The Strategy Logic Saga

In open-system verification, a fundamental area of research is the study of modal logics for strategic reasoning in the setting of multi-agent games [3, 5]. An important contribution in this field has been the development of Alternating-Time Temporal Logic (ATL∗, for short), introduced by Alur, Henzinger, and Kupferman [1]. Formally, it is obtained as a generalization of the logic CTL∗ [4], where the path quantifiers there exists “E” and for all “A” are replaced with strategic modalities of the form “〈〈A〉〉” and “[[A]]”, for a set A of agents. These modalities are used to express cooperation and competition among agents in order to achieve a temporal goal. Several decision problems have been investigated about ATL∗; both its model-checking and satisfiability problems are decidable in 2EXPTIME [10], just like it is for CTL∗. Despite its powerful expressiveness, ATL∗ suffers from the strong limitation that strategies are treated only implicitly through modalities that refer to games between competing coalitions. To overcome this problem, Chatterjee, Henzinger, and Piterman introduced Strategy Logic (CHP-SL, for short) [2],which treats strategies in two-player turn-based games as first-order objects. The explicit treatment of strategies makes this logic very useful and more expressive than ATL∗, however, it still suffers from severe limitations. In particular, it is limited to two-player turn-based games and does not allow different players to share the same strategy, suggesting that strategies have yet to become truly first-class objects in this logic. For example, it is impossible to describe the classic strategy-stealing argument of many real-life combinatorial games. These considerations has led us to introduce and investigate a new Strategy Logic, denoted SL, as a more general framework than CHP-SL, for explicit reasoning about strategies in multi-agent concurrent games [8]. Syntactically, SL extends the logic LTL [9] by means of strategy quantifiers, the existential 〈〈x〉〉 and the universal [[x]], as well as agent binding (a, x), where a is an agent and x a variable. Intuitively, these elements can be read as “there exists a strategy x”, “for all strategies x”, and “bind agent a to the strategy associated with x”, respectively. The price that one has to pay for the expressiveness of SL w.r.t. ATL∗ is the lack of important model-theoretic properties and an increased complexity of related decision problems. In particular, in [6, 8], it was shown that SL does not have the bounded-tree model property and the satisfiability problem is highly undecidable, precisely, Σ1-HARD. Moreover, in [7], 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 it is decidable). The negative complexity results on the decision problems of SL 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 sublogics, one may understand why SL is computationally more difficult than ATL∗. The main fragments we have investigated and studied are Nested-Goal, Boolean-Goal, and One-Goal Strategy Logic, respectively denoted by SL[NG], SL[BG], and SL[1G]. They encompass formulas in a special prenex normal form having nested temporal goals, Boolean combinations of goals, and a single goal at a time, respectively. For goal we mean an SL formula of the type [ψ, where [ is a binding prefix of the form (α1, x1), . . . , (αn, xn) containing all the involved agents and ψ is an agent-full formula. In SL[1G], 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 prove that the satisfiability and model-checking problems for