Non-Deterministic Communication Complexity with Few Witnesses

We study non-deterministic communication protocols in which no input has too many witnesses. De ne nk(f) to be the minimum complexity of a non-deterministic protocol for the function f in which each input has at most k witnesses. We present two di erent lower bounds for nk(f). Our rst result shows that nk(f) is bounded below by (qc(f)=k) where c(f) is the deterministic complexity. Our second results bounds nk(f) by log(rk(Mf ))=k 1 where rk(Mf ) is the rank of the representing matrix of f . As a consequence, it follows that the communication complexity analogue of the Turing-complexity class FewP is equal to the analogue of the class P . 3