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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Prior's tense logic forms the core of linear temporal logic, with both past- and future-looking modalities. We present a sound and complete proof system for tense logic over ordinals. Technically, this is a hypersequent system, enriched with an ordering, clusters, and annotations. The system is designed with proof search algorithms in mind, and yields an optimal coNP complexity for the validity problem. It entails a small model property for tense logic over ordinals: every satisfiable formula has a model of order type at most omega^2. It also allows to answer the validity problem for ordinals below or exactly equal to a given one.

David Baelde, Anthony Lick, and Sylvain Schmitz. A Hypersequent Calculus with Clusters for Tense Logic over Ordinals. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 15:1-15:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{baelde_et_al:LIPIcs.FSTTCS.2018.15, author = {Baelde, David and Lick, Anthony and Schmitz, Sylvain}, title = {{A Hypersequent Calculus with Clusters for Tense Logic over Ordinals}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.15}, URN = {urn:nbn:de:0030-drops-99143}, doi = {10.4230/LIPIcs.FSTTCS.2018.15}, annote = {Keywords: modal logic, proof system, hypersequent} }

Document

**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

We investigate the proof theory of a modal fragment of XPath equipped with data (in)equality tests over finite data trees, i.e., over finite unranked trees where nodes are labelled with both a symbol from a finite alphabet and a single data value from an infinite domain. We present a sound and complete sequent calculus for this logic, which yields the optimal PSPACE complexity bound for its validity problem.

David Baelde, Simon Lunel, and Sylvain Schmitz. A Sequent Calculus for a Modal Logic on Finite Data Trees. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 32:1-32:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baelde_et_al:LIPIcs.CSL.2016.32, author = {Baelde, David and Lunel, Simon and Schmitz, Sylvain}, title = {{A Sequent Calculus for a Modal Logic on Finite Data Trees}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.32}, URN = {urn:nbn:de:0030-drops-65720}, doi = {10.4230/LIPIcs.CSL.2016.32}, annote = {Keywords: XPath, proof systems, modal logic, complexity} }

Document

**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly underdeveloped. In particular, very little is known about the computational behavior of such proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a proof system: cut elimination and focalization. The first one is only known to hold for restricted (purely additive) infinitary calculi, thanks to the work of Santocanale and Fortier; the second one has never been studied in infinitary systems. In this paper, we consider the infinitary proof system muMALLi for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results. We thus establish muMALLi as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.

David Baelde, Amina Doumane, and Alexis Saurin. Infinitary Proof Theory: the Multiplicative Additive Case. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 42:1-42:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baelde_et_al:LIPIcs.CSL.2016.42, author = {Baelde, David and Doumane, Amina and Saurin, Alexis}, title = {{Infinitary Proof Theory: the Multiplicative Additive Case}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {42:1--42:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.42}, URN = {urn:nbn:de:0030-drops-65825}, doi = {10.4230/LIPIcs.CSL.2016.42}, annote = {Keywords: Infinitary proofs, linear logic} }

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**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Various logics have been introduced in order to reason over (co)inductive specifications and, through the Curry-Howard correspondence, to study computation over inductive and coinductive data. The logic mu-MALL is one of those logics, extending multiplicative and additive linear logic with least and greatest fixed point operators.
In this paper, we investigate the semantics of mu-MALL proofs in (computational) ludics. This framework is built around the notion of design, which can be seen as an analogue of the strategies of game semantics. The infinitary nature of designs makes them particularly well suited for representing computations over infinite data.
We provide mu-MALL with a denotational semantics, interpreting proofs by designs and formulas by particular sets of designs called behaviours. Then we prove a completeness result for the class of "essentially finite designs", which are those designs performing a finite computation followed by a copycat. On the way to completeness, we investigate semantic inclusion, proving its decidability (given two formulas, we can decide whether the semantics of one is included in the other's) and completeness (if semantic inclusion holds, the corresponding implication is provable in mu-MALL).

David Baelde, Amina Doumane, and Alexis Saurin. Least and Greatest Fixed Points in Ludics. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 549-566, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{baelde_et_al:LIPIcs.CSL.2015.549, author = {Baelde, David and Doumane, Amina and Saurin, Alexis}, title = {{Least and Greatest Fixed Points in Ludics}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {549--566}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.549}, URN = {urn:nbn:de:0030-drops-54374}, doi = {10.4230/LIPIcs.CSL.2015.549}, annote = {Keywords: proof theory, fixed points, linear logic, ludics, game semantics, completeness, circular proofs, infinitary proof systems} }

Document

**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

Security protocols are concurrent processes that communicate using cryptography with the aim of achieving various security properties. Recent work on their formal verification has brought procedures and tools for deciding trace equivalence properties (e.g. anonymity, unlinkability, vote secrecy) for a bounded number of sessions. However, these procedures are based on a naive symbolic exploration of all traces of the considered processes which, unsurprisingly, greatly limits the scalability and practical impact of the verification tools.
In this paper, we mitigate this difficulty by developing partial order reduction techniques for the verification of security protocols. We provide reduced transition systems that optimally eliminate redundant traces, and which are adequate for model-checking trace equivalence properties of protocols by means of symbolic execution. We have implemented our reductions in the tool Apte, and demonstrated that it achieves the expected speedup on various protocols.

David Baelde, Stéphanie Delaune, and Lucca Hirschi. Partial Order Reduction for Security Protocols. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 497-510, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{baelde_et_al:LIPIcs.CONCUR.2015.497, author = {Baelde, David and Delaune, St\'{e}phanie and Hirschi, Lucca}, title = {{Partial Order Reduction for Security Protocols}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {497--510}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.497}, URN = {urn:nbn:de:0030-drops-53946}, doi = {10.4230/LIPIcs.CONCUR.2015.497}, annote = {Keywords: Cryptographic protocols, verification, process algebra, trace equivalence} }

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