Lean Formalization of Completeness Proofs for Coalition Logic with and without Common Knowledge

Abstract: As computers become more commonplace, numerous indus- tries ranging from human-computer interaction to transportation, need to design systems that involve multiple computers and human users. De- signing algorithms that meet goals in such multi-agent systems (MAS) is an active research area in Artificial Intelligence. Given the complexity of this problem, modelling is a vital tool to verify design specifications. We build the foundation for such modelling by formalizing a logic that can describe the actions of coalitions of agents: Coalition Logic (CL). An additional challenge in MAS, is that each agent often has access to distinct information. To capture this, we also formalize one extension of CL that can model the individual and common knowledge of agents: Coalition Logic with Common Knowledge (CLC).

We formalize both the syntax and semantics of these logics in the proof assistant Lean. In order to demonstrate that these definitions relate to one another as expected, and can be used in non-trivial proofs, we prove soundness and completeness for both logics. Our implementation em- phasises generalizability, by being intentional about constraints in our definitions, leading us to extend CLC to allow common knowledge of the empty set. We also prioritise re-usability by introducing generic Lean classes for types of formulas. These classes can, for instance, prove propo- sitional lemmas for any extension of propositional logic. We hope that these two aspects will make our work extensible for applications in MAS. To this effect, we outline a model-checking algorithm that could check requirements of some MAS design.