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**Published in:** LIPIcs, Volume 278, 30th International Symposium on Temporal Representation and Reasoning (TIME 2023)

We present the Calculus of Temporal Influence, a simple logical calculus that allows reasoning about the behaviour of real-valued functions over time by making assertions that bound their values or the values of their derivatives. The motivation for the design of such a proof system comes from the need to provide the background computational machinery for tools that support learning in experimental subjects in secondary-education classrooms. The end goal is a tool that allows school pupils to formalise hypotheses about phenomena in natural sciences, such that their validity with respect to some formal experiment model can be checked automatically. The Calculus of Temporal Influence provides a language for formal statements and the mechanisms for reasoning about valid logical consequences. It extends (and deviates in parts from) previous work introducing the Calculus of (Non-Temporal) Influence by integrating the ability to model temporal effects in such experiments. We show that reasoning in the calculus is sound with respect to a natural formal semantics, that logical consequence is at least semi-decidable, and that one obtains polynomial-time decidability for a natural stratification of the problem.

Florian Bruse, Marit Kastaun, Martin Lange, and Sören Möller. The Calculus of Temporal Influence. In 30th International Symposium on Temporal Representation and Reasoning (TIME 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 278, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bruse_et_al:LIPIcs.TIME.2023.10, author = {Bruse, Florian and Kastaun, Marit and Lange, Martin and M\"{o}ller, S\"{o}ren}, title = {{The Calculus of Temporal Influence}}, booktitle = {30th International Symposium on Temporal Representation and Reasoning (TIME 2023)}, pages = {10:1--10:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-298-3}, ISSN = {1868-8969}, year = {2023}, volume = {278}, editor = {Artikis, Alexander and Bruse, Florian and Hunsberger, Luke}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2023.10}, URN = {urn:nbn:de:0030-drops-191009}, doi = {10.4230/LIPIcs.TIME.2023.10}, annote = {Keywords: temporal reasoning, formal models, continuous functions, polynomial decidability} }

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**Published in:** LIPIcs, Volume 247, 29th International Symposium on Temporal Representation and Reasoning (TIME 2022)

Timed Recursive CTL (TRCTL) was recently proposed as a merger of two extensions of the well-known branching-time logic CTL: Timed CTL on one hand is interpreted over real-time systems like timed automata, and Recursive CTL (RecCTL) on the other hand obtains high expressiveness through the introduction of a recursion operator. Model checking for the resulting logic is known to be 2-EXPTIME-complete.
The aim of this paper is to investigate the possibility to obtain a fragment of lower complexity without losing too much expressive power. It is obtained by a syntactic property called "tail-recursiveness" that restricts the way that recursive formulas can be built. This restriction is known to decrease the complexity of model checking by half an exponential in the untimed setting. We show that this also works in the real-time world: model checking for the tail-recursive fragment of TRCTL is EXPSPACE-complete. The upper bound is obtained by a standard untiming construction via region graphs, and rests on the known complexity of tail-recursive fragments of higher-order modal logics. The lower bound is established by a reduction from a suitable tiling problem.

Florian Bruse, Martin Lange, and Etienne Lozes. The Tail-Recursive Fragment of Timed Recursive CTL. In 29th International Symposium on Temporal Representation and Reasoning (TIME 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 247, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bruse_et_al:LIPIcs.TIME.2022.5, author = {Bruse, Florian and Lange, Martin and Lozes, Etienne}, title = {{The Tail-Recursive Fragment of Timed Recursive CTL}}, booktitle = {29th International Symposium on Temporal Representation and Reasoning (TIME 2022)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-262-4}, ISSN = {1868-8969}, year = {2022}, volume = {247}, editor = {Artikis, Alexander and Posenato, Roberto and Tonetta, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2022.5}, URN = {urn:nbn:de:0030-drops-172527}, doi = {10.4230/LIPIcs.TIME.2022.5}, annote = {Keywords: formal specification, temporal logic, real-time systems} }

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**Published in:** LIPIcs, Volume 206, 28th International Symposium on Temporal Representation and Reasoning (TIME 2021)

We introduce Timed Recursive CTL, a merger of two extensions of the well-known branching-time logic CTL: Timed CTL is interpreted over real-time systems like timed automata; Recursive CTL introduces a powerful recursion operator which takes the expressiveness of this logic CTL well beyond that of regular properties. The result is an expressive logic for real-time properties. We show that its model checking problem is decidable over timed automata, namely 2-EXPTIME-complete.

Florian Bruse and Martin Lange. Model Checking Timed Recursive CTL. In 28th International Symposium on Temporal Representation and Reasoning (TIME 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 206, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bruse_et_al:LIPIcs.TIME.2021.12, author = {Bruse, Florian and Lange, Martin}, title = {{Model Checking Timed Recursive CTL}}, booktitle = {28th International Symposium on Temporal Representation and Reasoning (TIME 2021)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-206-8}, ISSN = {1868-8969}, year = {2021}, volume = {206}, editor = {Combi, Carlo and Eder, Johann and Reynolds, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2021.12}, URN = {urn:nbn:de:0030-drops-147880}, doi = {10.4230/LIPIcs.TIME.2021.12}, annote = {Keywords: formal specification, temporal logic, real-time systems} }

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

The modal μ-calculus can only express bisimulation-invariant properties. It is a simple consequence of Kleene’s Fixpoint Theorem that on structures with finite bisimulation quotients, the fixpoint iteration of any formula converges after finitely many steps. We show that the converse does not hold: we construct a word with an infinite bisimulation quotient that is locally regular so that the iteration for any fixpoint formula of the modal μ-calculus on it converges after finitely many steps. This entails decidability of μ-calculus model-checking over this word. We also show that the reason for the discrepancy between infinite bisimulation quotients and trans-finite fixpoint convergence lies in the fact that the μ-calculus can only express regular properties.

Florian Bruse, Marco Sälzer, and Martin Lange. Finite Convergence of μ-Calculus Fixpoints on Genuinely Infinite Structures. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bruse_et_al:LIPIcs.MFCS.2021.24, author = {Bruse, Florian and S\"{a}lzer, Marco and Lange, Martin}, title = {{Finite Convergence of \mu-Calculus Fixpoints on Genuinely Infinite Structures}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {24:1--24:19}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.24}, URN = {urn:nbn:de:0030-drops-144643}, doi = {10.4230/LIPIcs.MFCS.2021.24}, annote = {Keywords: temporal logic, fixpoint iteration, bisimulation} }

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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

Fixpoint Logic with Chop (FLC) extends the modal μ-calculus with an operator for sequential composition between predicate transformers. This makes it an expressive modal fixpoint logic which is capable of formalising many non-regular program properties. Its satisfiability problem is highly undecidable. Here we define Visibly Pushdown Fixpoint Logic with Chop, a fragment in which fixpoint formulas are required to be of a certain form resembling visibly pushdown grammars. We give a sound and complete game-theoretic characterisation of FLC’s satisfiability problem and show that the games corresponding to formulas from this fragment are stair-parity games and therefore effectively solvable, resulting in 2EXPTIME-completeness of this fragment. The lower bound is inherited from PDL over Recursive Programs, which is structurally similar but considerably weaker in expressive power.

Florian Bruse and Martin Lange. A Decidable Non-Regular Modal Fixpoint Logic. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bruse_et_al:LIPIcs.CONCUR.2021.23, author = {Bruse, Florian and Lange, Martin}, title = {{A Decidable Non-Regular Modal Fixpoint Logic}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.23}, URN = {urn:nbn:de:0030-drops-144003}, doi = {10.4230/LIPIcs.CONCUR.2021.23}, annote = {Keywords: formal specification, temporal logic, expressive power} }

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**Published in:** LIPIcs, Volume 178, 27th International Symposium on Temporal Representation and Reasoning (TIME 2020)

We introduce extensions of the standard temporal logics CTL and LTL with a recursion operator that takes propositional arguments. Unlike other proposals for modal fixpoint logics of high expressive power, we obtain logics that retain some of the appealing pragmatic advantages of CTL and LTL, yet have expressive power beyond that of the modal μ-calculus or MSO. We advocate these logics by showing how the recursion operator can be used to express interesting non-regular properties. We also study decidability and complexity issues of the standard decision problems.

Florian Bruse and Martin Lange. Temporal Logic with Recursion. In 27th International Symposium on Temporal Representation and Reasoning (TIME 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 178, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bruse_et_al:LIPIcs.TIME.2020.6, author = {Bruse, Florian and Lange, Martin}, title = {{Temporal Logic with Recursion}}, booktitle = {27th International Symposium on Temporal Representation and Reasoning (TIME 2020)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-167-2}, ISSN = {1868-8969}, year = {2020}, volume = {178}, editor = {Mu\~{n}oz-Velasco, Emilio and Ozaki, Ana and Theobald, Martin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2020.6}, URN = {urn:nbn:de:0030-drops-129748}, doi = {10.4230/LIPIcs.TIME.2020.6}, annote = {Keywords: formal specification, temporal logic, expressive power} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

We study the following natural variation on the classical universality problem: given a language L(M) represented by M (e.g., a DFA/RE/NFA/PDA), does there exist an integer ? ≥ 0 such that Σ^? ⊆ L(M)? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such ? can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such ? is not bounded by any computable function of the number of states. In the case of a DFA, we show that the problem is NP-complete, and e^{√{n log n} (1+o(1))} is an asymptotically tight upper bound for the smallest such ?, where n is the number of states. Finally, we prove that in all these cases, the problem becomes computationally easier when the length ? is also given in binary in the input: it is polynomially solvable for a DFA, PSPACE-complete for an NFA, and co-NEXPTIME-complete for a PDA.

Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, and Marek Szykuła. Existential Length Universality. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2020.16, author = {Gawrychowski, Pawe{\l} and Lange, Martin and Rampersad, Narad and Shallit, Jeffrey and Szyku{\l}a, Marek}, title = {{Existential Length Universality}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.16}, URN = {urn:nbn:de:0030-drops-118770}, doi = {10.4230/LIPIcs.STACS.2020.16}, annote = {Keywords: decision problem, deterministic automaton, nondeterministic automaton, pushdown automaton, regular expression, regular language, universality} }

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**Published in:** LIPIcs, Volume 120, 25th International Symposium on Temporal Representation and Reasoning (TIME 2018)

Hybrid branching-time logics are a powerful extension of branching-time logics like CTL, CTL^* or even the modal mu-calculus through the addition of binders, jumps and variable tests. Their expressiveness is not restricted by bisimulation-invariance anymore. Hence, they do not retain the tree model property, and the finite model property is equally lost. Their satisfiability problems are typically undecidable, their model checking problems (on finite models) are decidable with complexities ranging from polynomial to non-elementary time. In this paper we study the expressive power of such hybrid branching-time logics beyond some earlier results about their invariance under hybrid bisimulations. In particular, we aim to extend the hierarchy of non-hybrid branching-time logics CTL, CTL^+, CTL^* and the modal mu-calculus to their hybrid extensions. We show that most separation results can be transferred to the hybrid world, even though the required techniques become a bit more involved. We also present some collapse results for restricted classes of models that are especially worth investigating, namely linear, tree-shaped and finite models.

Daniel Kernberger and Martin Lange. On the Expressive Power of Hybrid Branching-Time Logics. In 25th International Symposium on Temporal Representation and Reasoning (TIME 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 120, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kernberger_et_al:LIPIcs.TIME.2018.16, author = {Kernberger, Daniel and Lange, Martin}, title = {{On the Expressive Power of Hybrid Branching-Time Logics}}, booktitle = {25th International Symposium on Temporal Representation and Reasoning (TIME 2018)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-089-7}, ISSN = {1868-8969}, year = {2018}, volume = {120}, editor = {Alechina, Natasha and N{\o}rv\r{a}g, Kjetil and Penczek, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2018.16}, URN = {urn:nbn:de:0030-drops-97816}, doi = {10.4230/LIPIcs.TIME.2018.16}, annote = {Keywords: branching-time, mu-calculus, hybrid logics, expressiveness} }

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**Published in:** LIPIcs, Volume 90, 24th International Symposium on Temporal Representation and Reasoning (TIME 2017)

We consider the hybridisation of the mu-calculus through the addition of nominals, binder and jump. Especially the use of the binder differentiates our approach from earlier hybridisations of the mu-calculus and also results in a more involved formal semantics. We then investigate the model checking problem and obtain ExpTime-completeness for the full logic and the same complexity as the modal mu-calculus for a fixed number of variables. We also show that this logic is invariant under hybrid bisimulation and use this result to show that - contrary to the non-hybrid case - the hybrid extension of the full branching time logic CTL* is not a fragment of the fully hybrid mu-calculus.

Daniel Kernberger and Martin Lange. The Fully Hybrid mu-Calculus. In 24th International Symposium on Temporal Representation and Reasoning (TIME 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 90, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kernberger_et_al:LIPIcs.TIME.2017.17, author = {Kernberger, Daniel and Lange, Martin}, title = {{The Fully Hybrid mu-Calculus}}, booktitle = {24th International Symposium on Temporal Representation and Reasoning (TIME 2017)}, pages = {17:1--17:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-052-1}, ISSN = {1868-8969}, year = {2017}, volume = {90}, editor = {Schewe, Sven and Schneider, Thomas and Wijsen, Jef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2017.17}, URN = {urn:nbn:de:0030-drops-79244}, doi = {10.4230/LIPIcs.TIME.2017.17}, annote = {Keywords: mu-calculus, hybrid logics, model checking, bisimulation invariance} }

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