Formal Semantics for Kolmogorov-Arnold Network Representations of Operational Games

This paper develops a formal semantic framework for Kolmogorov-Arnold Network (KAN) representations of operational games. Operational games, as generalizations of strategic and dynamic games, provide powerful tools for modeling complex multi-agent interactions, but their analysis using traditional methods is limited due to the high dimensionality of state and strategy spaces. Drawing upon the constructive logical systems approach and the Kolmogorov-Arnold representation theorem, we establish rigorous mathematical foundations for embedding operational game dynamics within KAN architectures. The framework encompasses structural, operational, and denotational semantics, facilitating theoretical analysis of convergence properties, expressiveness, and computational complexity. We establish convergence theorems for KAN-based representations of equilibrium strategies and demonstrate how KANs' learnable edge functions provide unique advantages for modeling complex multi-agent systems governed by operational game dynamics. Theoretical results establish conditions for representation adequacy, computational tractability, and semantic preservation under aggregation operations. The proposed framework advances the integration of operational game theory with KAN methodologies, providing a foundation for empirical implementation and theoretical extension in domains with complex strategic interactions.