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. In this paper, we first precisely characterize the complexity of ATL model-checking over Alternating Transition Systems and Concurrent Game Structures when the number of agents is not fixed. We prove that it is ∆ P 2- and ∆ P 3-complete, depending on the underlying multi-agent model (ATS and CGS resp.). We also consider the same problems for some extensions of ATL. We then consider expressiveness issues. We show how ATS and CGS are related and provide translations between these models w.r.t. alternating bisimulation. We also prove that the standard definition of ATL (built on modalities “Next”, “Always ” and “Until”) cannot express the duals of its modalities: it is necessary to explicitely add the modality “Release”. 1.

models whose verification results transfer to the abstracted models for a logic with unrestricted use of negation and quantification. This framework is novel in that its models have quantitative or probabilistic observables and state transitions. Properties of a quantitative temporal logic have measurable denotations in these models. For probabilistic models such denotations approximate the probabilistic semantics of full LTL. We show how predicate-based abstractions specify abstract quantitative and probabilistic models with finite state space. 1

ATL + is a variant of alternating-time temporal logic that does not have the expressive power of full ATL ∗ , but still allows for expressing some natural properties of agents. It has been believed that verification with ATL + is Δ P 3-complete for both memoryless agents and players who can memorize the whole history of the game. In this paper, we show that the latter result is not correct. That is, we prove that model checking ATL + for agents that use strategies with memory is in fact PSPACE-complete. On a more optimistic note, we show that fairness constraints can be added to ATL + without further increasing the complexity of model checking, which makes ATL + an attractive alternative to the full