4 Search Results for "Veltri, Niccolò"

Document
DOI: 10.4230/LIPIcs.ITP.2023.20
Constructive Final Semantics of Finite Bags

Authors: Philipp Joram and Niccolò Veltri

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract
Finitely-branching and unlabelled dynamical systems are typically modelled as coalgebras for the finite powerset functor. If states are reachable in multiple ways, coalgebras for the finite bag functor provide a more faithful representation. The final coalgebra of this functor is employed as a denotational domain for the evaluation of such systems. Elements of the final coalgebra are non-wellfounded trees with finite unordered branching, representing the evolution of systems starting from a given initial state. This paper is dedicated to the construction of the final coalgebra of the finite bag functor in homotopy type theory (HoTT). We first compare various equivalent definitions of finite bags employing higher inductive types, both as sets and as groupoids (in the sense of HoTT). We then analyze a few well-known, classical set-theoretic constructions of final coalgebras in our constructive setting. We show that, in the case of set-based definitions of finite bags, some constructions are intrinsically classical, in the sense that they are equivalent to some weak form of excluded middle. Nevertheless, a type satisfying the universal property of the final coalgebra can be constructed in HoTT employing the groupoid-based definition of finite bags. We conclude by discussing generalizations of our constructions to the wider class of analytic functors.
Cite as

Philipp Joram and Niccolò Veltri. Constructive Final Semantics of Finite Bags. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{joram_et_al:LIPIcs.ITP.2023.20,
  author =	{Joram, Philipp and Veltri, Niccol\`{o}},
  title =	{{Constructive Final Semantics of Finite Bags}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.20},
  URN =		{urn:nbn:de:0030-drops-183954},
  doi =		{10.4230/LIPIcs.ITP.2023.20},
  annote =	{Keywords: finite bags, final coalgebra, homotopy type theory, Cubical Agda}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2021.22
Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor

Authors: Niccolò Veltri

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

Abstract
The finite powerset functor is a construct frequently employed for the specification of nondeterministic transition systems as coalgebras. The final coalgebra of the finite powerset functor, whose elements characterize the dynamical behavior of transition systems, is a well-understood object which enjoys many equivalent presentations in set-theoretic foundations based on classical logic. In this paper, we discuss various constructions of the final coalgebra of the finite powerset functor in constructive type theory, and we formalize our results in the Cubical Agda proof assistant. Using setoids, the final coalgebra of the finite powerset functor can be defined from the final coalgebra of the list functor. Using types instead of setoids, as it is common in homotopy type theory, one can specify the finite powerset datatype as a higher inductive type and define its final coalgebra as a coinductive type. Another construction is obtained by quotienting the final coalgebra of the list functor, but the proof of finality requires the assumption of the axiom of choice. We conclude the paper with an analysis of a classical construction by James Worrell, and show that its adaptation to our constructive setting requires the presence of classical axioms such as countable choice and the lesser limited principle of omniscience.
Cite as

Niccolò Veltri. Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{veltri:LIPIcs.FSCD.2021.22,
  author =	{Veltri, Niccol\`{o}},
  title =	{{Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.22},
  URN =		{urn:nbn:de:0030-drops-142601},
  doi =		{10.4230/LIPIcs.FSCD.2021.22},
  annote =	{Keywords: type theory, finite powerset, final coalgebra, Cubical Agda}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2019.5
Bicategories in Univalent Foundations

Authors: Benedikt Ahrens, Dan Frumin, Marco Maggesi, and Niels van der Weide

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

Abstract
We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of those, we develop the notion of "displayed bicategories", an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. Displayed bicategories allow us to construct univalent bicategories in a modular fashion. To demonstrate the applicability of this notion, we prove several bicategories are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Our work is formalized in the UniMath library of univalent mathematics.
Cite as

Benedikt Ahrens, Dan Frumin, Marco Maggesi, and Niels van der Weide. Bicategories in Univalent Foundations. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ahrens_et_al:LIPIcs.FSCD.2019.5,
  author =	{Ahrens, Benedikt and Frumin, Dan and Maggesi, Marco and van der Weide, Niels},
  title =	{{Bicategories in Univalent Foundations}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{5:1--5:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.5},
  URN =		{urn:nbn:de:0030-drops-105124},
  doi =		{10.4230/LIPIcs.FSCD.2019.5},
  annote =	{Keywords: bicategory theory, univalent mathematics, dependent type theory, Coq}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2019.32
Guarded Recursion in Agda via Sized Types

Authors: Niccolò Veltri and Niels van der Weide

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

Abstract
In type theory, programming and reasoning with possibly non-terminating programs and potentially infinite objects is achieved using coinductive types. Recursively defined programs of these types need to be productive to guarantee the consistency of the type system. Proof assistants such as Agda and Coq traditionally employ strict syntactic productivity checks, which often make programming with coinductive types convoluted. One way to overcome this issue is by encoding productivity at the level of types so that the type system forbids the implementation of non-productive corecursive programs. In this paper we compare two different approaches to type-based productivity: guarded recursion and sized types. More specifically, we show how to simulate guarded recursion in Agda using sized types. We formalize the syntax of a simple type theory for guarded recursion, which is a variant of Atkey and McBride’s calculus for productive coprogramming. Then we give a denotational semantics using presheaves over the preorder of sizes. Sized types are fundamentally used to interpret the characteristic features of guarded recursion, notably the fixpoint combinator.
Cite as

Niccolò Veltri and Niels van der Weide. Guarded Recursion in Agda via Sized Types. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{veltri_et_al:LIPIcs.FSCD.2019.32,
  author =	{Veltri, Niccol\`{o} and van der Weide, Niels},
  title =	{{Guarded Recursion in Agda via Sized Types}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{32:1--32:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.32},
  URN =		{urn:nbn:de:0030-drops-105391},
  doi =		{10.4230/LIPIcs.FSCD.2019.32},
  annote =	{Keywords: guarded recursion, type theory, semantics, coinduction, sized types}
}
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