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Documents authored by Hirschowitz, André

Document
DOI: 10.4230/LIPIcs.FSCD.2020.12
Modules over Monads and Operational Semantics

Authors: André Hirschowitz, Tom Hirschowitz, and Ambroise Lafont

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2019.6
Modular Specification of Monads Through Higher-Order Presentations

Authors: Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, and Marco Maggesi

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.CSL.2018.4
High-Level Signatures and Initial Semantics

Authors: Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, and Marco Maggesi

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.TLCA.2015.226
Wild omega-Categories for the Homotopy Hypothesis in Type Theory

Authors: André Hirschowitz, Tom Hirschowitz, and Nicolas Tabareau

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

Abstract
In classical homotopy theory, the homotopy hypothesis asserts that the fundamental varpi-groupoid construction induces an equivalence between topological spaces and weak varpi-groupoids. In the light of Voevodsky's univalent foundations program, which puts forward an interpretation of types as topological spaces, we consider the question of transposing the homotopy hypothesis to type theory. Indeed such a transposition could stand as a new approach to specifying higher inductive types. Since the formalisation of general weak varpi-groupoids in type theory is a difficult task, we only take a first step towards this goal, which consists in exploring a shortcut through strict varpi-categories. The first outcome is a satisfactory type-theoretic notion of strict varpi-category, which has hsets of cells in all dimensions. For this notion, defining the 'fundamental strict varpi-category' of a type seems out of reach. The second outcome is an 'incoherently strict' notion of type-theoretic varpi-category, which admits arbitrary types of cells in all dimensions. These are the 'wild' varpi-categories of the title. They allow the definition of a 'fundamental wild varpi-category' map, which leads to our (partial) homotopy hypothesis for type theory (stating an adjunction, not an equivalence). All of our results have been formalised in the Coq proof assistant. Our formalisation makes systematic use of the machinery of coinductive types.
Cite as

André Hirschowitz, Tom Hirschowitz, and Nicolas Tabareau. Wild omega-Categories for the Homotopy Hypothesis in Type Theory. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 226-240, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hirschowitz_et_al:LIPIcs.TLCA.2015.226,
  author =	{Hirschowitz, Andr\'{e} and Hirschowitz, Tom and Tabareau, Nicolas},
  title =	{{Wild omega-Categories for the Homotopy Hypothesis in Type Theory}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{226--240},
  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.2015.226},
  URN =		{urn:nbn:de:0030-drops-51669},
  doi =		{10.4230/LIPIcs.TLCA.2015.226},
  annote =	{Keywords: Homotopy Type theory; Omega-categories; Coinduction; Homotopy hypothesis}
}
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