Document Open Access Logo

Bounded Relativization

Authors Shuichi Hirahara, Zhenjian Lu, Hanlin Ren

PDF
Thumbnail PDF

File

LIPIcs.CCC.2023.6.pdf
  • Filesize: 1.22 MB
  • 45 pages

Document Identifiers

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Zhenjian Lu
  • University of Oxford, UK
Hanlin Ren
  • University of Oxford, UK

Funding

  • Hirahara, Shuichi: Supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Ren, Hanlin: Received support from DIMACS through grant number CCF-1836666 from the National Science Foundation.

Acknowledgements

We thank Rahul Santhanam for helpful discussions and for pointing out that the oracle in Theorem 52 can be constructed in EXPH, using [Shafi Goldwasser et al., 2021] only as a black-box. We thank Lijie Chen for helpful discussions regarding [Lijie Chen et al., 2022], Ian Mertz for discussions about the statement ({*}), and Ryan Williams for useful discussions about time-space tradeoffs for SAT. We thank Lijie Chen (again) and an anonymous CCC reviewer for pointing out an error in a previous version of this paper. Part of this work was completed when the authors are visiting the Simons Institute for the Theory of Computing, participating in the Meta-Complexity program.

Cite AsGet BibTex

Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren. Bounded Relativization. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 6:1-6:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.6

Abstract

Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ℭ, we say that a statement is ℭ-relativizing if the statement holds relative to every oracle 𝒪 ∈ ℭ. It is easy to see that every result that relativizes also ℭ-relativizes for every complexity class ℭ. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0, BPE^{MCSP}/2^{εn} ⊈ SIZE[2ⁿ/n]. We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021). Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ≠ L. For example: - Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ≠ BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ≠ L. - Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ≠ L. - Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ≠ L. In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ≠ EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Circuit complexity
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • relativization
  • circuit lower bound
  • derandomization
  • explicit construction
  • pseudodeterministic algorithms
  • interactive proofs

Metrics

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

Related Versions

References

  1. Scott Aaronson. Oracles are subtle but not malicious. In Conference on Computational Complexity (CCC), pages 340-354. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/CCC.2006.32.
  2. Scott Aaronson. The teaser. https://scottaaronson.blog/?p=3054, 2017. Accessed: Feb 6, 2023.
  3. Scott Aaronson and Avi Wigderson. Algebrization: A new barrier in complexity theory. ACM Trans. Comput. Theory, 1(1):2:1-2:54, 2009. URL: https://doi.org/10.1145/1490270.1490272.
  4. Eric Allender. Oracles versus proof techniques that do not relativize. In SIGAL International Symposium on Algorithms, volume 450 of Lecture Notes in Computer Science, pages 39-52. Springer, 1990. URL: https://doi.org/10.1007/3-540-52921-7_54.
  5. 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.
  6. Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. URL: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264.
  7. Sanjeev Arora, Russell Impagliazzo, and Umesh Vazirani. Relativizing versus nonrelativizing techniques: the role of local checkability. Manuscript, 1992. URL: https://people.eecs.berkeley.edu/~vazirani/pubs/relativizing.pdf.
  8. Barış Aydınlıoğlu and Eric Bach. Affine relativization: Unifying the algebrization and relativization barriers. ACM Trans. Comput. Theory, 10(1):1:1-1:67, 2018. URL: https://doi.org/10.1145/3170704.
  9. László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Comput. Complex., 1:3-40, 1991. URL: https://doi.org/10.1007/BF01200056.
  10. 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.
  11. Theodore P. Baker, John Gill, and Robert Solovay. Relativizations of the P = ? NP question. SIAM J. Comput., 4(4):431-442, 1975. URL: https://doi.org/10.1137/0204037.
  12. Harry Buhrman, Lance Fortnow, and Rahul Santhanam. Unconditional lower bounds against advice. In International Colloquium on Automata, Languages and Programming (ICALP), pages 195-209, 2009. URL: https://doi.org/10.1007/978-3-642-02927-1_18.
  13. Harry Buhrman, Lance Fortnow, and Thomas Thierauf. Nonrelativizing separations. In Conference on Computational Complexity (CCC), pages 8-12, 1998. URL: https://doi.org/10.1109/CCC.1998.694585.
  14. Harry Buhrman and Leen Torenvliet. Randomness is hard. SIAM J. Comput., 30(5):1485-1501, 2000. URL: https://doi.org/10.1137/S0097539799360148.
  15. Samuel R. Buss and R. Ryan Williams. Limits on alternation trading proofs for time-space lower bounds. Comput. Complex., 24(3):533-600, 2015. URL: https://doi.org/10.1007/s00037-015-0104-9.
  16. Richard Chang, Benny Chor, Oded Goldreich, Juris Hartmanis, Johan Håstad, Desh Ranjan, and Pankaj Rohatgi. The random oracle hypothesis is false. J. Comput. Syst. Sci., 49(1):24-39, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80084-4.
  17. Lijie Chen, Xin Lyu, and R. Ryan Williams. Almost-everywhere circuit lower bounds from non-trivial derandomization. In Symposium on Foundations of Computer Science (FOCS), pages 1-12. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00009.
  18. Lijie Chen, Dylan M. McKay, Cody D. Murray, and R. Ryan Williams. Relations and equivalences between circuit lower bounds and Karp-Lipton theorems. In Computational Complexity Conference (CCC), volume 137 of LIPIcs, pages 30:1-30:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.30.
  19. Lijie Chen, Ron D. Rothblum, and Roei Tell. Unstructured hardness to average-case randomness. In Symposium on Foundations of Computer Science (FOCS), pages 429-437. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00048.
  20. Lijie Chen, Ron D. Rothblum, Roei Tell, and Eylon Yogev. On exponential-time hypotheses, derandomization, and circuit lower bounds. In Symposium on Foundations of Computer Science (FOCS), pages 13-23. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00010.
  21. Lijie Chen and Roei Tell. Hardness vs randomness, revised: Uniform, non-black-box, and instance-wise. In Symposium on Foundations of Computer Science (FOCS), pages 125-136. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00021.
  22. Scott Diehl and Dieter van Melkebeek. Time-space lower bounds for the polynomial-time hierarchy on randomized machines. SIAM J. Comput., 36(3):563-594, 2006. URL: https://doi.org/10.1137/050642228.
  23. Lance Fortnow. The role of relativization in complexity theory. Bull. EATCS, 52:229-243, 1994. Google Scholar
  24. Lance Fortnow. Time-space tradeoffs for satisfiability. J. Comput. Syst. Sci., 60(2):337-353, 2000. URL: https://doi.org/10.1006/jcss.1999.1671.
  25. Lance Fortnow, Richard J. Lipton, Dieter van Melkebeek, and Anastasios Viglas. Time-space lower bounds for satisfiability. J. ACM, 52(6):835-865, 2005. URL: https://doi.org/10.1145/1101821.1101822.
  26. Lance Fortnow and Rahul Santhanam. Hierarchy theorems for probabilistic polynomial time. In Symposium on Foundations of Computer Science (FOCS), pages 316-324, 2004. URL: https://doi.org/10.1109/FOCS.2004.33.
  27. Lance Fortnow, Rahul Santhanam, and R. Ryan Williams. Fixed-polynomial size circuit bounds. In Computational Complexity Conference (CCC), pages 19-26. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/CCC.2009.21.
  28. Lance Fortnow and Michael Sipser. Are there interactive protocols for coNP languages? Inf. Process. Lett., 28(5):249-251, 1988. URL: https://doi.org/10.1016/0020-0190(88)90199-8.
  29. Gudmund Skovbjerg Frandsen and Peter Bro Miltersen. Reviewing bounds on the circuit size of the hardest functions. Inf. Process. Lett., 95(2):354-357, 2005. URL: https://doi.org/10.1016/j.ipl.2005.03.009.
  30. Oded Goldreich. Computational complexity - A conceptual perspective. Cambridge University Press, 2008. Google Scholar
  31. Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the pseudo-deterministic query complexity of NP search problems. In Computational Complexity Conference (CCC), volume 200 of LIPIcs, pages 36:1-36:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.36.
  32. Hans Heller. On relativized exponential and probabilistic complexity classes. Inf. Control., 71(3):231-243, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80012-2.
  33. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within NP. In Symposium on Foundations of Computer Science (FOCS), pages 247-258, 2018. URL: https://doi.org/10.1109/FOCS.2018.00032.
  34. Shuichi Hirahara. Non-disjoint promise problems from meta-computational view of pseudorandom generator constructions. In Computational Complexity Conference (CCC), volume 169 of LIPIcs, pages 20:1-20:47. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.20.
  35. Shuichi Hirahara. Unexpected hardness results for Kolmogorov complexity under uniform reductions. In Symposium on Theory of Computing (STOC), pages 1038-1051, 2020. URL: https://doi.org/10.1145/3357713.3384251.
  36. Shuichi Hirahara. Symmetry of information from meta-complexity. In Computational Complexity Conference (CCC), volume 234 of LIPIcs, pages 26:1-26:41. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.26.
  37. Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren. Bounded relativization. Electron. Colloquium Comput. Complex., 30:70, 2023. URL: https://eccc.weizmann.ac.il/report/2023/070.
  38. Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. An axiomatic approach to algebrization. In Symposium on Theory of Computing (STOC), pages 695-704. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536509.
  39. Russell Impagliazzo, Valentine Kabanets, and Ilya Volkovich. The power of natural properties as oracles. In Computational Complexity Conference (CCC), volume 102 of LIPIcs, pages 7:1-7:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.7.
  40. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: exponential time vs. probabilistic polynomial time. J. Comput. Syst. Sci., 65(4):672-694, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00024-7.
  41. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Symposium on Theory of Computing (STOC), pages 220-229. ACM, 1997. URL: https://doi.org/10.1145/258533.258590.
  42. 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.
  43. Emil Jerábek. Dual weak pigeonhole principle, Boolean complexity, and derandomization. Ann. Pure Appl. Log., 129(1-3):1-37, 2004. URL: https://doi.org/10.1016/j.apal.2003.12.003.
  44. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP^* = RE. CoRR, abs/2001.04383, 2020. URL: https://doi.org/10.48550/arXiv.2001.04383.
  45. Valentine Kabanets. Easiness assumptions and hardness tests: Trading time for zero error. J. Comput. Syst. Sci., 63(2):236-252, 2001. URL: https://doi.org/10.1006/jcss.2001.1763.
  46. Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In Symposium on Theory of Computing (STOC), pages 73-79. ACM, 2000. URL: https://doi.org/10.1145/335305.335314.
  47. Ravi Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Inf. Control., 55(1-3):40-56, 1982. URL: https://doi.org/10.1016/S0019-9958(82)90382-5.
  48. Richard M. Karp and Richard J. Lipton. Some connections between nonuniform and uniform complexity classes. In Symposium on Theory of Computing (STOC), pages 302-309. ACM, 1980. URL: https://doi.org/10.1145/800141.804678.
  49. Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos H. Papadimitriou. Total functions in the polynomial hierarchy. In Innovations in Theoretical Computer Science (ITCS), volume 185 of LIPIcs, pages 44:1-44:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.44.
  50. Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL: https://doi.org/10.1137/S0097539700389652.
  51. Oliver Korten. The hardest explicit construction. In Symposium on Foundations of Computer Science (FOCS), pages 433-444. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00051.
  52. Leonid A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61(1):15-37, 1984. URL: https://doi.org/10.1016/S0019-9958(84)80060-1.
  53. Richard J. Lipton and Anastasios Viglas. On the complexity of SAT. In Symposium on Foundations of Computer Science (FOCS), pages 459-464. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814618.
  54. Zhenjian Lu, Igor Carboni Oliveira, and Rahul Santhanam. Pseudodeterministic algorithms and the structure of probabilistic time. In Symposium on Theory of Computing (STOC), pages 303-316. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451085.
  55. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, 1992. URL: https://doi.org/10.1145/146585.146605.
  56. Oleg B Lupanov. On the synthesis of switching circuits. Doklady Akademii Nauk SSSR, 119(1):23-26, 1958. Google Scholar
  57. Peter Bro Miltersen, N. V. Vinodchandran, and Osamu Watanabe. Super-polynomial versus half-exponential circuit size in the exponential hierarchy. In International Computing and Combinatorics Conference (COCOON), pages 210-220, 1999. URL: https://doi.org/10.1007/3-540-48686-0_21.
  58. Cody D. Murray and R. Ryan Williams. Easiness amplification and uniform circuit lower bounds. In Computational Complexity Conference (CCC), volume 79 of LIPIcs, pages 8:1-8:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.8.
  59. 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.
  60. Igor C. Oliveira. Randomness and intractability in Kolmogorov complexity. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 32:1-32:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.32.
  61. Igor C. Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In Computational Complexity Conference (CCC), pages 18:1-18:49, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.18.
  62. Igor C. Oliveira and Rahul Santhanam. Pseudodeterministic constructions in subexponential time. In Symposium on Theory of Computing (STOC), pages 665-677, 2017. URL: https://doi.org/10.1145/3055399.3055500.
  63. Hanlin Ren and Rahul Santhanam. A relativization perspective on meta-complexity. In International Symposium on Theoretical Aspects of Computer Science (STACS), volume 219 of LIPIcs, pages 54:1-54:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.54.
  64. Hanlin Ren, Rahul Santhanam, and Zhikun Wang. On the range avoidance problem for circuits. In FOCS, pages 640-650. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00067.
  65. Rahul Santhanam. Circuit lower bounds for Merlin-Arthur classes. SIAM J. Comput., 39(3):1038-1061, 2009. URL: https://doi.org/10.1137/070702680.
  66. Adi Shamir. IP = PSPACE. J. ACM, 39(4):869-877, 1992. URL: https://doi.org/10.1145/146585.146609.
  67. Claude E. Shannon. The synthesis of two-terminal switching circuits. Bell Syst. Tech. J., 28(1):59-98, 1949. URL: https://doi.org/10.1002/j.1538-7305.1949.tb03624.x.
  68. Michael Sipser. A complexity theoretic approach to randomness. In Symposium on Theory of Computing (STOC), pages 330-335, 1983. URL: https://doi.org/10.1145/800061.808762.
  69. Larry J. Stockmeyer. The complexity of approximate counting (preliminary version). In Symposium on Theory of Computing (STOC), pages 118-126. ACM, 1983. URL: https://doi.org/10.1145/800061.808740.
  70. 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.
  71. Christopher Umans. Pseudo-random generators for all hardnesses. J. Comput. Syst. Sci., 67(2):419-440, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00046-1.
  72. Salil P. Vadhan. Pseudorandomness. Found. Trends Theor. Comput. Sci., 7(1-3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  73. N. V. Vinodchandran. A note on the circuit complexity of PP. Theor. Comput. Sci., 347(1-2):415-418, 2005. URL: https://doi.org/10.1016/j.tcs.2005.07.032.
  74. Emanuele Viola. On approximate majority and probabilistic time. In Conference on Computational Complexity (CCC), pages 155-168. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/CCC.2007.16.
  75. Nikhil Vyas and R. Ryan Williams. On oracles and algorithmic methods for proving lower bounds. In Innovations in Theoretical Computer Science Conference (ITCS), volume 251 of LIPIcs, pages 99:1-99:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.99.
  76. R. Ryan Williams. Inductive time-space lower bounds for SAT and related problems. Comput. Complex., 15(4):433-470, 2006. URL: https://doi.org/10.1007/s00037-007-0221-1.
  77. R. Ryan Williams. Time-space tradeoffs for counting NP solutions modulo integers. Comput. Complex., 17(2):179-219, 2008. URL: https://doi.org/10.1007/s00037-008-0248-y.
  78. R. Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM J. Comput., 42(3):1218-1244, 2013. URL: https://doi.org/10.1137/10080703X.
  79. R. Ryan Williams. Nonuniform ACC circuit lower bounds. J. ACM, 61(1):2:1-2:32, 2014. URL: https://doi.org/10.1145/2559903.
  80. Christopher B. Wilson. Relativized circuit complexity. J. Comput. Syst. Sci., 31(2):169-181, 1985. URL: https://doi.org/10.1016/0022-0000(85)90040-6.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact