Physical artificial intelligence (AI) systems involve distributed sensing agents with embedded AI models that must coordinate to perceive, reason, and act in networked environments. Transmitting raw sensor data incurs significant communication overhead, latency, and redundancy. While semantic communication (SC) mitigates these challenges by transmitting task-relevant information, existing deep learning-based joint source-channel coding approaches exhibit limited adaptability, poor out-of-distribution generalization, and scalability challenges. To address these limitations, this paper proposes a framework for compositional semantic communication (CSC), enabling heterogeneous physical AI sources to transmit semantic representations (SRs) that compose meaningfully at a base station (BS) or edge server for remote inference. First, a category-theoretic measure of compositional semantics is developed to quantify each device's contribution to inference tasks beyond mutual information. Second, Grothendieck topologies and presheaves formalize semantic composition across devices, ensuring consistency and task relevance. Building on these foundations, multi-device coordination is formulated as a Stackelberg game in which devices commit to encoding strategies and the BS optimally composes received SRs. An ADMM-based algorithm computes equilibrium signaling strategies. Equilibrium existence is established under mild conditions and is Pareto optimal when compositional information yields increasing collective benefit. Simulation results demonstrate that the proposed approach achieves up to 17% bandwidth reduction and 53% lower end-to-end latency than cooperative multi-agent, distributed gradient descent, and uniform-selection CSC baselines while maintaining 85% inference accuracy across diverse autonomous driving scenarios.

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