HyperLTL Satisfiability Is Highly Undecidable, HyperCTL$^$ is Even Harder
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...
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| Published in: | Logical methods in computer science Vol. 21, Issue 1 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science Association
01.01.2025
Logical Methods in Computer Science e.V |
| Subjects: | |
| ISSN: | 1860-5974, 1860-5974 |
| Online Access: | Get full text |
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| Summary: | 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.
Comment: arXiv admin note: substantial text overlap with arXiv:2105.04176 |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-21(1:3)2025 |