Hyperproperties, which generalize trace properties by relating multiple traces, are widely studied in informationflow security. Recently, a number of logics for hyperproperties have been proposed, and there is a need to understand their decidability and relative expressiveness. The new logics have been obtained from standard logics with two principal extensions: temporal logics, like LTL and CTL * , have been generalized to hyperproperties by adding variables for traces or paths. First-order and second-order logics, like monadic first-order logic of order and MSO, have been extended with the equallevel predicate. We study the impact of the two extensions across the spectrum of linear-time and branching-time logics, in particular for logics with quantification over propositions. The resulting hierarchy of hyperlogics differs significantly from the classical hierarchy, suggesting that the equal-level predicate adds more expressiveness than trace and path variables. Within the hierarchy of hyperlogics, we identify new boundaries on the decidability of the satisfiability problem. Specifically, we show that while HyperQPTL and HyperCTL * are both undecidable in general, formulas within their ∃ * ∀ * fragments are decidable.
We study the expressiveness and reactive synthesis problem of HyperQPTL, a logic that specifies ω-regular hyperproperties. HyperQPTL is an extension of linear-time temporal logic (LTL) with explicit trace and propositional quantification and therefore truly combines trace relations and ω-regularity. As such, HyperQPTL can express promptness, which states that there is a common bound on the number of steps up to which an event must have happened. We demonstrate how the HyperQPTL formulation of promptness differs from the type of promptness expressible in the logic Prompt-LTL. Furthermore, we study the realizability problem of HyperQPTL by identifying decidable fragments, where one decidable fragment contains formulas for promptness. We show that, in contrast to the satisfiability problem of HyperQPTL, propositional quantification has an immediate impact on the decidability of the realizability problem. We present a reduction to the realizability problem of HyperLTL, which immediately yields a bounded synthesis procedure. We implemented the synthesis procedure for HyperQPTL in the bounded synthesis tool BoSy. Our experimental results show that a range of arbiter satisfying promptness can be synthesized.
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We study the expressivity and the model checking problem of linear temporal logic with team semantics (TeamLTL). In contrast to LTL, TeamLTL is capable of defining hyperproperties, i.e., properties which relate multiple execution traces. Logics for hyperproperties have so far been mostly obtained by extending temporal logics like LTL and QPTL with trace quantification, resulting in HyperLTL and HyperQPTL. We study the expressiveness of TeamLTL (and its extensions with downward closed generalised atoms A and connectives such as Boolean disjunction ) in comparison to HyperLTL and HyperQPTL. Thereby, we also obtain a number of model checking results for TeamLTL, a question which is so far an open problem. The two types of logics follow a fundamentally different approach to hyperproperties and are are of incomparable expressiveness. We establish that the universally quantified fragment of HyperLTL subsumes the so-called k-coherent fragment of TeamLTL(A, ). This also implies that the model checking problem is decidable for the fragment. We show decidability of model checking of the so-called left-flat fragment of TeamLTL(A, ) via a translation to a decidable fragment of HyperQPTL. Finally, we show that the model checking problem of TeamLTL with Boolean disjunction and inclusion atom is undecidable.
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