HyperLTL Satisfiability Is Highly Undecidable, HyperCTL${}^{\ast }$ is Even Harder

Abstract

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 $\hat{I}{\pounds }_1^1$-complete and HyperCTL* satisfiability is $\hat{I}{\pounds }_1^2$-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove $\hat{I}{\pounds }_1^2$-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 ${\hat{I}}_1^1$-complete.

arXiv admin note: substantial text overlap with arXiv:2105.04176