Temporal logics for the specification of information-flow properties are able to express relations between multiple executions of a system. The two most important such logics are HyperLTL and HyperCTL*, which generalise LTL and CTL* by trace quantification. It is known that this expressiveness comes at a price, i.e.\ satisfiability is undecidable for both logics. In this paper we settle the exact complexity of these problems, showing that both are in fact highly undecidable: we prove that HyperLTL satisfiability is $\Sigma_1^1$-complete and HyperCTL* satisfiability is $\Sigma_1^2$-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove $\Sigma_1^2$-membership for HyperCTL*, we prove that every satisfiable HyperCTL* sentence has a model that is equinumerous to the continuum, the first upper bound of this kind. We also prove this bound to be tight. Furthermore, we prove that both countable and finitely-branching satisfiability for HyperCTL* are as hard as truth in second-order arithmetic, i.e.\ still highly undecidable. Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is $\Pi_1^1$-complete.

翻译：用于规约信息流属性的时序逻辑能够表达系统多次执行之间的关系。其中最重要的两种逻辑是HyperLTL和HyperCTL*，它们通过迹量化推广了LTL和CTL*。已知这种表达能力的提升需要付出代价，即两种逻辑的可满足性均不可判定。本文确定了这些问题的精确复杂度，证明两者实际上具有高度不可判定性：我们证明HyperLTL可满足性是$\Sigma_1^1$-完全的，而HyperCTL*可满足性是$\Sigma_1^2$-完全的。这相较于先前已知的下界是显著提升，并且是首次提出的上界。为证明HyperCTL*的$\Sigma_1^2$归属度，我们证明每个可满足的HyperCTL*语句都存在一个与连续统等势的模型，这是此类问题的首个上界结果。我们还证明该界限是紧致的。此外，我们证明HyperCTL*的可数可满足性与有限分支可满足性均与二阶算术的真值问题同等困难，即仍具有高度不可判定性。最后，我们证明HyperLTL量词交替层级的每一层级的成员问题都是$\Pi_1^1$-完全的。