Interactive information complexity and its applications

Given a two-party Boolean function f : {0, 1} n {0, 1} n {0, 1} that maps input (x, y) to f (x, y), communication complexity studies how many communicated bits must be exchanged between two players, Alice who knows only x, and Bob who knows only y, in order for them to jointly compute f (x, y).Since Andrew Yao defined the communication model in the late 1970s, communication complexity has steadily developed without the influence of information theory, which was founded by Claude Shannon in the late 1940s to study coding theory.However, with the introduction of information theory into communication complexity, in recent decades, a new research topic in computational complexity theory has emerged: information complexity.Within Yao's communication model, information complexity studies how much information a protocol reveals about the players' input.Allowing for a degree of error > 0 when computing a Boolean function potentially requires less information revealed.For example, any Boolean-valued function can be computed with an error 1/2 by a random guess, that has essentially no communication and reveals no information about the inputs.This thesis studies how information complexity changes as one allows for different errors when computing a Boolean function.The two main questions studied are:(1). (small error) How much information can be saved by allowing a small error > 0, as compared to cases when no error is allowed at all?(2). (large error) How much information must be revealed in order to have an error of at most 1/2 -?We systematically study these two questions for arbitrary functions, obtaining virtually complete answers for both.For Question (1), we show that at least (h( )) and at most i [f, , ]: the communication task of computing f with an error at most when inputs are sampled according to a distribution , page 17.IC (): the internal information cost of a protocol with respect to an input distribution , page 18. IC (T ): the internal information complexity of a communication task T with respect to an input distribution , page 18. IC ext (): the external information cost of a protocol with respect to an input distribution , page 19.IC ext (T ): the external information complexity of a communication task T with respect to an input distribution , page 19.IC (f ): the internal information complexity of performing the task [f, 0] with respect to an input distribution , page 19.IC (f, ): the internal information complexity of performing the task [f, ] with respect to an input distribution , page 19.IC ext (f, ): the external information complexity of performing the task [f, ] with respect to an input distribution , page 20.IC (f, , ): the internal information complexity of performing the task [f, , ] with respect to an input distribution , page 19.IC ext (f, , ): the external information complexity of performing the task [f, , ] with respect to an input distribution , page 20.IC(f, ): the prior-free information complexity of f with an error , page 20.IC D (f, ): the prior-free distributional information complexity of f with an error , page 20.(X Y): the set of probability distributions on X Y, page 30.p a (t): Pr[ = t|a] where is the random transcript of and a is an input, page 46.

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