Complexity Zoo:List of Communication Complexity Classes

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Complexity classes by letter: Symbols - A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z

Lists of related classes: Communication Complexity - Hierarchies - Nonuniform

Communication complexity deals with how much data must be exchanged between parties cooperating to compute a function whose input is split amongst them. Many computational complexity classes have communication complexity analogues. For convenience, we list here those analogues present in the Zoo.

In the literature, these names sometimes refer to classes of total functions, and sometimes refer to classes of partial functions (promise problems); for some classes this makes a big difference! Also, in the literature, these class names are sometimes overloaded to refer to the corresponding communication complexity measure (e.g., P^cc(f) may refer to the deterministic communication complexity of f, while P^cc also stands for the class of all f for which P^cc(f) is at most polylog(n).)

Often, inclusions between communication complexity classes, such as MA^cc⊆AM^cc hold for exactly the same reasons as they do for the corresponding classical complexity classes.

For surveys on two-party communication complexity classes (as well as sources of many of the original definitions), see [BFS86], [HR90], [GPW16b].

Two-party classes:

⊕P^cc - AM^cc - BPP^cc - coNP^cc - MA^cc - NP^cc - P^cc - PH^cc - P^NPcc - PostBPP^cc - PP^cc - PSPACE^cc - RP^cc - SBP^cc - UAM^cc - UP^cc - UPostBPP^cc - UPP^cc - USBP^cc - UWAPP^cc - WAPP^cc - ZAM^cc - ZPP^cc

Multi-party number-on-forehead classes:

BPP_$k$^cc - NP_$k$^cc - RP_$k$^cc - P_$k$^cc -