.
  28. Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In Proc. 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 73-79, 2000. URL: https://doi.org/10.1145/335305.335314.
  29. Ker-I Ko. On the complexity of learning minimum time-bounded Turing machines. SIAM Journal of Computing, 20(5):962-986, 1991. URL: https://doi.org/10.1137/0220059.
  30. Leonid A. Levin. Hardness of search problems. Accessed 12-June-2021. URL: https://www.cs.bu.edu/fac/lnd/research/hard.htm.
  31. Leonid A. Levin. Universal sequential search problems. Problemy peredachi informatsii, 9(3):115-116, 1973. Google Scholar
  32. Yanyi Liu and Rafael Pass. On one-way functions and Kolmogorov complexity. In Proc. 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1243-1254, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00118.
  33. Cody D. Murray and R. Ryan Williams. On the (non) NP-hardness of computing circuit complexity. Theory of Computing, 13(1):1-22, 2017. URL: https://doi.org/10.4086/toc.2017.v013a004.
  34. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  35. Igor Carboni Oliveira. Randomness and intractability in Kolmogorov complexity. In Proc. 46th International Colloquium on Automata, Languages and Programming (ICALP), volume 132 of LIPIcs, pages 32:1-32:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.32.
  36. Igor Carboni Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In Proc. 32nd Computational Complexity Conference (CCC), volume 79 of LIPIcs, pages 18:1-18:49, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.18.
  37. Rafail Ostrovsky and Avi Wigderson. One-way functions are essential for non-trivial zero-knowledge. In Proc. Second Israel Symposium on Theory of Computing Systems, (ISTCS), pages 3-17, 1993. URL: https://doi.org/10.1109/ISTCS.1993.253489.
  38. Alexander A. Razborov and Steven Rudich. Natural proofs. Journal of Computer and System Sciences, 55(1):24-35, 1997. URL: https://doi.org/10.1006/jcss.1997.1494.
  39. Hanlin Ren and Rahul Santhanam. Hardness of KT characterizes parallel cryptography. In Proc. 36th Computational Complexity Conference (CCC), volume 200 of LIPIcs, pages 35:1-35:58, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.35.
  40. Hanlin Ren and Rahul Santhanam. A relativization perspective on meta-complexity. Electron. Colloquium Comput. Complex., page 89, 2021. URL: https://eccc.weizmann.ac.il/report/2021/089.
  41. Michael Saks and Rahul Santhanam. Circuit lower bounds from NP-hardness of MCSP under Turing reductions. In Proc. 35th Computational Complexity Conference (CCC), volume 169 of LIPIcs, pages 26:1-26:13, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.26.
  42. Rahul Santhanam. Pseudorandomness and the minimum circuit size problem. In Proc. 11th Conference on Innovations in Theoretical Computer Science (ITCS), volume 151 of LIPIcs, pages 68:1-68:26, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.68.
  43. 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.
  44. 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.
  45. Hoeteck Wee. Finding Pessiland. In Proc. 3rd Theory of Cryptography Conference (TCC), volume 3876 of Lecture Notes in Computer Science, pages 429-442, 2006. URL: https://doi.org/10.1007/11681878_22.
  46. David Xiao. On basing ZK ≠ BPP on the hardness of PAC learning. In Proc. 24th Annual IEEE Conference on Computational Complexity (CCC), pages 304-315, 2009. URL: https://doi.org/10.1109/CCC.2009.11.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail