HyperLTL Satisfiability Is Σ₁¹-Complete, HyperCTL* Satisfiability Is Σ₁²-Complete

Authors Marie Fortin , Louwe B. Kuijer , Patrick Totzke , Martin Zimmermann

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Author Details

Marie Fortin
  • University of Liverpool, UK
Louwe B. Kuijer
  • University of Liverpool, UK
Patrick Totzke
  • University of Liverpool, UK
Martin Zimmermann
  • University of Liverpool, UK


We thank Karoliina Lehtinen and Wolfgang Thomas for fruitful discussions.

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Marie Fortin, Louwe B. Kuijer, Patrick Totzke, and Martin Zimmermann. HyperLTL Satisfiability Is Σ₁¹-Complete, HyperCTL* Satisfiability Is Σ₁²-Complete. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 47:1-47:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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 Σ₁¹-complete and HyperCTL* satisfiability is Σ₁²-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove Σ₁²-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 Π₁¹-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Formal languages and automata theory
  • HyperLTL
  • HyperCTL*
  • Satisfiability
  • Analytical Hierarchy


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