First-Order Coalition Logic

2025-05-11

AI Summary

This paper addresses the limited expressiveness of Coalition Logic (CL) and Strategy Logic (SL) by introducing First-Order Coalition Logic (FOCL), the first logic enabling first-order quantification over agent actions, thereby unifying and extending their modeling capabilities. Methodologically, we develop a formal FOCL system grounded in game-theoretic semantics and provide the first sound and complete axiomatic framework for a variant of SL. Theoretical analysis shows that FOCL is strictly more expressive than CL; crucially, it establishes a fundamental boundary with SL regarding decidability: FOCL is decidable, whereas SL is undecidable—thereby reviving the long-standing challenge of recursive axiomatization for SL. Our main contributions are threefold: (i) a complete axiomatic system for FOCL; (ii) a rigorous demonstration of its superior expressive power over CL; and (iii) a precise characterization of the essential differences between FOCL and SL concerning both decidability and axiomatizability.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We introduce First-Order Coalition Logic ($mathsf{FOCL}$), which combines key intuitions behind Coalition Logic ($mathsf{CL}$) and Strategy Logic ($mathsf{SL}$). Specifically, $mathsf{FOCL}$ allows for arbitrary quantification over actions of agents. $mathsf{FOCL}$ is interesting for several reasons. First, we show that $mathsf{FOCL}$ is strictly more expressive than existing coalition logics. Second, we provide a sound and complete axiomatisation of $mathsf{FOCL}$, which, to the best of our knowledge, is the first axiomatisation of any variant of $mathsf{SL}$ in the literature. Finally, while discussing the satisfiability problem for $mathsf{FOCL}$, we reopen the question of the recursive axiomatisability of $mathsf{SL}$.

Problem

• Combines Coalition Logic and Strategy Logic intuitions
• Enables arbitrary quantification over agent actions
• Investigates expressiveness and axiomatization of new logic

Innovation

• Combines Coalition Logic and Strategy Logic
• Allows arbitrary quantification over agent actions
• Provides sound and complete axiomatisation for FOCL