Strategy Logic Fragments Fabio Mogavero, Aniello Murano and Giuseppe Perelli Università degli Studi di Napoli "Federico II", Napoli, Italy. Moshe Y. Vardi Rice University, Houston, Texas, USA. 1 Extended Abstract Strategy Logic (SL, for short) has been recently introduced by Mogavero, Murano, and Vardi as a formalism for reasoning explicitly about strategies, as first-order objects, in multi-agent concurrent games [3]. This logic turns to be very powerful, strictly subsuming all major previously studied modal logics for strategic reasoning, including ATL,ATL * , and the like. The price that one has to pay for the expressive- ness of S L w.r.t. ATL * is the lack of important model-theoretic properties and an increased complexity of related decision problems. In particular, in [1, 3], it was shown that SL does not have the bounded-tree model property and the satisfiability problem is highly undecidable, precisely, Σ 1 1 -HARD. Moreover, in [2], it was shown that the model checking problem is nonelementary-complete (we recall that also for CHP-SL it is known to be nonelementary, while it is open the question whether the related satisfiability problem is decidable or not). The negative complexity results on the decision problems of S L with respect ATL * , provide motivations for an investigation of decidable fragments of SL, strictly subsuming ATL * , with a better complexity. In particular, by means of these sublog- ics, one may understand why SL is computationally more difficult than ATL * . The main fragments we have introduced and studied are Nested-Goal, Boolean- Goal, and One-Goal Strategy Logic, respectively denoted by SL[NG],SL[BG], and SL[1 G]. They encompass formulas in a special prenex normal form having nested temporal goals, Boolean combinations of goals, and a single goal at a time, re- spectively. For goal we mean an SL formula of the type ♭ψ, where ♭ is a binding prefix of the form (α 1 ,x 1 ),..., (α n ,x n ) containing all the involved agents and ψ is an agent-full formula. In S L[1 G], each temporal formula ψ is prefixed by a quantification-binding prefix ℘♭ that quantifies over a tuple of strategies and binds them to all agents. As main results about these fragments, we have proved that the satisfiability and model-checking problems for SL[1 G] are 2EXPTIME-COMPLETE, thus not harder than the one for ATL * . On the contrary, for SL[NG], the model checking problem is 1