# Analysis of Semantic Communication for Logic-based Hypothesis Deduction

## TL;DR

This paper analyzes semantic communication in logic-based hypothesis deduction, formulating an optimal message selection strategy that improves inference accuracy under resource constraints, validated through theoretical analysis and experiments.

## Contribution

It introduces a novel approach to semantic communication for FOL-based deduction, deriving optimal strategies using constrained resource allocation and Bayesian hypothesis testing.

## Key findings

- Achieves near-optimal hypothesis identification within resource limits.
- Demonstrates reduced error compared to random and prior methods.
- Provides theoretical convergence analysis and validation through experiments.

## Abstract

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.