Interactive Coding for Markovian Protocols

We address the problem of simulating an arbitrary Markovian interactive protocol over binary symmetric channels with crossover probability $\varepsilon$. 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 the special case where there is no dependence on previous outputs (no interaction), the maximal achievable rate is given by the (one-way) Shannon capacity $1-h(\varepsilon)$. For Markovian protocols, we first show that a rate of $\frac{2}{3}(1-h(\varepsilon))$ can be trivially achieved. We then describe a more involved coding scheme and provide a closed-form lower bound for its rate at any noise level $\varepsilon$. Specifically, we show that this scheme outperforms the trivial one for any $\varepsilon<0.044$, and achieves a rate higher than $\frac{1-h(\varepsilon)}{1+h(\varepsilon)+h\left(<\varepsilon(2-\varepsilon)>\right)}=1-\Theta(h(\varepsilon))$ as $\varepsilon\to 0$, which is order-wise the best possible. This should be juxtaposed with a result of Kol and Raz that shows the capacity for interactive protocols with alternating rounds is lower bounded by $1-O(\sqrt{h(\varepsilon)})$.