In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between agents and express the correctness of the system no matter how the environment behaves. An important contribution in this context is given by modal logics for strategic ability, in the setting of multiagent games, such as ATL, ATL * , and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which we denote here by CHP-SL, with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-SL is obtained by using first-order quantifications over strategies and has been investigated in the very specific setting of two-agents turned-based games, where a nonelementary model-checking algorithm has been provided. While CHP-SL is a very expressive logic, we claim that it does not fully capture the strategic aspects of multiagent systems.In this article, we introduce and study a more general strategy logic, denoted SL, for reasoning about strategies in multiagent concurrent games. As a key aspect, strategies in SL are not intrinsically glued to a specific agent, but an explicit binding operator allows an agent to bind to a strategy variable. This allows agents to share strategies or reuse one previously adopted. We prove that SL strictly includes CHP-SL, while maintaining a decidable model-checking problem. In particular, the algorithm we propose is computationally not harder than the best one known for CHP-SL. Moreover, we prove that such a problem for SL is NONELEMENTARY. This negative result has spurred us to investigate syntactic fragments of SL, strictly subsuming ATL * , with the hope of obtaining an elementary model-checking problem. Among others, we introduce and study the sublogics SL [NG], SL [BG], and SL[1G]. They encompass formulas in a special prenex normal form having, respectively, nested temporal goals, Boolean combinations of goals, and, a single goal at a time. Intuitively, for a goal, we mean a sequence of bindings, one for each agent, followed by an LTL formula. We prove that the model-checking problem for SL[1G] is 2EXPTIME-COMPLETE, thus not harder than the one for ATL * . In contrast, SL[NG] turns out to be NONELEMENTARY-HARD, strengthening the corresponding result for SL. Regarding SL[BG], we show that it includes CHP-SL and its model-checking is decidable with a 2EXPTIMElower-bound.It is worth enlightening that to achieve the positive results about SL[1G], we introduce a fundamental property of the semantics of this logic, called behavioral, which allows to strongly simplify the reasoning about strategies. Indeed, in a nonbehavioral logic such as SL [BG] and the subsuming ones, to satisfy a formula, one has to take into account that a move of an agent, at a given moment of a play, may depend on the moves taken by any agent in another counterfactual play.

Model checking is a powerful method widely explored in formal verification. Given a model of a system, e.g., a Kripke structure, and a formula specifying its expected behaviour, one can verify whether the system meets the behaviour by checking the formula against the model. Classically, system behaviour is expressed by a formula of a temporal logic, such as LTL and the like. These logics are “point-wise” interpreted, as they describe how the system evolves state-by-state. However, there are relevant properties, such as those constraining the temporal relations between pairs of temporally extended events or involving temporal aggregations, which are inherently “interval-based”, and thus asking for an interval temporal logic. In this paper, we give a formalization of the model checking problem in an interval logic setting. First, we provide an interpretation of formulas of Halpern and Shoham’s interval temporal logic HS over finite Kripke structures, which allows one to check interval properties of computations. Then, we prove that the model checking problem for HS against finite Kripke structures is decidable by a suitable small model theorem, and we provide a lower bound to its computational complexity

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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. This logic turns out 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 expressiveness of SL is the lack of important model-theoretic properties and an increased complexity of decision problems. In particular, SL does not have the bounded-tree model property and the related satisfiability problem is highly undecidable while for ATL * it is 2EXPTIME-COMPLETE. An obvious question that arises is then what makes ATL * decidable. Understanding this should enable us to identify decidable fragments of SL. We focus, in this work, on the limitation of ATL * to allow only one temporal goal for each strategic assertion and study the fragment of SL with the same restriction. Specifically, we introduce and study the syntactic fragment One-Goal Strategy Logic (SL[1G], for short), which consists of formulas in prenex normal form having a single temporal goal at a time for every strategy quantification of agents. We show that SL[1G] is strictly more expressive than ATL * . Our main result is that SL[1G] has the bounded tree-model property and its satisfiability problem is 2EXPTIME-COMPLETE, as it is for ATL * .

Model checking has come of age. A number of techniques are increasingly used in industrial setting to verify hardware and software systems, both against models and concrete implementations. While it is generally accepted that obstacles still remain, notably handling infinite state systems efficiently, much of current work involves refining and improving existing techniques such as predicate abstraction

ABSTRACT. The fully enriched µ-calculus is the extension of the propositional µ-calculus with inverse programs, graded modalities, and nominals. While satisfiability in several expressive fragments of the fully enriched µ-calculus is known to be decidable and EXPTIME-complete, it has recently been proved that the full calculus is undecidable. In this paper, we study the fragments of the fully enriched µ-calculus that are obtained by dropping at least one of the additional constructs. We show that, in all fragments obtained in this way, satisfiability is decidable and EXPTIME-complete. Thus, we identify a family of decidable logics that are maximal (and incomparable) in expressive power. Our results are obtained by introducing two new automata models, showing that their emptiness problems are EXPTIME-complete, and then reducing satisfiability in the relevant logics to these problems. The automata models we introduce are two-way graded alternating parity automata over infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite forests. The former are a common generalization of two incomparable automata models from the literature. The latter extend alternating automata in a similar way as the fully enriched µ-calculus extends the standard µ-calculus.

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ABSTRACT. The fully enriched µ-calculus is the extension of the propositional µ-calculus with inverse programs, graded modalities, and nominals. While satisfiability in several expressive fragments of the fully enriched µ-calculus is known to be decidable and EXPTIME-complete, it has recently been proved that the full calculus is undecidable. In this paper, we study the fragments of the fully enriched µ-calculus that are obtained by dropping at least one of the additional constructs. We show that, in all fragments obtained in this way, satisfiability is decidable and EXPTIME-complete. Thus, we identify a family of decidable logics that are maximal (and incomparable) in expressive power. Our results are obtained by introducing two new automata models, showing that their emptiness problems are EXPTIME-complete, and then reducing satisfiability in the relevant logics to these problems. The automata models we introduce are two-way graded alternating parity automata over infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite forests. The former are a common generalization of two incomparable automata models from the literature. The latter extend alternating automata in a similar way as the fully enriched µ-calculus extends the standard µ-calculus.