二进制交互式一阶马尔可夫协议的Shannon容量
Shannon Capacity is Achievable for Binary Interactive First-Order Markovian Protocols
摘要：
研究了在一对具有交叉概率$\varepsilon$的二进制对称信道上模拟任意二进制交互式一阶马尔可夫协议的问题。我们感兴趣的是可靠模拟的可实现速率，即，在表征通信中的最小可能爆破时，使得可以获得消失的错误概率（在协议长度中）。而对于一般的交互式协议，每一方的输出可能取决于其对应方的所有先前输出，在（一阶）马尔可夫协议中，这种依赖性仅限于最后观察到的输出。以前的工作在该领域讨论了更广泛的家庭的协议，但评估可实现的速率仅在极限，其中$\varepsilon\to0$。在本文中，我们证明了单向香农容量，$1-h(\varepsilon)$，可以实现任何二进制一阶马尔可夫协议。这个令人惊讶的结果，是尽我们所知，第一个例子中，非平凡的交互式协议可以模拟香农容量。我们给出了两种容量实现编码方案，它们都将协议划分为独立的块，并实现了垂直块编码。第一种方案是基于随机分离成具有可变长度的块。第二种方案基于确定性的块分离，并有效地预测它们的最后一次传输。预测可以被看作是一个二进制指针跳跃游戏，我们表明，最后一步可以计算与$O(\log m)$位，其中$m$是在游戏中的轮数。最后，我们讨论了可能的扩展结果的高阶模型的工作。
英文摘要：
We address the problem of simulating an arbitrary binary interactive first-order Markovian protocol over a pair of 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. Previous works in the field discuss broader families of protocols but assess the achievable rates only at the limit where $\varepsilon\to0$. In this paper, we prove that the one-way Shannon capacity, $1-h(\varepsilon)$, 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. We give two capacity achieving coding schemes, which both divide the protocol into independent blocks, and implement vertical block coding. The first scheme is based on a random separation into blocks with variable lengths. The second scheme is based on a deterministic separation into blocks, and efficiently predicting their last transmission. The prediction can be regarded as a binary pointer jumping game, for which we show that the final step can be calculated with $O(\log m)$ bits, where $m$ is the number of rounds in the game. We conclude the work by discussing possible extensions of the result to higher order models.
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