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Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE

Authors Joshua Cook , Dana Moshkovitz

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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Circuit complexity
  • Theory of computation → Interactive proof systems
Keywords
  • MA
  • PCP
  • Circuit Complexity

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Abstract

We prove that for some constant a > 1, for all k ≤ a, MATIME[n^{k+o(1)}]/1 ⊄ SIZE[O(n^k)], for some specific o(1) function. This is a super linear polynomial circuit lower bound.
Previously, Santhanam [Santhanam, 2007] showed that there exists a constant c > 1 such that for all k > 1: MATIME[n^{ck}]/1 ⊄ SIZE[O(n^k)]. Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c. Using ideas from Murray and Williams [Murray and Williams, 2018], one can show for all k > 1: MATIME [n^{10 k²}]/1 ⊄ SIZE[O(n^k)].
To prove this result, we construct the first PCP for SPACE[n] with quasi-linear verifier time: our PCP has a Õ(n) time verifier, Õ(n) space prover, O(log(n)) queries, and polynomial alphabet size. Prior to this work, PCPs for SPACE[O(n)] had verifiers that run in Ω(n²) time. This PCP also proves that NE has MIP verifiers which run in time Õ(n).

Cite As Get BibTex

Joshua Cook and Dana Moshkovitz. Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.55

Author Details

Joshua Cook
  • Department of Computer Science, University of Texas Austin, TX, USA
Dana Moshkovitz
  • Department of Computer Science, University of Texas Austin, TX, USA

Funding

This material is based upon work supported by the National Science Foundation under grant number 1705028.

References

  1. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, USA, 1st edition, 2009. Google Scholar
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, May 1998. URL: https://doi.org/10.1145/278298.278306.
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of np. J. ACM, 45(1):70-122, January 1998. URL: https://doi.org/10.1145/273865.273901.
  4. L. Babai, L. Fortnow, and C. Lund. Nondeterministic exponential time has two-prover interactive protocols. In Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pages 16-25 vol.1, 1990. URL: https://doi.org/10.1109/FSCS.1990.89520.
  5. E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan. Short pcps verifiable in polylogarithmic time. In 20th Annual IEEE Conference on Computational Complexity (CCC'05), pages 120-134, 2005. URL: https://doi.org/10.1109/CCC.2005.27.
  6. Eli Ben-Sasson, Alessandro Chiesa, Daniel Genkin, and Eran Tromer. On the concrete efficiency of probabilistically-checkable proofs. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 585-594. Association for Computing Machinery, 2013. URL: https://doi.org/10.1145/2488608.2488681.
  7. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. Robust pcps of proximity, shorter pcps and applications to coding. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 1-10. Association for Computing Machinery, 2004. URL: https://doi.org/10.1145/1007352.1007361.
  8. Eli Ben-Sasson and Emanuele Viola. Short pcps with projection queries. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 163-173. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_14.
  9. Joshua Cook and Dana Moshkovitz. Tighter ma/1 circuit lower bounds from verifier efficient pcps for pspace, 2022. URL: https://eccc.weizmann.ac.il/report/2022/014/.
  10. Stephen A. Cook. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC '71, pages 151-158, New York, NY, USA, 1971. Association for Computing Machinery. URL: https://doi.org/10.1145/800157.805047.
  11. Stephen A. Cook. A hierarchy for nondeterministic time complexity. In Proceedings of the Fourth Annual ACM Symposium on Theory of Computing, STOC '72, pages 187-192. Association for Computing Machinery, 1972. URL: https://doi.org/10.1145/800152.804913.
  12. Irit Dinur and Prahladh Harsha. Composition of low-error 2-query pcps using decodable pcps. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 472-481, 2009. URL: https://doi.org/10.1109/FOCS.2009.8.
  13. Irit Dinur and Omer Reingold. Assignment testers: towards a combinatorial proof of the pcp-theorem. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 155-164, 2004. URL: https://doi.org/10.1109/FOCS.2004.16.
  14. Dean Doron, Dana Moshkovitz, Justin Oh, and David Zuckerman. Nearly optimal pseudorandomness from hardness. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1057-1068. IEEE, 2020. Google Scholar
  15. Lance Fortnow and Rahul Santhanam. Robust simulations and significant separations. In Proceedings of the 38th International Colloquim Conference on Automata, Languages and Programming - Volume Part I, ICALP'11, pages 569-580. Springer-Verlag, 2011. Google Scholar
  16. Lance Fortnow, Rahul Santhanam, and Luca Trevisan. Hierarchies for semantic classes. In Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC '05, pages 348-355. Association for Computing Machinery, 2005. URL: https://doi.org/10.1145/1060590.1060642.
  17. Lance Fortnow, Rahul Santhanam, and Ryan Williams. Fixed-polynomial size circuit bounds. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 19-26, 2009. URL: https://doi.org/10.1109/CCC.2009.21.
  18. J. Hartmanis and R. E. Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117:285-306, 1965. URL: http://www.jstor.org/stable/1994208.
  19. Thomas Holenstein. Parallel repetition: Simplifications and the no-signaling case. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 411-419. Association for Computing Machinery, 2007. URL: https://doi.org/10.1145/1250790.1250852.
  20. Justin Holmgren and Ron Rothblum. Delegating computations with (almost) minimal time and space overhead. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 124-135, 2018. URL: https://doi.org/10.1109/FOCS.2018.00021.
  21. Leonid Levin. Universal'nyie perebornyie zadachi (universal search problems, in russian). Problemy Peredachi Informatsii, 9:265-266, 1973. Google Scholar
  22. Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, October 1992. URL: https://doi.org/10.1145/146585.146605.
  23. Or Meir. Combinatorial pcps with efficient verifiers. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 463-471, 2009. URL: https://doi.org/10.1109/FOCS.2009.10.
  24. Peter Bro Miltersen, N. V. Vinodchandran, and Osamu Watanabe. Super-polynomial versus half-exponential circuit size in the exponential hierarchy. In Proceedings of the 5th Annual International Conference on Computing and Combinatorics, COCOON'99, pages 210-220. Springer-Verlag, 1999. Google Scholar
  25. Dana Moshkovitz and Ran Raz. Two query pcp with sub-constant error. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 314-323, 2008. URL: https://doi.org/10.1109/FOCS.2008.60.
  26. Cody Murray and Ryan Williams. Circuit lower bounds for nondeterministic quasi-polytime: An easy witness lemma for np and nqp. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 890-901. Association for Computing Machinery, 2018. URL: https://doi.org/10.1145/3188745.3188910.
  27. Anup Rao. Parallel repetition in projection games and a concentration bound. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 1-10. Association for Computing Machinery, 2008. URL: https://doi.org/10.1145/1374376.1374378.
  28. Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, 1998. URL: https://doi.org/10.1137/S0097539795280895.
  29. Rahul Santhanam. Circuit lower bounds for merlin-arthur classes. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 275-283. Association for Computing Machinery, 2007. URL: https://doi.org/10.1145/1250790.1250832.
  30. Joel I. Seiferas, Michael J. Fischer, and Albert R. Meyer. Separating nondeterministic time complexity classes. J. ACM, 25(1):146-167, 1978. URL: https://doi.org/10.1145/322047.322061.
  31. Adi Shamir. Ip = pspace. J. ACM, 39(4):869-877, October 1992. URL: https://doi.org/10.1145/146585.146609.
  32. D. van Melkebeek and K. Pervyshev. A generic time hierarchy for semantic models with one bit of advice. In 21st Annual IEEE Conference on Computational Complexity (CCC'06), pages 14 pp.-144, 2006. URL: https://doi.org/10.1109/CCC.2006.7.
  33. Ryan Williams. Non-uniform acc circuit lower bounds. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 115-125, 2011. URL: https://doi.org/10.1109/CCC.2011.36.
  34. Stanislav Žák. A turing machine time hierarchy. Theoretical Computer Science, 26(3):327-333, 1983. URL: https://doi.org/10.1016/0304-3975(83)90015-4.
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