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Equality Alone Does not Simulate Randomness

Authors Arkadev Chattopadhyay, Shachar Lovett, Marc Vinyals

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Arkadev Chattopadhyay
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
Shachar Lovett
  • Department of Computer Science and Engineering, University of California, San Diego, USA
Marc Vinyals
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India


We are grateful to Bruno Loff, Jaikumar Radhakrishnan and especially Suhail Sherif for helpful discussions in the early stages of this work. We thank Sagnik Mukhopadhyay and anonymous reviewers for providing useful feedback to an earlier version of this manuscript. We also thank the Simons Institute for the Theory of Computing at Berkeley where some of this work took place.

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Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality Alone Does not Simulate Randomness. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it. Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Communication lower bound
  • derandomization


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