The Strength of Equality Oracles in Communication

Authors Toniann Pitassi, Morgan Shirley, Adi Shraibman

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Author Details

Toniann Pitassi
  • Columbia University, New York, NY, USA
Morgan Shirley
  • University of Toronto, Canada
Adi Shraibman
  • The Academic College of Tel Aviv-Yaffo, Israel


We thank Arkadev Chattopadhyay, Lianna Hambardzumyan, Aleksandar Nikolov, and Suhail Sherif for helpful discussions.

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Toniann Pitassi, Morgan Shirley, and Adi Shraibman. The Strength of Equality Oracles in Communication. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 89:1-89:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


It is well-known that randomized communication protocols are more powerful than deterministic protocols. In particular the Equality function requires Ω(n) deterministic communication complexity but has efficient randomized protocols. Previous work of Chattopadhyay, Lovett and Vinyals shows that randomized communication is strictly stronger than what can be solved by deterministic protocols equipped with an Equality oracle. Despite this separation, we are far from understanding the exact strength of Equality oracles in the context of communication complexity. In this work we focus on nondeterminisic communication equipped with an Equality oracle, which is a subclass of Merlin-Arthur communication. We show that this inclusion is strict by proving that the previously-studied Integer Inner Product function, which can be efficiently computed even with bounded-error randomness, cannot be computed using sublinear communication in the nondeterministic Equality model. To prove this we give a new matrix-theoretic characterization of the nondeterministic Equality model: specifically, there is a tight connection between this model and a covering number based on the blocky matrices of Hambardzumyan, Hatami, and Hatami, as well as a natural variant of the Gamma-2 factorization norm. Similar equivalences are shown for the unambiguous nondeterministic model with Equality oracles. A bonus result arises from these proofs: for the studied communication models, a single Equality oracle call suffices without loss of generality. Our results allow us to prove a separation between deterministic and unambiguous nondeterminism in the presence of Equality oracles. This stands in contrast to the result of Yannakakis which shows that these models are polynomially-related without oracles. We suggest a number of intriguing open questions along this direction of inquiry, as well as others that arise from our work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Factorization norm
  • blocky rank
  • Merlin-Arthur


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  1. Scott Aaronson and Avi Wigderson. Algebrization: A New Barrier in Complexity Theory. ACM Trans. Comput. Theory, 1(1):2:1-2:54, February 2009. URL:
  2. Alfred V. Aho, Jeffrey D. Ullman, and Mihalis Yannakakis. On notions of information transfer in VLSI circuits. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC '83, pages 133-139, New York, NY, USA, December 1983. Association for Computing Machinery. URL:
  3. Daniel Avraham and Amir Yehudayoff. On blocky ranks of matrices. Technical Report 137, Electronic Colloquium on Computational Complexity, 2022. URL:
  4. Laszlo Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory. In 27th Annual Symposium on Foundations of Computer Science (Sfcs 1986), pages 337-347, October 1986. URL:
  5. Béla Bollobás. On generalized graphs. Acta Mathematica Academiae Scientiarum Hungaricae, 16(3):447-452, September 1965. URL:
  6. Nicolas Bousquet, Aurélie Lagoutte, and Stéphan Thomassé. Clique versus independent set. European Journal of Combinatorics, 40:73-92, August 2014. URL:
  7. Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture. In Mikołaj Bojańczy and Chandra Chekuri, editors, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), volume 213 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:16, Dagstuhl, Germany, 2021. Schloss Dagstuhl endash Leibniz-Zentrum für Informatik. URL:
  8. Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality Alone Does Not Simulate Randomness. Technical Report 206, Electronic Colloquium on Computational Complexity, 2018. URL:
  9. Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality Alone Does not Simulate Randomness. In Amir Shpilka, editor, 34th Computational Complexity Conference (CCC 2019), volume 137 of Leibniz International Proceedings in Informatics (LIPIcs), pages 14:1-14:11, Dagstuhl, Germany, 2019. Schloss Dagstuhlendash Leibniz-Zentrum fuer Informatik. URL:
  10. Arkadev Chattopadhyay, Nikhil S. Mande, and Suhail Sherif. The Log-Approximate-Rank Conjecture Is False. J. ACM, 67(4):23:1-23:28, June 2020. URL:
  11. Dmitry Gavinsky. The Layer Complexity of Arthur-Merlin-like Communication. Theory of Computing, 17(8):1-28, September 2021. URL:
  12. Mika Göös. Lower Bounds for Clique vs. Independent Set. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1066-1076, October 2015. URL:
  13. Mika Göös, Toniann Pitassi, and Thomas Watson. The Landscape of Communication Complexity Classes. comput. complex., 27(2):245-304, June 2018. URL:
  14. Mika Göös and Thomas Watson. Communication Complexity of Set-Disjointness for All Probabilities. Theory of Computing, 12(1):1-23, August 2016. URL:
  15. Lianna Hambardzumyan, Hamed Hatami, and Pooya Hatami. Dimension-free bounds and structural results in communication complexity. Isr. J. Math., October 2022. URL:
  16. Lane A. Hemaspaandra and Heribert Vollmer. The satanic notations: Counting classes beyond #P and other definitional adventures. SIGACT News, 26(1):2-13, March 1995. URL:
  17. Hartmut Klauck. Lower Bounds for Quantum Communication Complexity. SIAM J. Comput., 37(1):20-46, January 2007. URL:
  18. Matthias Krause. Geometric arguments yield better bounds for threshold circuits and distributed computing. Theoretical Computer Science, 156(1):99-117, March 1996. URL:
  19. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  20. Troy Lee and Adi Shraibman. An Approximation Algorithm for Approximation Rank. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 351-357, July 2009. URL:
  21. Troy Lee and Adi Shraibman. Lower Bounds in Communication Complexity. Found. Trends Theor. Comput. Sci., 3(4):263-399, October 2009. URL:
  22. Troy Lee, Adi Shraibman, and Robert Špalek. A Direct Product Theorem for Discrepancy. In 2008 23rd Annual IEEE Conference on Computational Complexity, pages 71-80, June 2008. URL:
  23. Nati Linial and Adi Shraibman. Learning Complexity vs Communication Complexity. Combinatorics, Probability and Computing, 18(1-2):227-245, March 2009. URL:
  24. Nati Linial and Adi Shraibman. Lower bounds in communication complexity based on factorization norms. Random Structures & Algorithms, 34(3):368-394, 2009. URL:
  25. Noam Nisan. The Communication Complexity of Threshold Gates. In Combinatorics, Paul Erdős Is Eighty, pages 301-315, 1994. Google Scholar
  26. Periklis Papakonstantinou, Dominik Scheder, and Hao Song. Overlays and Limited Memory Communication. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 298-308, June 2014. URL:
  27. Michal Parnas, Dana Ron, and Adi Shraibman. The Boolean rank of the uniform intersection matrix and a family of its submatrices. Linear Algebra and its Applications, 574:67-83, August 2019. URL:
  28. Toniann Pitassi, Morgan Shirley, and Thomas Watson. Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity. comput. complex., 30(2):1-48, July 2021. URL:
  29. Anup Rao and Amir Yehudayoff. Communication Complexity and Applications. Cambridge University Press, Cambridge, 2020. URL:
  30. Mihalis Yannakakis. Expressing combinatorial optimization problems by Linear Programs. Journal of Computer and System Sciences, 43(3):441-466, December 1991. URL:
  31. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (Preliminary Report). In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC '79, pages 209-213, New York, NY, USA, April 1979. Association for Computing Machinery. URL:
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