Communication on noisy channels: a coding theorem for computation

Abstract

Communication is critical to distributed computing, parallel computing, or any situation in which automata interact-hence its significance as a resource in computation. In view of the likelihood of errors occurring in a lengthy interaction, it is desirable to incorporate this possibility in the model of communication. The author relates the noisy channel and the standard (noise less channel) complexities of a communication problem by establishing a `two-way' or interactive analogue of Shanon's coding theorem: every noiseless channel protocol can be simulated by a private-coin noisy channel protocol whose time bound is proportional to the original (noiseless) time bound and inversely proportional to the capacity of the channel, while the protocol errs with vanishing probability. The method involves simulating the original protocol while implementing a hierarchical system of progress checks which ensure that errors of any magnitude in the simulation are, with high probability, rapidly eliminated.