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 xed (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 di erent: 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.