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**Published in:** LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Inductive-inductive types (IITs) are a generalisation of inductive types in type theory. They allow the mutual definition of types with multiple sorts where later sorts can be indexed by previous ones. An example is the Chapman-style syntax of type theory with conversion relations for each sort where e.g. the sort of types is indexed by contexts. In this paper we show that if a model of extensional type theory (ETT) supports indexed W-types, then it supports finitely branching IITs. We use a small internal type theory called the theory of signatures to specify IITs. We show that if a model of ETT supports the syntax for the theory of signatures, then it supports all IITs. We construct this syntax from indexed W-types using preterms and typing relations and prove its initiality following Streicher. The construction of the syntax and its initiality proof were formalised in Agda.

Ambrus Kaposi, András Kovács, and Ambroise Lafont. For Finitary Induction-Induction, Induction Is Enough. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 6:1-6:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaposi_et_al:LIPIcs.TYPES.2019.6, author = {Kaposi, Ambrus and Kov\'{a}cs, Andr\'{a}s and Lafont, Ambroise}, title = {{For Finitary Induction-Induction, Induction Is Enough}}, booktitle = {25th International Conference on Types for Proofs and Programs (TYPES 2019)}, pages = {6:1--6:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-158-0}, ISSN = {1868-8969}, year = {2020}, volume = {175}, editor = {Bezem, Marc and Mahboubi, Assia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.6}, URN = {urn:nbn:de:0030-drops-130707}, doi = {10.4230/LIPIcs.TYPES.2019.6}, annote = {Keywords: type theory, inductive types, inductive-inductive types} }

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**Published in:** LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as ̅λμ-calculus, π-calculus, Positive GSOS specifications, differential λ-calculus, and the big-step, simply-typed, call-by-value λ-calculus. Finally, we design a suitable notion of signature for transition monads.

André Hirschowitz, Tom Hirschowitz, and Ambroise Lafont. Modules over Monads and Operational Semantics. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 12:1-12:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hirschowitz_et_al:LIPIcs.FSCD.2020.12, author = {Hirschowitz, Andr\'{e} and Hirschowitz, Tom and Lafont, Ambroise}, title = {{Modules over Monads and Operational Semantics}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {12:1--12:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.12}, URN = {urn:nbn:de:0030-drops-123341}, doi = {10.4230/LIPIcs.FSCD.2020.12}, annote = {Keywords: Operational semantics, Category theory} }

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**Published in:** LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

In their work on second-order equational logic, Fiore and Hur have studied presentations of simply typed languages by generating binding constructions and equations among them. To each pair consisting of a binding signature and a set of equations, they associate a category of "models", and they give a monadicity result which implies that this category has an initial object, which is the language presented by the pair.
In the present work, we propose, for the untyped setting, a variant of their approach where monads and modules over them are the central notions. More precisely, we study, for monads over sets, presentations by generating ("higher-order") operations and equations among them. We consider a notion of 2-signature which allows to specify a monad with a family of binding operations subject to a family of equations, as is the case for the paradigmatic example of the lambda calculus, specified by its two standard constructions (application and abstraction) subject to beta- and eta-equalities. Such a 2-signature is hence a pair (Sigma,E) of a binding signature Sigma and a family E of equations for Sigma. This notion of 2-signature has been introduced earlier by Ahrens in a slightly different context.
We associate, to each 2-signature (Sigma,E), a category of "models of (Sigma,E)"; and we say that a 2-signature is "effective" if this category has an initial object; the monad underlying this (essentially unique) object is the "monad specified by the 2-signature". Not every 2-signature is effective; we identify a class of 2-signatures, which we call "algebraic", that are effective.
Importantly, our 2-signatures together with their models enjoy "modularity": when we glue (algebraic) 2-signatures together, their initial models are glued accordingly.
We provide a computer formalization for our main results.

Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, and Marco Maggesi. Modular Specification of Monads Through Higher-Order Presentations. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ahrens_et_al:LIPIcs.FSCD.2019.6, author = {Ahrens, Benedikt and Hirschowitz, Andr\'{e} and Lafont, Ambroise and Maggesi, Marco}, title = {{Modular Specification of Monads Through Higher-Order Presentations}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {6:1--6:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-107-8}, ISSN = {1868-8969}, year = {2019}, volume = {131}, editor = {Geuvers, Herman}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.6}, URN = {urn:nbn:de:0030-drops-105136}, doi = {10.4230/LIPIcs.FSCD.2019.6}, annote = {Keywords: free monads, presentation of monads, initial semantics, signatures, syntax, monadic substitution, computer-checked proofs} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We present a device for specifying and reasoning about syntax for datatypes, programming languages, and logic calculi. More precisely, we consider a general notion of "signature" for specifying syntactic constructions. Our signatures subsume classical algebraic signatures (i.e., signatures for languages with variable binding, such as the pure lambda calculus) and extend to much more general examples.
In the spirit of Initial Semantics, we define the "syntax generated by a signature" to be the initial object - if it exists - in a suitable category of models. Our notions of signature and syntax are suited for compositionality and provide, beyond the desired algebra of terms, a well-behaved substitution and the associated inductive/recursive principles.
Our signatures are "general" in the sense that the existence of an associated syntax is not automatically guaranteed. In this work, we identify a large and simple class of signatures which do generate a syntax.
This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi, which, in turn, was directly inspired by some earlier work of Ghani-Uustalu-Hamana and Matthes-Uustalu.
The main results presented in the paper are computer-checked within the UniMath system.

Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, and Marco Maggesi. High-Level Signatures and Initial Semantics. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ahrens_et_al:LIPIcs.CSL.2018.4, author = {Ahrens, Benedikt and Hirschowitz, Andr\'{e} and Lafont, Ambroise and Maggesi, Marco}, title = {{High-Level Signatures and Initial Semantics}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {4:1--4:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.4}, URN = {urn:nbn:de:0030-drops-96713}, doi = {10.4230/LIPIcs.CSL.2018.4}, annote = {Keywords: initial semantics, signatures, syntax, monadic substitution, computer-checked proofs} }

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