Embedding Coalition Logic in the Minimal Normal Multi-Modal Logic with Intersection

We present a satisfiability-preserving embedding of coalition logic into a normal modal logic. An advantage of standard, normal, modal logics is a well understood theoretical foundation and the availability of tools for automated verification and reasoning. The target logic is multi-modal K with intersection of modalities, interpreted over standard Kripke models corresponding to game structures. There is a restriction: we consider only game structures that are injective. We argue that this is a minor limitation, e.g., because coalition logic cannot discern between injective and non-injective game structures. We give a complete axiomatisation of the corresponding models, as well as a characterisation of key complexity problems. We also prove a representation theorem identifying the effectivity functions corresponding to injective games.