We introduce an extension of Strategy Logic for the imperfect-information setting, called SL ii and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of “hierarchical instances” for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SL ii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises the decidability of distributed synthesis for systems with hierarchical information. It allows us to easily derive new decidability results concerning strategic problems under imperfect information such as the existence of Nash equilibria or rational synthesis. To establish this result, we go through an intermediary, “low-level” logic much more adapted to automata techniques. QCTL * is an extension of CTL * with second-order quantification over atomic propositions that has been used to study strategic logics with perfect information. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. The simple syntax of the resulting logic, QCTL * ii , allows us to provide a conceptually neat reduction of SL ii to QCTL * ii that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of QCTL * ii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable.

We introduce an extension of Strategy logic for the imperfect-information setting, called SL ii , and study its modelchecking problem. As this logic naturally captures multi-player games with imperfect information, the problem turns out to be undecidable. We introduce a syntactical class of "hierarchical instances" for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model. We prove that model-checking SL ii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises previous ones, such as decidability of multi-player games with imperfect information and hierarchical observations, and decidability of distributed synthesis for hierarchical systems.To establish the decidability result, we introduce and study QCTL * ii , an extension of QCTL (itself an extension of CTL with second-order quantification over atomic propositions) by parameterising its quantifiers with observations. The simple syntax of QCTL * ii allows us to provide a conceptually neat reduction of SL ii to QCTL * ii that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of QCTL * ii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automatatheoretic approach, that it is decidable. The decidability result for SL ii follows since the reduction maps hierarchical instances of SL ii to hierarchical formulas of QCTL * ii .

Temporal logics are extensively used for the specification of on-going behaviours of reactive systems. Two significant developments in this area are the extension of traditional temporal logics with modalities that enable the specification of on-going strategic behaviours in multi-agent systems, and the transition of temporal logics to a quantitative setting, where different satisfaction values enable the specifier to formalise concepts such as certainty or quality. We introduce and study SL[F]-a quantitative extension of SL (Strategy Logic), one of the most natural and expressive logics describing strategic behaviours. The satisfaction value of an SL[F] formula is a real value in [0, 1], reflecting "how much" or "how well" the strategic on-going objectives of the underlying agents are satisfied. We demonstrate the applications of SL[F] in quantitative reasoning about multi-agent systems, by showing how it can express concepts of stability in multi-agent systems, and how it generalises some fuzzy temporal logics. We also provide a model-checking algorithm for our logic, based on a quantitative extension of Quantified CTL ⋆ .

Abstract. In the literature, two powerful temporal logic formalisms have been proposed for expressing information-flow security requirements, that in general, go beyond regular properties. One is classic, based on the knowledge modalities of epistemic logic. The other one, the so-called hyper logic, is more recent and subsumes many proposals from the literature. In an attempt to better understand how these logics compare with each other, we consider the logic KCTL * (the extension of CTL * with knowledge modalities and synchronous perfect recall semantics) and HyperCTL * . We first establish that KCTL * and HyperCTL * are expressively incomparable. Then, we introduce a natural linear past extension of HyperCTL * , called HyperCTL * lp , that unifies KCTL * and HyperCTL * . We show that the model-checking problem for HyperCTL * lp is decidable, and we provide its exact computational complexity in terms of a new measure of path quantifiers' alternation. For this, we settle open complexity issues for unrestricted quantified propositional temporal logic.