Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity
We investigate the power of randomness in two-party communicationcomplexity. In particular, we study the model where theparties can make a constant number of queries to a function that has anefficient one-sided-error randomized protocol. The complexity classesdefined by this model comprise the Randomized Boolean Hierarchy,which is analogous to the Boolean Hierarchy but defined with one-sidederrorrandomness instead of nondeterminism. Our techniques connectthe Nondeterministic and Randomized Boolean Hierarchies, and we providea complete picture of the relationships among complexity classeswithin and across these two hierarchies. In particular, we prove thatthe Randomized Boolean Hierarchy does not collapse, and we prove aquery-to-communication lifting theorem for all levels of the NondeterministicBoolean Hierarchy and use it to resolve an open problem statedin the paper by Halstenberg and Reischuk (CCC 1988) which initiatedthe study of this hierarchy.
Pitassi, T., Shirley, M., & Watson, T. (2021). Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity. Computational Complexity, 30 (2) https://doi.org/10.1007/s00037-021-00210-5