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The Complexity of Second-Order HyperLTL

Authors Hadar Frenkel , Martin Zimmermann

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ACM Subject Classification
  • Theory of computation → Verification by model checking
  • Theory of computation → Logic and verification
Keywords
  • HyperLTL
  • Satisfiability
  • Model-checking

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Abstract

We determine the complexity of second-order HyperLTL satisfiability, finite-state satisfiability, and model-checking: All three are equivalent to truth in third-order arithmetic. 
We also consider two fragments of second-order HyperLTL that have been introduced with the aim to facilitate effective model-checking by restricting the sets one can quantify over. The first one restricts second-order quantification to smallest/largest sets that satisfy a guard while the second one restricts second-order quantification further to least fixed points of (first-order) HyperLTL definable functions. All three problems for the first fragment are still equivalent to truth in third-order arithmetic while satisfiability for the second fragment is Σ₁¹-complete, i.e., only as hard as for (first-order) HyperLTL and therefore much less complex. Finally, finite-state satisfiability and model-checking are in Σ₂² and are Σ₁¹-hard, and thus also less complex than for full second-order HyperLTL.

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Hadar Frenkel and Martin Zimmermann. The Complexity of Second-Order HyperLTL. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 10:1-10:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.10

Author Details

Hadar Frenkel
  • Bar-Ilan University, Ramat Gan, Israel
Martin Zimmermann
  • Aalborg University, Denmark

Funding

  • Zimmermann, Martin: Supported by DIREC – Digital Research Centre Denmark.

Acknowledgements

This work was initiated by a discussion at Dagstuhl Seminar 23391 "The Futures of Reactive Synthesis" and some results were obtained at Dagstuhl Seminar 24111 "Logics for Dependence and Independence: Expressivity and Complexity". We are grateful to Gaëtan Regaud for finding and fixing a bug in the proof of Theorem 18 and to the reviewers for their detailed and valuable feedback, which improved the paper considerably.

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