Reasoning about Joint Action and Coalitional Ability in K n with Intersection

In this paper we point out that standard PDL-like logics with intersection are useful for reasoning about game structures. In particular, they can express coalitional ability operators known from coalition logic and ATL. An advantage of standard, normal, modal logics is a well understood theoretical foundation and the availability of tools for automated verification and reasoning. We study a minimal variant, multi-modal K with intersection of modalities, interpreted over models corresponding to game structures. There is a restriction: we consider only game structures that are injective. 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.