Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

Authors Toniann Pitassi, Morgan Shirley, Thomas Watson



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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

Acknowledgements

We thank Benjamin Rossman for helpful comments and discussions.

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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

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.

Subject Classification

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

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