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Simulation Theorems via Pseudo-random Properties


We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999), to any gadget which satisfies a certain hitting property. We prove that inner product and gap-Hamming satisfy this property, and as a corollary, we obtain a deterministic simulation theorem for these gadgets, where the gadget’s input size is logarithmic in the input size of the outer function. This yields the first deterministic simulation theorem with a logarithmic gadget size, answering an open question posed by Göös, Pitassi & Watson (in: Proceedings of the 56th FOCS, 2015).

Our result also implies the previous results for the indexing gadget, with better parameters than was previously known. Moreover, a simulation theorem with logarithmic-sized gadget implies a quadratic separation in the deterministic communication complexity and the logarithm of the 1-partition number, no matter how high the 1-partition number is with respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson (2015).


  • Boaz Barak, Mark Braverman, Xi Chen & Anup Rao (2013). How to compress interactive communication. SIAM Journal on Computing 42(3), 1327–1363.

  • Paul Beame, Trinh Huynh & Toniann Pitassi (2010). Hardness amplification in proof complexity. In Proceedings of the 42nd STOC, 87–96

  • Paul Beame, Toniann Pitassi, Nathan Segerlind & Avi Wigderson (2005). A direct sum theorem for corruption and the multiparty NOF communication complexity of set disjointness. In Proceedings of the 20th CCC, 52–66

  • Maria Luisa Bonet, Juan Luis Esteban, Nicola Galesi & Jan Johannsen (2000). On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems. SIAM Journal on Computing 30(5), 1462–1484. ISSN 0097-5397. URL

    MathSciNet  Article  Google Scholar 

  • Mark Braverman & Anup Rao: Information Equals Amortized Communication. IEEE Transactions on Information Theory 60(10), 6058–6069 (2014)

    MathSciNet  Article  Google Scholar 

  • Mark Braverman, Anup Rao, Omri Weinstein & Amir Yehudayoff (2013a). Direct Product via Round-Preserving Compression. In Proceedings of the 40th ICALP, 232–243

  • Mark Braverman, Anup Rao, Omri Weinstein & Amir Yehudayoff (2013b). Direct Products in Communication Complexity. In Proceedings of the 54th FOCS, 746–755

  • Joshua Brody, Harry Buhrman, Michal Kouckỳ, Bruno Loff, Florian Speelman & Nikolay Vereshchagin (2013). Towards a reverse newman's theorem in interactive information complexity. In Proceedings of the 28th CCC, 24–33

  • Arkadev Chattopadhyay (2007). Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits. In Proceedings of the 48th FOCS, 449–458

  • Arkadev Chattopadhyay (2009). Circuits, Communication and Polynomials. Ph.D. thesis, McGill University

  • Arkadev Chattopadhyay & Anil Ada (2008). Multiparty Communication Complexity of Disjointness. Technical Report TR08-002, Electronic Colloquium on Computational Complexity (ECCC). URL

  • Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff & Sagnik Mukhopadhyay (2017a). Lower Bounds for Elimination via Weak Regularity. In Proceedings of the 34th STACS, 21:1–21:14

  • Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir & Toniann Pitassi (2019). Query-to-communication lifting for BPP using inner product. arXiv:1904.13056

  • Chattopadhyay, Arkadev, Koucký, Michal, Loff, Bruno, Mukhopadhyay, Sagnik: Composition and Simulation Theorems via Pseudo-random Properties. Electronic Colloquium on Computational Complexity (ECCC) 24, 14 (2017b)

    MATH  Google Scholar 

  • David, Matei, Pitassi, Toniann, Viola, Emanuele: Improved separations between nondeterministic and randomized multiparty communication. ACM Transactions on Computation Theory 1(2), (2009)

    Article  Google Scholar 

  • Drucker, Andrew: Improved direct product theorems for randomized query complexity. Computational Complexity 21(2), 197–244 (2012)

    MathSciNet  Article  Google Scholar 

  • Peter Frankl & Zoltán Füredi: A short proof for a theorem of Harper about Hamming-spheres. Discrete Mathematics 34(3), 311–313 (1981)

    MathSciNet  Article  Google Scholar 

  • Mika Göös, Pritish Kamath, Toniann Pitassi & Thomas Watson (2017a). Query-to-communication Lifting for \(\sf P\it ^{\sf NP\it }\). In Proceedings of the 32nd CCC

  • Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson & David Zuckerman (2015). Rectangles are nonnegative juntas. In Proceedings of the 47th STOC, 257–266. ACM

  • Mika Göös & Toniann Pitassi (2014). Communication lower bounds via critical block sensitivity. In Proceedings of the 46th STOC, 847–856

  • Mika Göös, Toniann Pitassi & Thomas Watson (2015). Deterministic communication vs. partition number. In Proceedings of the 56th FOCS

  • Mika Göös, Toniann Pitassi & Thomas Watson (2017b). Query-to-Communication Lifting for BPP. In Proceedings of the 58th FOCS

  • Harper, L.H.: Optimal numberings and isoperimetric problems on graphs. Journal of Combinatorial Theory 1(3), 385–393 (1966)

    MathSciNet  Article  Google Scholar 

  • Prahladh Harsha, Rahul Jain, David McAllester & Jaikumar Radhakrishnan (2007). The communication complexity of correlation. In Proceedings of the 22nd CCC, 10–23

  • Hamed Hatami, Kaave Hosseini & Shachar Lovett (2018). Structure of Protocols for XOR Functions. SIAM J. Comput. 47(1), 208–217. URL

    MathSciNet  Article  Google Scholar 

  • Trinh Huynh & Jakob Nordstrom (2012). On the virtue of succinct proofs: Amplifying communication complexity hardness to time-space trade-offs in proof complexity. In Proceedings of the 44th STOC, 233–248

  • Russell Impagliazzo (1995). Hard-Core Distributions for Somewhat Hard Problems. In Proceedings of the 36th FOCS, 538–545

  • Jain, Rahul: New strong direct product results in communication complexity. Journal of the ACM 62(3), 20 (2015)

    MathSciNet  Article  Google Scholar 

  • Rahul Jain, Hartmut Klauck & Ashwin Nayak (2008). Direct product theorems for classical communication complexity via subdistribution bounds. In Proceedings of the 40th STOC, 599–608

  • Rahul Jain, Attila Pereszlényi & Penghui Yao (2012). A direct product theorem for the two-party bounded-round public-coin communication complexity. In Proceedings of the 53rd FOCS, 167–176

  • Rahul Jain, Jaikumar Radhakrishnan & Pranab Sen (2003). A direct sum theorem in communication complexity via message compression. In Proceedings of the 20th ICALP, 300–315

  • Rahul Jain & Penghui Yao (2012). A strong direct product theorem in terms of the smooth rectangle bound. Technical report, arXiv:1209.0263

  • Jan Johannsen (2001). Depth Lower Bounds for Monotone Semi-Unbounded Fan-in Circuits. ITA 35(3), 277–286. URL

    MathSciNet  Article  Google Scholar 

  • Mauricio Karchmer & Avi Wigderson: Monotone Circuits for Connectivity Require Super-Logarithmic Depth. SIAM Journal on Discrete Mathematics 3(2), 255–265 (1990)

    MathSciNet  Article  Google Scholar 

  • Kerenidis, Iordanis, Laplante, Sophie, Lerays, Virginie, Roland, Jérémie, Xiao, David: Lower bounds on information complexity via zero-communication protocols and applications. SIAM Journal on Computing 44(5), 1550–1572 (2015)

    MathSciNet  Article  Google Scholar 

  • Alexander Kozachinskiy (2018). Raz-McKenzie simulation: new gadget and unimprovability of Thickness Lemma. In Proceedings of the 43rd MFCS

  • Eyal Kushilevitz & Noam Nisan (1997). Communication complexity. Cambridge University Press. ISBN 978-0-521-56067-2

  • Troy Lee, Adi Shraibman & Robert Spalek (2008). A Direct Product Theorem for Discrepancy. In Proceedings of the 23rd CCC, 71–80

  • Troy Lee & Shengyu Zhang (2010). Composition theorems in communication complexity. In Proceedings of the 27th ICALP, 475–489. Springer

  • Levin, Leonid A.: One-way functions and pseudorandom generators. Combinatorica 7(4), 357–363 (1987)

    MathSciNet  Article  Google Scholar 

  • Bruno Loff & Sagnik Mukhopadhyay (2019). Lifting Theorems for Equality. In Proceedings of the 36th STACS, 50:1–50:19. URL

  • Raghu Meka & Toniann Pitassi (editors) (2017). Hardness Escalation in Communication Complexity and Query Complexity, Workshop at 58th FOCS. URL

  • Jakob Nordström (2016). Private communication

  • Denis Pankratov (2012). Direct sum questions in classical communication complexity. Ph.D. thesis, Masters thesis, University of Chicago

  • Anup Rao & Amir Yehudayoff (2015). Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness. In Proceedings of the 30th CCC, 88–101

  • Ran Raz & Pierre McKenzie: Separation of the Monotone NC Hierarchy. Combinatorica 19(3), 403–435 (1999)

    MathSciNet  Article  Google Scholar 

  • Susanna F. de Rezende, Jakob Nordström & Marc Vinyals (2016). How Limited Interaction Hinders Real Communication. In Proceedings of the 56th FOCS

  • Robert Robere, Toniann Pitassi, Benjamin Rossman & Stephen A Cook (2016). Exponential lower bounds for monotone span programs. In Proceedings of the 57th FOCS, 406–415

  • Shaltiel, Ronen: Towards proving strong direct product theorems. Computational Complexity 12(1–2), 1–22 (2003)

    MathSciNet  Article  Google Scholar 

  • Alexander A. Sherstov (2009). Separating AC0 from Depth-2 Majority Circuits. SIAM Journal on Computing 38(6), 2113–2129. URL

    MathSciNet  Article  Google Scholar 

  • Sherstov, Alexander A.: The pattern matrix method. SIAM Journal on Computing 40(6), 1969–2000 (2011)

    MathSciNet  Article  Google Scholar 

  • Alexander A. Sherstov (2012a). The multiparty communication complexity of set disjointness. In Proceedings of the 44th STOC, 525–548

  • Sherstov, Alexander A.: Strong direct product theorems for quantum communication and query complexity. SIAM Journal on Computing 41(5), 1122–1165 (2012b)

    MathSciNet  Article  Google Scholar 

  • Alexander A. Sherstov (2013). Communication lower bounds using directional derivatives. In Proceedings of the 45th STOC, 921–930

  • Yaoyun Shi & Yufan Zhu: Quantum communication complexity of block-composed functions. Quantum Information & Computation 9(5), 444–460 (2009)

    MathSciNet  MATH  Google Scholar 

  • Dmitry Sokolov (2017). Dag-like communication and its applications. In International Computer Science Symposium in Russia, 294–307

    Chapter  Google Scholar 

  • Emanuele Viola & Avi Wigderson: Norms, XOR Lemmas, and Lower Bounds for Polynomials and Protocols. Theory of Computing 4(1), 137–168 (2008)

    MathSciNet  Article  Google Scholar 

  • Thomas Watson (2017). A \(\sf ZPP\it ^{\sf NP\it }\) Lifting Theorem. Unpublished preprint

  • Xiaodi Wu, Penghui Yao & Henry Yuen (2017). Raz-McKenzie simulation with the inner product gadget. Technical Report TR17-010, Electronic Colloquium on Computational Complexity (ECCC). URL

  • Yannakakis, Mihalis: Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 43(3), 441–466 (1991)

    MathSciNet  Article  Google Scholar 

  • Andrew Chi-Chih Yao (1979). Some Complexity Questions Related to Distributive Computing (Preliminary Report). In Proceedings of the 11h STOC, 209–213

  • Andrew Chi-Chih Yao (1982). Theory and Applications of Trapdoor Functions (Extended Abstract). In Proceedings of the 23rd FOCS, 80–91

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Part of the research for this work was done at the Institut Henri Poincaré, as part of the workshop Nexus of Information and Computation Theories.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 616787. The first author was partially supported by a Ramanujan Fellowship of the DST, India, and the last author is partially supported by a TCS fellowship and by European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 715672.

The research leading to these results has also received funding from the Foundation for Science and Technology (FCT), Portugal, grant number SFRH/BPD/116010/2016. This work is partially funded by the ERDF through the COMPETE 2020 programme within project POCI-01-0145-FEDER-006961, and by National Funds through the FCT as part of project UID/EEA/50014/2013.

We are thankful to the anonymous referees for their thorough reading and numerous suggestions for improving the paper.

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Correspondence to Sagnik Mukhopadhyay.

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Chattopadhyay, A., Koucký, M., Loff, B. et al. Simulation Theorems via Pseudo-random Properties. comput. complex. 28, 617–659 (2019).

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  • Communication complexity
  • lifting theorem
  • simulation theorem
  • Inner-product
  • gap-Hamming

Subject classification

  • Theory of computation
  • Communication Complexity