This work presents an analysis of semantic communication in the context of First-Order Logic (FOL)-based deduction. Specifically, the receiver holds a set of hypotheses about the State of the World (SotW), while the transmitter has incomplete evidence about the true SotW but lacks access to the ground truth. The transmitter aims to communicate limited information to help the receiver identify the hypothesis most consistent with true SotW. We formulate the objective as approximating the posterior distribution of the transmitter at the receiver. Using Stirling's approximation, this reduces to a constrained, finite-horizon resource allocation problem. Applying the Karush-Kuhn-Tucker conditions yields a truncated water-filling solution. Despite the problem's non-convexity, symmetry and permutation invariance ensure global optimality. Based on this, we design message selection strategies, both for single- and multi- round communication, and model the receiver's inference as an $m$-ary Bayesian hypothesis testing problem. Under the Maximum A Posteriori (MAP) rule, our communication strategy achieves optimal performance within budget constraints. We further analyze convergence rates and validate the theoretical findings through experiments, demonstrating reduced error over random selection and prior methods.

翻译：本文对基于一阶逻辑推理的语义通信进行了分析。具体而言，接收端持有关于世界状态的一组假设，而发送端虽掌握关于真实世界状态的不完整证据，却无法获知真实情况。发送端的目标是传递有限信息，以协助接收端识别与真实世界状态最相符的假设。我们将该目标形式化为在接收端近似发送端的后验分布。利用斯特林近似，该问题可简化为一个有限时域的约束性资源分配问题。应用Karush-Kuhn-Tucker条件可得到截断注水解。尽管该问题具有非凸性，但其对称性与置换不变性保证了全局最优性。基于此，我们设计了适用于单轮及多轮通信的消息选择策略，并将接收端的推理建模为一个$m$元贝叶斯假设检验问题。在最大后验概率准则下，所提出的通信策略在预算约束内实现了最优性能。我们进一步分析了收敛速率，并通过实验验证了理论结果，证明了该方法相较于随机选择及现有方法具有更低的误差。