Fundamental limits incorporating logical reasoning into Shannon's information theory

A cornerstone of Shannon's famous information theory is the idea of decoupling the meaning of a message from its efficient transmission. In this article, we propose an extension of Shannon's communication model where the sender and receiver are assumed to have reasoning capabilities. In such a setting, we gain insights by coupling the fields of information theory and mathematical logic. Under the assumption that a message is coming from a stochastic source, Shannon's theory establishes that the fundamental compression limit is the entropy of the source. Imagine, however, that we were not interested in the message itself, but rather, we were focused on the logical conclusions that one could derive from it. In this work, we obtain a closed-form expression for a fundamental compression limit in the presence of reasoning capabilities. Our expression is valid under various assumptions about what the sender and receiver know, providing initial answers to key questions such as how much the fundamental limit varies as one narrows or widens the amount of knowledge that is being transferred from sender to receiver, or how, surprisingly, such a fundamental limit remains the same even if the sender is unaware of what it is that a receiver already knows. We also offer practical algorithms that are empirically demonstrated to be significantly more efficient than alternatives that do not account for the existence of reasoning capabilities.