This article introduces a theory of communication that covers the following generic scenario: Alice knows more than Bob about a certain set of logic propositions and Alice and Bob wish to communicate as efficiently as possible with the shared goal that, following their communication, Bob should be able to deduce a particular logic proposition that Alice knows to be true. We assume that our logic system is propositional logic, and we build on top of one of the legendary works in this area, namely the work of Carnap and Bar-Hillel on a theory of semantic information. Our main contribution is a collection of theorems studying various different assumptions on what Alice and Bob know and what their goal is. These theorems all provide sharp upper and lower bounds phrased in terms of an entropy-like function that we call $\Lambda$, in reference to its apparent connection to problems of communication involving logic. It turns out that when the goal is to communicate only a portion of the knowledge that Alice possesses, the optimum communication cost is lower than most people seem to assume, yet unavoidably, such optimum communication strategies end up allowing Bob to prove even more things than originally intended. Another interesting outcome is that in some scenarios, Alice need not know the logic statements that Bob knows in order to attain asymptotically the same communication efficiency as if she knew the statement, in a nod to the famous Slepian-Wolf and Wyner-Ziv results from source coding theory. Our work also introduces practical codes, which are comprised of a combination of linear codes and enumerative source codes, which turn out to be asymptotically optimal for some scenarios.

翻译：本文提出了一种通信理论，涵盖以下通用场景：爱丽丝比鲍勃更了解一组逻辑命题，而爱丽丝和鲍勃希望以尽可能高效的方式进行通信，其共同目标是：在通信结束后，鲍勃应能推导出爱丽丝已知为真的特定逻辑命题。我们假设逻辑系统为命题逻辑，并以此领域经典著作——卡尔纳普与巴尔-希勒尔关于语义信息理论的研究为基础。主要贡献在于：我们针对爱丽丝和鲍勃的知识状态及其目标的多种不同假设，提出了一系列定理。这些定理均给出了以称为Λ的类熵函数表达的严格上下界，该函数因与涉及逻辑的通信问题存在明显关联而得名。研究发现，当目标仅需传递爱丽丝所掌握部分知识时，最优通信成本低于多数人的假设，但此类最优通信策略不可避免地会使鲍勃推导出超出原始意图的更多结论。另一有趣结论是：在特定场景中，爱丽丝无需知晓鲍勃已知的逻辑语句，即可在渐近意义上达到与之等价的通信效率——这呼应了信源编码理论中著名的Slepian-Wolf与Wyner-Ziv结论。本文还引入由线性码与枚举信源码组合构成的实用编码方案，该方案在若干场景中渐近最优。