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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Quantified LTL (QLTL) extends the temporal logic LTL with quantifications over atomic propositions. Several semantics exist to handle these quantifications, depending on the definition of executions over which formulas are interpreted: either infinite sequences of subsets of atomic propositions (aka the "tree semantics") or infinite sequences of control states combined with a labelling function that associates atomic propositions to the control states (aka the "structure semantics"). The main difference being that in the latter different occurrences of a control state should be labelled similarly. The tree semantics has been intensively studied from the complexity and expressivity point of view (especially in the work of Sistla [Sistla, 1983; Sistla et al., 1987]) for which the satisfiability and model-checking problems are known to be TOWER-complete. For the structure semantics, French has shown that the satisfiability problem is undecidable [French, 2003]. We study here the model-checking problem for QLTL under this semantics and prove that it is EXPSPACE-complete. We also show that the complexity drops down to PSPACE-complete for two specific cases of structures, namely path and flat ones.

François Laroussinie, Loriane Leclercq, and Arnaud Sangnier. QLTL Model-Checking. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{laroussinie_et_al:LIPIcs.CSL.2024.35, author = {Laroussinie, Fran\c{c}ois and Leclercq, Loriane and Sangnier, Arnaud}, title = {{QLTL Model-Checking}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {35:1--35:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.35}, URN = {urn:nbn:de:0030-drops-196783}, doi = {10.4230/LIPIcs.CSL.2024.35}, annote = {Keywords: Quantified Linear-time temporal logic, Model-cheking, Verification, Automata theory} }

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**Published in:** LIPIcs, Volume 147, 26th International Symposium on Temporal Representation and Reasoning (TIME 2019)

QCTL extends the temporal logic CTL with quantifications over atomic propositions. This extension is known to be very expressive: QCTL allows us to express complex properties over Kripke structures (it is as expressive as MSO). Several semantics exist for the quantifications: here, we work with the structure semantics, where the extra propositions label the Kripke structure (and not its execution tree), and the model-checking problem is known to be PSPACE-complete in this framework. We propose a model-checking algorithm for QCTL based on a reduction to QBF. We consider several reduction strategies, and we compare them with a prototype (based on the SMT-solver Z3) on several examples.

Akash Hossain and François Laroussinie. From Quantified CTL to QBF. In 26th International Symposium on Temporal Representation and Reasoning (TIME 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 147, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hossain_et_al:LIPIcs.TIME.2019.11, author = {Hossain, Akash and Laroussinie, Fran\c{c}ois}, title = {{From Quantified CTL to QBF}}, booktitle = {26th International Symposium on Temporal Representation and Reasoning (TIME 2019)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-127-6}, ISSN = {1868-8969}, year = {2019}, volume = {147}, editor = {Gamper, Johann and Pinchinat, Sophie and Sciavicco, Guido}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2019.11}, URN = {urn:nbn:de:0030-drops-113699}, doi = {10.4230/LIPIcs.TIME.2019.11}, annote = {Keywords: Model-checking, Quantified CTL, QBF solvers, SAT based model-checking} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

QCTL extends the temporal logic CTL with quantification over atomic propositions. While the algorithmic questions for QCTL and its fragments with limited quantification depth are well-understood (e.g. satisfiability of QkCTL, with at most k nested blocks of quantifiers, is (k+1)-EXPTIME-complete), very few results are known about the expressiveness of this logic.
We address such expressiveness questions in this paper. We first consider the distinguishing power of these logics (i.e., their ability to separate models), their relationship with behavioural equivalences, and their ability to capture the behaviours of finite Kripke structures with so-called characteristic formulas. We then consider their expressive power (i.e., their ability to express a property), showing that in terms of expressiveness the hierarchy QkCTL collapses at level 2 (in other terms, any QCTL formula can be expressed using at most two nested blocks of quantifiers).

Amélie David, Francois Laroussinie, and Nicolas Markey. On the Expressiveness of QCTL. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{david_et_al:LIPIcs.CONCUR.2016.28, author = {David, Am\'{e}lie and Laroussinie, Francois and Markey, Nicolas}, title = {{On the Expressiveness of QCTL}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.28}, URN = {urn:nbn:de:0030-drops-61643}, doi = {10.4230/LIPIcs.CONCUR.2016.28}, annote = {Keywords: Specification, Verification, Temporal Logic, Expressiveness, Tree automata} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We study the alternating-time temporal logics ATL and ATL* extended with strategy contexts: these make agents commit to their strategies during the evaluation of formulas, contrary to plain ATL and ATL* where strategy quantifiers reset previously selected strategies.
We illustrate the important expressive power of strategy contexts by proving that they make the extended logics, namely ATLsc and ATLsc*, equally expressive: any formula in ATLsc* can be translated into an equivalent, linear-size ATLsc formula. Despite the high expressiveness of these logics, we~prove that their model-checking problems remain decidable by~designing a tree-automata-based algorithm for model-checking ATLsc* on the full class of $n$-player concurrent game structures.

Arnaud Da Costa, François Laroussinie, and Nicolas Markey. ATL with Strategy Contexts: Expressiveness and Model Checking. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 120-132, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dacosta_et_al:LIPIcs.FSTTCS.2010.120, author = {Da Costa, Arnaud and Laroussinie, Fran\c{c}ois and Markey, Nicolas}, title = {{ATL with Strategy Contexts: Expressiveness and Model Checking}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {120--132}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.120}, URN = {urn:nbn:de:0030-drops-28589}, doi = {10.4230/LIPIcs.FSTTCS.2010.120}, annote = {Keywords: alternating temporal logic, agent, strategy quantifier} }

Document

**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Quantified LTL (QLTL) extends the temporal logic LTL with quantifications over atomic propositions. Several semantics exist to handle these quantifications, depending on the definition of executions over which formulas are interpreted: either infinite sequences of subsets of atomic propositions (aka the "tree semantics") or infinite sequences of control states combined with a labelling function that associates atomic propositions to the control states (aka the "structure semantics"). The main difference being that in the latter different occurrences of a control state should be labelled similarly. The tree semantics has been intensively studied from the complexity and expressivity point of view (especially in the work of Sistla [Sistla, 1983; Sistla et al., 1987]) for which the satisfiability and model-checking problems are known to be TOWER-complete. For the structure semantics, French has shown that the satisfiability problem is undecidable [French, 2003]. We study here the model-checking problem for QLTL under this semantics and prove that it is EXPSPACE-complete. We also show that the complexity drops down to PSPACE-complete for two specific cases of structures, namely path and flat ones.

François Laroussinie, Loriane Leclercq, and Arnaud Sangnier. QLTL Model-Checking. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{laroussinie_et_al:LIPIcs.CSL.2024.35, author = {Laroussinie, Fran\c{c}ois and Leclercq, Loriane and Sangnier, Arnaud}, title = {{QLTL Model-Checking}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {35:1--35:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.35}, URN = {urn:nbn:de:0030-drops-196783}, doi = {10.4230/LIPIcs.CSL.2024.35}, annote = {Keywords: Quantified Linear-time temporal logic, Model-cheking, Verification, Automata theory} }

Document

**Published in:** LIPIcs, Volume 147, 26th International Symposium on Temporal Representation and Reasoning (TIME 2019)

QCTL extends the temporal logic CTL with quantifications over atomic propositions. This extension is known to be very expressive: QCTL allows us to express complex properties over Kripke structures (it is as expressive as MSO). Several semantics exist for the quantifications: here, we work with the structure semantics, where the extra propositions label the Kripke structure (and not its execution tree), and the model-checking problem is known to be PSPACE-complete in this framework. We propose a model-checking algorithm for QCTL based on a reduction to QBF. We consider several reduction strategies, and we compare them with a prototype (based on the SMT-solver Z3) on several examples.

Akash Hossain and François Laroussinie. From Quantified CTL to QBF. In 26th International Symposium on Temporal Representation and Reasoning (TIME 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 147, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hossain_et_al:LIPIcs.TIME.2019.11, author = {Hossain, Akash and Laroussinie, Fran\c{c}ois}, title = {{From Quantified CTL to QBF}}, booktitle = {26th International Symposium on Temporal Representation and Reasoning (TIME 2019)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-127-6}, ISSN = {1868-8969}, year = {2019}, volume = {147}, editor = {Gamper, Johann and Pinchinat, Sophie and Sciavicco, Guido}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2019.11}, URN = {urn:nbn:de:0030-drops-113699}, doi = {10.4230/LIPIcs.TIME.2019.11}, annote = {Keywords: Model-checking, Quantified CTL, QBF solvers, SAT based model-checking} }

Document

**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

QCTL extends the temporal logic CTL with quantification over atomic propositions. While the algorithmic questions for QCTL and its fragments with limited quantification depth are well-understood (e.g. satisfiability of QkCTL, with at most k nested blocks of quantifiers, is (k+1)-EXPTIME-complete), very few results are known about the expressiveness of this logic.
We address such expressiveness questions in this paper. We first consider the distinguishing power of these logics (i.e., their ability to separate models), their relationship with behavioural equivalences, and their ability to capture the behaviours of finite Kripke structures with so-called characteristic formulas. We then consider their expressive power (i.e., their ability to express a property), showing that in terms of expressiveness the hierarchy QkCTL collapses at level 2 (in other terms, any QCTL formula can be expressed using at most two nested blocks of quantifiers).

Amélie David, Francois Laroussinie, and Nicolas Markey. On the Expressiveness of QCTL. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{david_et_al:LIPIcs.CONCUR.2016.28, author = {David, Am\'{e}lie and Laroussinie, Francois and Markey, Nicolas}, title = {{On the Expressiveness of QCTL}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.28}, URN = {urn:nbn:de:0030-drops-61643}, doi = {10.4230/LIPIcs.CONCUR.2016.28}, annote = {Keywords: Specification, Verification, Temporal Logic, Expressiveness, Tree automata} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We study the alternating-time temporal logics ATL and ATL* extended with strategy contexts: these make agents commit to their strategies during the evaluation of formulas, contrary to plain ATL and ATL* where strategy quantifiers reset previously selected strategies.
We illustrate the important expressive power of strategy contexts by proving that they make the extended logics, namely ATLsc and ATLsc*, equally expressive: any formula in ATLsc* can be translated into an equivalent, linear-size ATLsc formula. Despite the high expressiveness of these logics, we~prove that their model-checking problems remain decidable by~designing a tree-automata-based algorithm for model-checking ATLsc* on the full class of $n$-player concurrent game structures.

Arnaud Da Costa, François Laroussinie, and Nicolas Markey. ATL with Strategy Contexts: Expressiveness and Model Checking. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 120-132, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dacosta_et_al:LIPIcs.FSTTCS.2010.120, author = {Da Costa, Arnaud and Laroussinie, Fran\c{c}ois and Markey, Nicolas}, title = {{ATL with Strategy Contexts: Expressiveness and Model Checking}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {120--132}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.120}, URN = {urn:nbn:de:0030-drops-28589}, doi = {10.4230/LIPIcs.FSTTCS.2010.120}, annote = {Keywords: alternating temporal logic, agent, strategy quantifier} }