HyperLTL Satisfiability is $\Sigma_1^1$-complete, HyperCTL* Satisfiability is $\Sigma_1^2$-complete

TL;DR

This paper precisely characterizes the high undecidability levels of satisfiability for HyperLTL and HyperCTL*, showing they are $ ext{Sigma}_1^1$-complete and $ ext{Sigma}_1^2$-complete respectively, significantly advancing understanding of their complexity.

Contribution

It establishes the exact high undecidability complexity of HyperLTL and HyperCTL* satisfiability, providing the first upper bounds and tightness results.

Findings

HyperLTL satisfiability is $ ext{Sigma}_1^1$-complete.

HyperCTL* satisfiability is $ ext{Sigma}_1^2$-complete.

Every satisfiable HyperCTL* sentence has a model of continuum size.

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 $\Sigma_1^1$-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 prove this bound to be tight. Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is $\Pi_1^1$-complete.