On the P versus NP intersected with co-NP question in communication complexity

We consider the analog of the P versus NP ∩ co-NP question for the classical two-party communication protocols where polynomial time is replaced by poly-logarithmic communication: if both a boolean function f and its negation ¬ f have small (poly-logarithmic in the number of variables) nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition model of communication this question was answered by Aho, Ullman and Yannakakis in 1983: here P = NP ∩ co-NP.We show that in the best partition model of communication the situation is entirely different: here P is a proper subset even of RP ∩ co-RP. This, in particular, resolves an open question raised by Papadimitriou and Sipser in 1982.

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