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DOI: 10.4230/LIPIcs.MFCS.2021.47

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

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

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

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 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.

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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)

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@InProceedings{fortin_et_al:LIPIcs.MFCS.2021.47,
  author =	{Fortin, Marie and Kuijer, Louwe B. and Totzke, Patrick and Zimmermann, Martin},
  title =	{{HyperLTL Satisfiability Is \Sigma₁¹-Complete, HyperCTL* Satisfiability Is \Sigma₁²-Complete}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{47:1--47:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.47},
  URN =		{urn:nbn:de:0030-drops-144870},
  doi =		{10.4230/LIPIcs.MFCS.2021.47},
  annote =	{Keywords: HyperLTL, HyperCTL*, Satisfiability, Analytical Hierarchy}
}

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

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