Complexity classes in communication complexity theory (preliminary version) Laszl6 Babai Eotvos University, Budapest and the University of Chicago Peter Frankl c. N. R. S., Paris Janos Simon University of Chicago A.bstraet. We take a complexity theoretic view of A. C. Yao's theory of communication complexity. A rich struc- ture of natural complexity classes is introduced. Besides providing a more structured approach to the complex- ity of a variety of concrete problems of interest to VLSI, the main objective is to exploit the analogy between Tur- ing 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. Surpris- ingly, pcc = Np cC n coNp cC is known [AUY]. We develop the definitions of PSPACE cC and of the polynomial tinle hierarchy in CC. Notions of reducibility are introduced and a natural complete member in each class is found. BPp cC E;c n H;c [Si2] remains valid. We solve the question that BPp cc Np cC by proving an O( vIR) lower bound for the bounded-error complexity of the coNp cc _ complete problem "disjointness". Similar lower bounds follow for essentially any nontrivial monotone graph prop- erty. Another consequence is that the deterministically exponentially hard "equality" relation is not Np cC -hard with respect to oracle-protocol reductions. We prove that the distributional complexity of the dis- jointness problem is O( vIR log n) under any product mea- sure on {O, l}R X {O, 1 }R. This points to the difficulty of improving the O( vIR) lower bound for the B2PP com- plexity of "disjointness" . The variety of counting and probabilistic classes ap- pears to be greater than in the Turing machine versions. Many of the simplest graph problems (undirected reacha- bility, 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 E 2 c =F II 2 c. Another major problem is to show that PSPACE cC and the probabilistic class Upp cc are not comparable. 0272-5428/86/0000/0337$01.00 © 1986 IEEE 337 1. IntrodnetioD Motivated by VLSI applications, research in comnlunica- tion complexity has so far mainly focused on lower bounds for protocols computing specific functions. In this paper we take a look at communication COIll- plexity from the point of view of ("machine based") conl- plexity theory. We find a rich .structure of natural conl- plexity classes, providing a structured franlework for the classification of various concrete functions, by introducing notions of reducibility and highlighting complete prob- lems in diff'erent classes. This stucture may occasionally serve as a guide to finding lower bounds of significCl nce to VLSI, although this should not be the primary objec- tive of this theory. An example is given in Corollary 9.6; the recognition that simple graph problenls such as con- nectedness between a pair of points are PSPACEcc-hard has lead to an O(n) lower bound for the bounded-error proba.bilistic complexity of these problellls. The prinlary goal, however, is to gain insight into the nature of alternation, counting and probabilistic conlplex- ity in a context where the chances of progress might be greater than for the analogous questions in Turing ma- chine complexity. In the basic model, introduced by Yao [Yal], two com- municating parties (North and South) want to coopera- tively determine the value f(x, y) of a Boolean function f in 2n variables. Both North and South have com- plete information about f and unlimited computationa.l power but receive only half of the input (x and y, reap.) (x, y E {O, 1 }R). They excha.nge bits according to some protocol until one of them (South) declares the value of f(x, y). The objective is to Ininimize the nunlber of bits exchanged. This model and its bounded-error probabilistic ver- sion have proved a useful tool in obtaining area-tinle tradeoff's for VLSI computation ([Th], [Ya2]). Non- deterministic protocols were introduced by Lipton and Sedgewick [LS], nlainly because it was a.pparent that the known lower bound techniques for deterlninistic protocols worked for nondeterministic ones as well. (Although the