Shannon Capacity is Achievable for a Large Class of Interactive Markovian Protocols

We address the problem of simulating a binary interactive 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. We analyze the family of Mth-order Markovian protocols in which the transmission at every time depends only on the last M bits of the protocol. For M =1 (first-order Markovian) we prove that all protocols can be simulated at Shannon's capacity. For M > 1 we characterize large classes of protocols that can be simulated at Shannon's capacity.