In alternating-time temporal logic ATL*, agents with perfect recall assign choices to sequences of states, i.e., to possible finite histories of the game. However, when a nested strategic modality is interpreted, the new strategy does not take into account the previous sequence of events. It is as if agents collect their observations in the nested game again from scratch, thus effectively forgetting what they observed before. Intuitively, it does not fit the assumption of agents having perfect recall of the past.

Recently we have proposed a new semantics for ATL* where the past is not forgotten in nested games [8]. In this paper we give a formal treatment and show that the standard semantics of ATL* coincides with our new semantics in case of agents with perfect information. On the other hand, both kinds of semantics differ if agents have imperfect information about the state of the game. We compare the expressivity of the logics and their sets of validities. The latter characterize general properties of the underlying class of games.