The Complexity of Asynchronous HyperLTL

Hyperproperties express, e.g., information-flow properties of systems, which involves the simultaneous reasoning about multiple execution traces of a system. Consequently, HyperLTL, the most important specification logic for hyperproperties, extends LTL with quantification over traces. However, HyperLTL can only express synchronous hyperproperties. Recently, several logics for asynchronous hyperproperties have been proposed. Here, we focus on AHLTL, asynchronous HyperLTL, which extends HyperLTL with quantification over trajectories that control the relative speed at which time progresses on the quantified traces. Model-checking AHLTL is known to be undecidable while satisfiability is known to be $Σ_1^1$-hard, but the precise complexity of both problems is open. Here, we close these gaps and show that model-checking is equivalent to truth in second-order arithmetic while satisfiability is $Σ_1^1$-complete if the trajectory is existentially quantified and $Σ_1^1$-hard and in $Σ_2^1$ if the trajectory is universally quantified.