Document

**Published in:** LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

We investigate containers and polynomial functors in Quantitative Type Theory, and give initial algebra semantics of inductive data types in the presence of linearity. We show that reasoning by induction is supported, and equivalent to initiality, also in the linear setting.

Georgi Nakov and Fredrik Nordvall Forsberg. Quantitative Polynomial Functors. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{nakov_et_al:LIPIcs.TYPES.2021.10, author = {Nakov, Georgi and Nordvall Forsberg, Fredrik}, title = {{Quantitative Polynomial Functors}}, booktitle = {27th International Conference on Types for Proofs and Programs (TYPES 2021)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-254-9}, ISSN = {1868-8969}, year = {2022}, volume = {239}, editor = {Basold, Henning and Cockx, Jesper and Ghilezan, Silvia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.10}, URN = {urn:nbn:de:0030-drops-167795}, doi = {10.4230/LIPIcs.TYPES.2021.10}, annote = {Keywords: quantitative type theory, polynomial functors, inductive data types} }

Document

Early Ideas

**Published in:** LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

We investigate containers and polynomial functors in Quantitative Type Theory, and give initial algebra semantics of inductive data types in the presence of linearity. We show that reasoning by induction is supported, and equivalent to initiality, also in the linear setting.

Georgi Nakov and Fredrik Nordvall Forsberg. Quantitative Polynomial Functors (Early Ideas). In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 22:1-22:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{nakov_et_al:LIPIcs.CALCO.2021.22, author = {Nakov, Georgi and Nordvall Forsberg, Fredrik}, title = {{Quantitative Polynomial Functors}}, booktitle = {9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)}, pages = {22:1--22:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-212-9}, ISSN = {1868-8969}, year = {2021}, volume = {211}, editor = {Gadducci, Fabio 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.CALCO.2021.22}, URN = {urn:nbn:de:0030-drops-153774}, doi = {10.4230/LIPIcs.CALCO.2021.22}, annote = {Keywords: quantitative type theory, polynomial functors, inductive data types} }

Document

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

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.

Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu. Connecting Constructive Notions of Ordinals in Homotopy Type Theory. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kraus_et_al:LIPIcs.MFCS.2021.70, author = {Kraus, Nicolai and Nordvall Forsberg, Fredrik and Xu, Chuangjie}, title = {{Connecting Constructive Notions of Ordinals in Homotopy Type Theory}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {70:1--70:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.70}, URN = {urn:nbn:de:0030-drops-145100}, doi = {10.4230/LIPIcs.MFCS.2021.70}, annote = {Keywords: Constructive ordinals, Cantor normal forms, Brouwer trees} }

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Complete Volume

**Published in:** LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

LIPIcs, Volume 104, TYPES'17, Complete Volume

23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{abel_et_al:LIPIcs.TYPES.2017, title = {{LIPIcs, Volume 104, TYPES'17, Complete Volume}}, booktitle = {23rd International Conference on Types for Proofs and Programs (TYPES 2017)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-071-2}, ISSN = {1868-8969}, year = {2019}, volume = {104}, editor = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017}, URN = {urn:nbn:de:0030-drops-101671}, doi = {10.4230/LIPIcs.TYPES.2017}, annote = {Keywords: Theory of computation, Type theory, Proof theory, Program verification} }

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Front Matter

**Published in:** LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

Front Matter, Table of Contents, Preface, Conference Organization

23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abel_et_al:LIPIcs.TYPES.2017.0, author = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {23rd International Conference on Types for Proofs and Programs (TYPES 2017)}, pages = {0:i--0:x}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-071-2}, ISSN = {1868-8969}, year = {2019}, volume = {104}, editor = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017.0}, URN = {urn:nbn:de:0030-drops-100488}, doi = {10.4230/LIPIcs.TYPES.2017.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Dybjer and Setzer introduced the definitional principle of inductive-recursively defined families - i.e. of families (U : Set, T : U -> D) such that the inductive definition of U may depend on the recursively defined T --- by defining a type DS D E of codes. Each c : DS D E defines a functor [c] : Fam D -> Fam E, and
(U, T) = \mu [c] : Fam D is exhibited as the initial algebra of [c].
This paper considers the composition of DS-definable functors: Given F : Fam C -> Fam D and G : Fam D -> Fam E, is G \circ F : Fam C -> Fam E DS-definable, if F and G are? We show that this is the case if and only if powers of families are DS-definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF a subsytem of DS a subsystem of PN. Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN-functors exist by adapting Dybjer-Setzer's proof for DS.

Neil Ghani, Conor McBride, Fredrik Nordvall Forsberg, and Stephan Spahn. Variations on Inductive-Recursive Definitions. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ghani_et_al:LIPIcs.MFCS.2017.63, author = {Ghani, Neil and McBride, Conor and Nordvall Forsberg, Fredrik and Spahn, Stephan}, title = {{Variations on Inductive-Recursive Definitions}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {63:1--63:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.63}, URN = {urn:nbn:de:0030-drops-81184}, doi = {10.4230/LIPIcs.MFCS.2017.63}, annote = {Keywords: Type Theory, induction-recursion, initial-algebra semantics} }

Document

**Published in:** LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

We provide a categorical framework for models of a type theory that has special types for physical quantities. The types are indexed by the physical dimensions that they involve. Fibrations are used to organize this index structure in the models of the type theory. We develop some informative models of this type theory: firstly, a model based on group actions, which captures invariance under scaling, and secondly, a way of constructing new models using relational parametricity.

Robert Atkey, Neil Ghani, Fredrik Nordvall Forsberg, Timothy Revell, and Sam Staton. Models for Polymorphism over Physical Dimension. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 45-59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{atkey_et_al:LIPIcs.TLCA.2015.45, author = {Atkey, Robert and Ghani, Neil and Nordvall Forsberg, Fredrik and Revell, Timothy and Staton, Sam}, title = {{Models for Polymorphism over Physical Dimension}}, booktitle = {13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)}, pages = {45--59}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-87-3}, ISSN = {1868-8969}, year = {2015}, volume = {38}, editor = {Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.45}, URN = {urn:nbn:de:0030-drops-51547}, doi = {10.4230/LIPIcs.TLCA.2015.45}, annote = {Keywords: Category Theory, Units of Measure, Dimension Types, Type Theory} }