Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

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 that has 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-sidederror 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.

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Acknowledgements

A previous version of this article appeared at ICALP 2020 (Pitassi et al. 2020). Toniann Pitassi was supported by NSERC, NSF Grant No. CCF-1900460 and the IAS School of Mathematics. Morgan Shirley was supported by NSERC. Thomas Watson was supported by NSF grants CCF-1657377 and CCF-1942742. We thank Benjamin Rossman for helpful comments and discussions.

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Pitassi, T., Shirley, M. & Watson, T. Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity. comput. complex. 30, 10 (2021). https://doi.org/10.1007/s00037-021-00210-5

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Keywords

  • Boolean hierarchies
  • Lifting theorems
  • Query complexity

Subject classification

  • 68Q11