Query-To-Communication Lifting for BPP Using Inner Product

Authors Arkadev Chattopadhyay, Yuval Filmus , Sajin Koroth , Or Meir , Toniann Pitassi



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.35.pdf
  • Filesize: 449 kB
  • 15 pages

Document Identifiers

Author Details

Arkadev Chattopadhyay
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
Yuval Filmus
  • Department of Computer Science, Technion Israel Institute of Technology, Haifa, Israel
Sajin Koroth
  • Department of Computer Science, University of Haifa, Haifa, Israel
Or Meir
  • Department of Computer Science, University of Haifa, Haifa, Israel
Toniann Pitassi
  • Department of Computer Science, University of Toronto, Canada

Acknowledgements

We thank Daniel Kane for some very enlightening conversations and suggestions. This work was done (in part) while the authors were visiting the Simons Institute for the Theory of Computing.

Cite AsGet BibTex

Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-To-Communication Lifting for BPP Using Inner Product. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.35

Abstract

We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. [Arkadev Chattopadhyay et al., 2017] and Wu et al. [Xiaodi Wu et al., 2017]. The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [Mika Göös et al., 2017], used the much larger indexing gadget. Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, Göös, Pitassi and Watson [Mika Göös et al., 2017] used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • lifting theorems
  • inner product
  • BPP Lifting
  • Deterministic Lifting

Metrics

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

References

  1. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 67-76, 2010. Google Scholar
  2. Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct Products in Communication Complexity. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 746-755, 2013. Google Scholar
  3. Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-to-communication lifting for BPP using inner product, April 2019. URL: http://arxiv.org/abs/1904.13056.
  4. Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Simulation Theorems via Pseudorandom Properties. CoRR, abs/1704.06807, 2017. URL: http://arxiv.org/abs/1704.06807.
  5. Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Simulation beats richness: new data-structure lower bounds. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 1013-1020, 2018. Google Scholar
  6. Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals. How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity). In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 295-304, 2016. Google Scholar
  7. Tomás Feder, Eyal Kushilevitz, Moni Naor, and Noam Nisan. Amortized Communication Complexity. SIAM J. Comput., 24(4):736-750, 1995. URL: http://dx.doi.org/10.1137/S0097539792235864.
  8. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 902-911, 2018. Google Scholar
  9. Mika Göös and T. S. Jayram. A Composition Theorem for Conical Juntas. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 5:1-5:16, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.5.
  10. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles Are Nonnegative Juntas. SIAM J. Comput., 45(5):1835-1869, 2016. Google Scholar
  11. Mika Göös and Toniann Pitassi. Communication Lower Bounds via Critical Block Sensitivity. SIAM J. Comput., 47(5):1778-1806, 2018. URL: http://dx.doi.org/10.1137/16M1082007.
  12. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic Communication vs. Partition Number. In Proceedings of IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 1077-1088, 2015. Google Scholar
  13. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for BPP. In Proceedings of IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 132-143, 2017. Google Scholar
  14. Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of Protocols for XOR Functions. SIAM J. Comput., 47(1):208-217, 2018. Google Scholar
  15. Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic Depth Lower Bounds via Direct Sum in Communication Coplexity. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference, Chicago, Illinois, USA, June 30 - July 3, 1991, pages 299-304, 1991. Google Scholar
  16. Hartmut Klauck. A strong direct product theorem for disjointness. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 77-86, 2010. Google Scholar
  17. Alexander Kozachinskiy. From Expanders to Hitting Distributions and Simulation Theorems. In 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, pages 4:1-4:15, 2018. Google Scholar
  18. Bruno Loff and Sagnik Mukhopadhyay. Lifting Theorems for Equality. Electronic Colloquium on Computational Complexity (ECCC), 25:175, 2018. Google Scholar
  19. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. Combinatorica, 19(3):403-435, 1999. Google Scholar
  20. Alexander A. Sherstov. The Pattern Matrix Method. SIAM J. Comput., 40(6):1969-2000, 2011. Google Scholar
  21. Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. Quantum Information & Computation, 9(5):444-460, 2009. Google Scholar
  22. Xiaodi Wu, Penghui Yao, and Henry S. Yuen. Raz-McKenzie simulation with the inner product gadget. Electronic Colloquium on Computational Complexity (ECCC), 24:10, 2017. URL: https://eccc.weizmann.ac.il/report/2017/010.
  23. Andrew C. Yao. Theory and Application of Trapdoor Functions. In Proceedings of IEEE 23rd Annual Symposium on Foundations of Computer Science (FOCS), pages 80-91, 1982. Google Scholar
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