Shannon Capacity is Achievable for Binary Interactive First-Order Markovian Protocols
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We address the problem of simulating an arbitrary binary interactive first-order Markovian protocol over a pair of binary symmetric channels with crossover probability ε. We are interested in the achievable rates of reliable simulation, i.e., in characterizing the smallest possible blowup in communications such that a vanishing error probability (in the protocol length) can be attained. Whereas for general interactive protocols the output of each party may depend on all previous outputs of its counterpart, in a (first-order) Markovian protocol this dependence is limited to the last observed output only. In this paper we prove that the one-way Shannon capacity, 1-h(ε), can be achieved for any binary first-order Markovian protocol. This surprising result, is to the best of our knowledge, the first example in which non-trivial interactive protocol can be simulated in the Shannon capacity. Our scheme is based on two simple notions: non-interactive simulation, block-wise interactive communication. Previous results in the field discuss different families of protocol and mostly assess the achievable rates at the limit where ε→0. We also show that for higher order Markovian protocols, if the transmission functions are drawn uniformly i.i.d, the probability of drawing a non-capacity achieving protocol goes to zero with n.
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