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**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear.

Cédric de Lacroix and Luigi Santocanale. Frobenius Structures in Star-Autonomous Categories. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{delacroix_et_al:LIPIcs.CSL.2023.18, author = {de Lacroix, C\'{e}dric and Santocanale, Luigi}, title = {{Frobenius Structures in Star-Autonomous Categories}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {18:1--18:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.18}, URN = {urn:nbn:de:0030-drops-174798}, doi = {10.4230/LIPIcs.CSL.2023.18}, annote = {Keywords: Quantale, Frobenius quantale, Girard quantale, associative algebra, star-autonomous category, nuclear object, adjoint} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

For a regular cardinal kappa, a formula of the modal mu-calculus is kappa-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of kappa-directed sets. We define the fragment C1 (x) of the modal mu-calculus and prove that all the formulas in this fragment are aleph_1-continuous. For each formula phi(x) of the modal mu-calculus, we construct a formula psi(x) in C1 (x) such that phi(x) is kappa-continuous, for some kappa, if and only if psi(x) is equivalent to phi(x). Consequently, we prove that (i) the problem whether a formula is kappa-continuous for some kappa is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment C0(x) studied by Fontaine and the fragment C1 (x). We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal mu-calculus. An ordinal alpha is the closure ordinal of a formula phi(x) if its interpretation on every model converges to its least fixed-point in at most alpha steps and if there is a model where the convergence occurs exactly in alpha steps. We prove that omega_1, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, omega, omega_1 by using the binary operator symbol + gives rise to a closure ordinal.

Maria João Gouveia and Luigi Santocanale. Aleph1 and the Modal mu-Calculus. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gouveia_et_al:LIPIcs.CSL.2017.38, author = {Gouveia, Maria Jo\~{a}o and Santocanale, Luigi}, title = {{Aleph1 and the Modal mu-Calculus}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {38:1--38:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.38}, URN = {urn:nbn:de:0030-drops-76926}, doi = {10.4230/LIPIcs.CSL.2017.38}, annote = {Keywords: Modal mu-calculus, regular cardinal, continuous function, aleph1, omega1, closure ordinal, ordinal sum} }

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**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

One of the authors introduced in [Santocanale, FoSSaCS, 2002] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [Santocanale, FoSSaCS, 2002] is cut-free; even if sound and complete for provability, it lacked an important property for the semantics of proofs, namely fullness w.r.t. the class of intended categorical models (called mu-bicomplete categories in [Santocanale, ITA, 2002]).
In this paper we fix this problem by adding the cut rule to the calculus and by modifying accordingly the syntactical constraint ensuring soundness of proofs. The enhanced proof system fully represents arrows of the canonical model (a free mu-bicomplete category). We also describe a cut-elimination procedure as a a model of computation arising from the above mentioned categorical operations. The procedure constructs a cut-free proof-tree with possibly infinite branches out of a finite circular proof with cuts.

Jérôme Fortier and Luigi Santocanale. Cuts for circular proofs: semantics and cut-elimination. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 248-262, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{fortier_et_al:LIPIcs.CSL.2013.248, author = {Fortier, J\'{e}r\^{o}me and Santocanale, Luigi}, title = {{Cuts for circular proofs: semantics and cut-elimination}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {248--262}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.248}, URN = {urn:nbn:de:0030-drops-42019}, doi = {10.4230/LIPIcs.CSL.2013.248}, annote = {Keywords: categorical proof-theory, fixpoints, initial and final (co)algebras, inductive and coinductive types} }