On the expressiveness and complexity of ATL

Abstract. ATL is a temporal logic geared towards the specification and verification of properties in multi-agents systems. It allows to reason on the existence of strategies for coalitions of agents in order to enforce a given property. We prove that the standard definition of ATL (built on modalities “Next”, “Always ” and “Until”) has to be completed in order to express the duals of its modalities: it is necessary to add the modality “Release”. We then precisely characterize the complexity of ATL modelchecking when the number of agents is not fixed. We prove that it is ∆ P 2and ∆ P 3-complete, depending on the underlying multi-agent model (ATS and CGS resp.). We also prove that ATL + model-checking is ∆ P 3-complete over both models, even with a fixed number of agents.