HyperLTL Satisfiability Is Highly Undecidable, HyperCTL* is Even Harder

2023-03-29

arXiv.org

AI Summary

This paper investigates the satisfiability problem for the higher-order temporal logics HyperLTL and HyperCTL*, aiming to characterize their inherent computational limits in modeling information-flow properties across multiple execution traces. Using reductions from higher-order logic, analysis within the hyperarithmetical hierarchy, and model-theoretic constructions, we establish: (i) HyperLTL satisfiability is Σ₁¹-complete; (ii) HyperCTL* satisfiability is Σ₁²-complete—yielding the first tight upper bound for HyperCTL*; (iii) HyperCTL* admits models of cardinality continuum, and this bound is optimal; (iv) each quantifier alternation level is Π₁¹-complete; and (v) counting satisfiability and finite-branching satisfiability are both equivalent to the truth problem of second-order arithmetic. These results provide the first complete complexity characterization of HyperCTL* satisfiability—including tight model-size bounds—and systematically delineate the fundamental expressiveness and decidability boundaries of HyperLTL and HyperCTL* across varying semantic models.

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 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.

Problem

• HyperLTL
• HyperCTL*
• Satisfiability

Innovation

• HyperCTL* Satisfaction Complexity
• Infinite Continuous Models Compatibility
• HyperLTL Quantified Alternation Difficulty