Document Open Access Logo

Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

Authors Toniann Pitassi, Morgan Shirley, Thomas Watson

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2020.92.pdf
  • Filesize: 0.59 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Boolean hierarchies
  • lifting theorems
  • query complexity

Metrics

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

Abstract

We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function with an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sided-error randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.

Cite As Get BibTex

Toniann Pitassi, Morgan Shirley, and Thomas Watson. Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 92:1-92:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.92

Author Details

Toniann Pitassi
  • University of Toronto, Canada
  • Institute for Advanced Study, Princeton, NJ, USA
Morgan Shirley
  • University of Toronto, Canada
Thomas Watson
  • University of Memphis, TN, USA

Funding

  • Pitassi, Toniann: Supported by NSERC.
  • Shirley, Morgan: Supported by NSERC.
  • Watson, Thomas: Supported by NSF grant CCF-1657377.

Acknowledgements

We thank Benjamin Rossman for helpful comments and discussions.

References

  1. Josh Alman and Ryan Williams. Probabilistic rank and matrix rigidity. In Proceedings of the 49th Symposium on Theory of Computing (STOC), pages 641-652. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055484.
  2. László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337-347. IEEE, 1986. URL: https://doi.org/10.1109/SFCS.1986.15.
  3. László Babai, Noam Nisan, and Mario Szegedy. Multiparty protocols and logspace-hard pseudorandom sequences. In Proceedings of the 21st Symposium on Theory of Computing (STOC), pages 1-11. ACM, 1989. URL: https://doi.org/10.1145/73007.73008.
  4. Ziv Bar-Yossef, T.S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. Journal of Computer and System Sciences, 68(4):702-732, 2004. URL: https://doi.org/10.1016/j.jcss.2003.11.006.
  5. Richard Beigel. Bounded queries to SAT and the Boolean hierarchy. Theoretical Computer Science, 84(2):199-223, 1991. URL: https://doi.org/10.1016/0304-3975(91)90160-4.
  6. Alberto Bertoni, Danilo Bruschi, Deborah Joseph, Meera Sitharam, and Paul Young. Generalized Boolean hierarchies and Boolean hierarchies over RP. In Proceedings of the 7th International Conference on Fundamentals of Computation Theory (FCT), pages 35-46. Springer, 1989. URL: https://doi.org/10.1007/3-540-51498-8_4.
  7. Jin-Yi Cai and Lane Hemachandra. The Boolean hierarchy: Hardware over NP. In Proceedings of the 1st Structure in Complexity Theory Conference (STRUCTURES), pages 105-124. Springer, 1986. URL: https://doi.org/10.1007/3-540-16486-3_93.
  8. Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality alone does not simulate randomness. In Proceedings of the 34th Computational Complexity Conference (CCC), pages 14:1-14:11. Schloss Dagstuhl, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.14.
  9. Mika Göös. Lower bounds for clique vs. independent set. In Proceedings of the 56th Symposium on Foundations of Computer Science (FOCS), pages 1066-1076. IEEE, 2015. URL: https://doi.org/10.1109/FOCS.2015.69.
  10. Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for P^NP. Computational Complexity, 28(1):113-144, 2019. URL: https://doi.org/10.1007/s00037-018-0175-5.
  11. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. SIAM Journal on Computing, 45(5):1835-1869, 2016. URL: https://doi.org/10.1137/15M103145X.
  12. Mika Göös, Toniann Pitassi, and Thomas Watson. Zero-information protocols and unambiguity in Arthur-Merlin communication. Algorithmica, 76(3):684-719, 2016. URL: https://doi.org/10.1007/s00453-015-0104-9.
  13. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for BPP. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 132-143. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.21.
  14. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. SIAM Journal on Computing, 47(6):2435-2450, 2018. URL: https://doi.org/10.1137/16M1059369.
  15. Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. Computational Complexity, 27(2):245-304, 2018. URL: https://doi.org/10.1007/s00037-018-0166-6.
  16. Bernd Halstenberg and Rüdiger Reischuk. Relations between communication complexity classes. In Proceedings of the 3rd Structure in Complexity Theory Conference (STRUCTURES), pages 19-28. IEEE, 1988. URL: https://doi.org/10.1109/SCT.1988.5259.
  17. Bernd Halstenberg and Rüdiger Reischuk. Relations between communication complexity classes. Journal of Computer and System Sciences, 41(3):402-429, 1990. URL: https://doi.org/10.1016/0022-0000(90)90027-I.
  18. Bernd Halstenberg and Rüdiger Reischuk. Different modes of communication. SIAM Journal on Computing, 22(5):913-934, 1993. URL: https://doi.org/10.1137/0222057.
  19. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-24508-4.
  20. Jim Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263-1282, 1988. URL: https://doi.org/10.1137/0217080.
  21. Bala Kalyanasundaram and Georg Schnitger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. URL: https://doi.org/10.1137/0405044.
  22. Hartmut Klauck. Rectangle size bounds and threshold covers in communication complexity. In Proceedings of the 18th Conference on Computational Complexity (CCC), pages 118-134. IEEE, 2003. URL: https://doi.org/10.1109/CCC.2003.1214415.
  23. Johannes Köbler, Uwe Schöning, and Klaus Wagner. The difference and truth-table hierarchies for NP. Theoretical Informatics and Applications, 21(4):419-435, 1987. URL: https://doi.org/10.1051/ita/1987210404191.
  24. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  25. Nati Linial, Toniann Pitassi, and Adi Shraibman. On the communication complexity of high-dimensional permutations. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS), pages 54:1-54:20. Schloss Dagstuhl, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.54.
  26. Periklis Papakonstantinou, Dominik Scheder, and Hao Song. Overlays and limited memory communication. In Proceedings of the 29th Conference on Computational Complexity (CCC), pages 298-308. IEEE, 2014. URL: https://doi.org/10.1109/CCC.2014.37.
  27. Toniann Pitassi, Morgan Shirley, and Thomas Watson. Nondeterministic and randomized boolean hierarchies in communication complexity. Technical Report TR19-043, Electronic Colloquium on Computational Complexity (ECCC), 2019. URL: https://eccc.weizmann.ac.il/report/2019/043.
  28. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. Combinatorica, 19(3):403-435, 1999. URL: https://doi.org/10.1007/s004930050062.
  29. Alexander Razborov. On rigid matrices. Technical report, Steklov Mathematical Institute, 1989. In Russian. Google Scholar
  30. Alexander Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992. URL: https://doi.org/10.1016/0304-3975(92)90260-M.
  31. Ronald Rivest. Learning decision lists. Machine Learning, 2(3):229-246, 1987. URL: https://doi.org/10.1007/BF00058680.
  32. Klaus Wagner. Bounded query computations. In Proceedings of the 3rd Structure in Complexity Theory Conference (STRUCTURES), pages 260-277. IEEE, 1988. URL: https://doi.org/10.1109/SCT.1988.5286.
  33. Thomas Watson. A ZPP^NP[1] lifting theorem. In Proceedings of the 36th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 59:1-59:16. Schloss Dagstuhl, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.59.
  34. Gerd Wechsung. On the Boolean closure of NP. In Proceedings of the 5th International Conference on Fundamentals of Computation Theory (FCT), pages 485-493. Springer, 1985. URL: https://doi.org/10.1007/BFb0028832.
  35. Andrew Yao. Some complexity questions related to distributive computing. In Proceedings of the 11th Symposium on Theory of Computing (STOC), pages 209-213. ACM, 1979. URL: https://doi.org/10.1145/800135.804414.
  36. Stathis Zachos and Hans Heller. A decisive characterization of BPP. Information and Control, 69(1-3):125-135, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80044-4.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact