Document Open Access Logo

On Disperser/Lifting Properties of the Index and Inner-Product Functions

Authors Paul Beame, Sajin Koroth

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2023.14.pdf
  • Filesize: 0.85 MB
  • 17 pages

Document Identifiers

Author Details

Paul Beame
  • University of Washington, Seattle, WA, USA
Sajin Koroth
  • University of Victoria, Victoria, BC, Canada

Funding

  • Beame, Paul: Research supported by NSF grant CCF-2006359.
  • Koroth, Sajin: Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Discovery Grant RGPIN-2022-05211.

Cite AsGet BibTex

Paul Beame and Sajin Koroth. On Disperser/Lifting Properties of the Index and Inner-Product Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.14

Abstract

Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated "lifted" function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang [Shachar Lovett et al., 2022] using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; - The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N. - The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N), also does not have this property when its size is less than log N. - Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. - Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Proof complexity
Keywords
  • Decision trees
  • communication complexity
  • lifting theorems
  • proof complexity

Metrics

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

References

  1. Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. Improved bounds for the sunflower lemma. Annals of Mathematics, 194(3):795-815, 2021. Google Scholar
  2. Paul Beame, Toniann Pitassi, and Nathan Segerlind. Lower bounds for Lovász-Schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput., 37(3):845-869, 2007. URL: https://doi.org/10.1137/060654645.
  3. Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149-169, 2001. URL: https://doi.org/10.1145/375827.375835.
  4. Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-to-communication lifting using low-discrepancy gadgets. SIAM J. Comput., 50(1):171-210, 2021. URL: https://doi.org/10.1137/19M1310153.
  5. Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Simulation theorems via pseudo-random properties. Comput. Complex., 28(4):617-659, 2019. URL: https://doi.org/10.1007/s00037-019-00190-7.
  6. Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. Parity decision tree lifting via stifling. In 14th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 10-13, 2022, Cambridge, MA, USA, volume 251 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. These proceedings. Google Scholar
  7. Stephen A. Cook and Robert A. Reckhow. The relative efficiency of propositional proof systems. J. Symb. Log., 44(1):36-50, 1979. URL: https://doi.org/10.2307/2273702.
  8. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. Theory Comput., 16:1-30, 2020. URL: https://doi.org/10.4086/toc.2020.v016a013.
  9. Mika Göös, Rahul Jain, and Thomas Watson. Extension complexity of independent set polytopes. SIAM J. Comput., 47(1):241-269, 2018. URL: https://doi.org/10.1137/16M109884X.
  10. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. SIAM J. Comput., 45(5):1835-1869, 2016. URL: https://doi.org/10.1137/15M103145X.
  11. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. SIAM J. Comput., 47(5):1778-1806, 2018. URL: https://doi.org/10.1137/16M1082007.
  12. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. SIAM J. Comput., 47(6):2435-2450, 2018. URL: https://doi.org/10.1137/16M1059369.
  13. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for BPP. SIAM J. Comput., 49(4), 2020. URL: https://doi.org/10.1137/17M115339X.
  14. Danny Harnik and Ran Raz. Higher lower bounds on monotone size. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 378-387. ACM, 2000. URL: https://doi.org/10.1145/335305.335349.
  15. Trinh Huynh and Jakob Nordström. On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 233-248. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214000.
  16. Dmitry Itsykson and Dmitry Sokolov. Lower bounds for splittings by linear combinations. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, volume 8635 of Lecture Notes in Computer Science, pages 372-383. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44465-8_32.
  17. Dmitry Itsykson and Dmitry Sokolov. Resolution over linear equations modulo two. Ann. Pure Appl. Log., 171(1), 2020. URL: https://doi.org/10.1016/j.apal.2019.102722.
  18. R. Kaas and J. M. Buhrman. Mean, median and mode in binomial distributions. Statistica Neerlandica, 34(1):13-18, 1980. URL: https://doi.org/10.1111/j.1467-9574.1980.tb00681.x.
  19. James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 567-576. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746599.
  20. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with sunflowers. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 104:1-104:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.104.
  21. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 234-243. IEEE Computer Society, 1997. URL: https://doi.org/10.1109/SFCS.1997.646112.
  22. Ran Raz and Iddo Tzameret. Resolution over linear equations and multilinear proofs. Ann. Pure Appl. Log., 155(3):194-224, 2008. URL: https://doi.org/10.1016/j.apal.2008.04.001.
  23. Robert Robere, Toniann Pitassi, Benjamin Rossman, and Stephen A. Cook. Exponential lower bounds for monotone span programs. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, New Brunswick, New Jersey, USA, pages 406-415. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.51.
  24. Alexander A. Sherstov. The pattern matrix method. SIAM J. Comput., 40(6):1969-2000, 2011. URL: https://doi.org/10.1137/080733644.
  25. Xiaodi Wu, Penghui Yao, and Henry S. Yuen. Raz-McKenzie simulation with the inner product gadget. Electron. Colloquium Comput. Complex., TR17-010, 2017. URL: http://arxiv.org/abs/TR17-010.
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