Complexity classes in communication complexity theory

We take a complexity theoretic view of A. C. Yao's theory of communication complexity. A rich structure of natural complexity classes is introduced. Besides providing a more structured approach to the complexity of a variety of concrete problems of interest to VLSI, the main objective is to exploit the analogy between Turing machine (TM) and communication complexity (CC) classes. The latter provide a more amicable environment for the study of questions analogous to the most notorious problems in TM complexity. Implicitly, CC classes corresponding to P, NP, coNP, BPP and PP have previously been considered. Surprisingly, pcc = Npcc ∩ coNPcc is known [AUY]. We develop the definitions of PSPACEcc and of the polynomial time hierarchy in CC. Notions of reducibility are introduced and a natural complete member in each class is found. BPPcc ⊆ Σ2cc ∩ Π2cc [Si2] remains valid. We solve the question that BPPcc ? NPcc by proving an Ω(√n) lower bound for the bounded-error complexity of the coNPcc- complete problem "disjointness". Similar lower bounds follow for essentially any nontrivial monotone graph property. Another consequence is that the deterministically exponentially hard "equality" relation is not NPcc-hard with respect to oracle-protocol reductions. We prove that the distributional complexity of the disjointness problem is O(√n log n) under any product measure on {0, 1}n × {0, 1}n. This points to the difficulty of improving the Ω(√n) lower bound for the B2PP complexity of "disjointness". The variety of counting and probabilistic classes appears to be greater than in the Turing machine versions. Many of the simplest graph problems (undirected reachability, planarity, bipartiteness, 2-CNF-satisfiability) turn out to be PSPACEcc-hard. The main open problem remains the separation of the hierarchy, more specifically, the conjecture that Σ2cc ≠ Π2cc. Another major problem is to show that PSPACEcc and the probabilistic class UPPcc are not comparable.

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