Relations between communication complexity classes

Abstract

We study the complexity of communication between two processors in terms of complexity classes as introduced by Babai, Frankl, and Simon ( in “Proceedings, 27th Annu. IEEE Sympos. on Found. of Comput. Sci., 1986,” pp. 337–347). They showed some analogies between Turing machine (TM) classes like P, NP , Σ k etc. and the corresponding communication complexity classes which we denote by C- P , C- NP , C- Σ k etc. This paper continues these investigations, in particular, the Boolean hierarchy C- BH and probabilistic classes are considered. We enlarge this correspondence by showing that C- Σ k+1 can also be characterized by a C- NP -protocol that uses only one question to a C- Σ k -oracle. For the probabilistic class C- PP it suffices to consider probabilistic protocols with moderately bounded error, this solves an open problem suggested in op. cit. In contrast to the case of TM complexity, we are able to show some proper inclusions like C- BH ∪ C- BPP ⊂ C- Σ 2 ∩ C- π 2 ν C- PP and C- PP ⊂ C- P c-# P . The nonuniformity of communication protocols allows us to show that the Boolean communication hierarchy does not collapse.