6 Search Results for "Melquiond, Guillaume"


Document
A Strong Call-By-Need Calculus

Authors: Thibaut Balabonski, Antoine Lanco, and Guillaume Melquiond

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


Abstract
We present a call-by-need λ-calculus that enables strong reduction (that is, reduction inside the body of abstractions) and guarantees that arguments are only evaluated if needed and at most once. This calculus uses explicit substitutions and subsumes the existing strong-call-by-need strategy, but allows for more reduction sequences, and often shorter ones, while preserving the neededness. The calculus is shown to be normalizing in a strong sense: Whenever a λ-term t admits a normal form n in the λ-calculus, then any reduction sequence from t in the calculus eventually reaches a representative of the normal form n. We also exhibit a restriction of this calculus that has the diamond property and that only performs reduction sequences of minimal length, which makes it systematically better than the existing strategy. We have used the Abella proof assistant to formalize part of this calculus, and discuss how this experiment affected its design.

Cite as

Thibaut Balabonski, Antoine Lanco, and Guillaume Melquiond. A Strong Call-By-Need Calculus. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 9:1-9:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{balabonski_et_al:LIPIcs.FSCD.2021.9,
  author =	{Balabonski, Thibaut and Lanco, Antoine and Melquiond, Guillaume},
  title =	{{A Strong Call-By-Need Calculus}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{9:1--9:22},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.9},
  URN =		{urn:nbn:de:0030-drops-142477},
  doi =		{10.4230/LIPIcs.FSCD.2021.9},
  annote =	{Keywords: strong reduction, call-by-need, evaluation strategy, normalization}
}
Document
Formal Proofs of Tarjan’s Strongly Connected Components Algorithm in Why3, Coq and Isabelle

Authors: Ran Chen, Cyril Cohen, Jean-Jacques Lévy, Stephan Merz, and Laurent Théry

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)


Abstract
Comparing provers on a formalization of the same problem is always a valuable exercise. In this paper, we present the formal proof of correctness of a non-trivial algorithm from graph theory that was carried out in three proof assistants: Why3, Coq, and Isabelle.

Cite as

Ran Chen, Cyril Cohen, Jean-Jacques Lévy, Stephan Merz, and Laurent Théry. Formal Proofs of Tarjan’s Strongly Connected Components Algorithm in Why3, Coq and Isabelle. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{chen_et_al:LIPIcs.ITP.2019.13,
  author =	{Chen, Ran and Cohen, Cyril and L\'{e}vy, Jean-Jacques and Merz, Stephan and Th\'{e}ry, Laurent},
  title =	{{Formal Proofs of Tarjan’s Strongly Connected Components Algorithm in Why3, Coq and Isabelle}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.13},
  URN =		{urn:nbn:de:0030-drops-110683},
  doi =		{10.4230/LIPIcs.ITP.2019.13},
  annote =	{Keywords: Mathematical logic, Formal proof, Graph algorithm, Program verification}
}
Document
Quantitative Continuity and Computable Analysis in Coq

Authors: Florian Steinberg, Laurent Théry, and Holger Thies

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)


Abstract
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant Coq and heavily relies on the Incone library for information theoretic continuity. This library is developed by one of the authors and the results of this paper extend the library. While full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the Incone library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The paper includes a brief description of the most important concepts of Incone and its sub libraries mf and Metric. The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that the countably infinite product of a space with itself is isomorphic to a space of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also describe many non-computational results that support the correctness of definitions from the library. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and represented space structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers.

Cite as

Florian Steinberg, Laurent Théry, and Holger Thies. Quantitative Continuity and Computable Analysis in Coq. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{steinberg_et_al:LIPIcs.ITP.2019.28,
  author =	{Steinberg, Florian and Th\'{e}ry, Laurent and Thies, Holger},
  title =	{{Quantitative Continuity and Computable Analysis in Coq}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{28:1--28:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.28},
  URN =		{urn:nbn:de:0030-drops-110830},
  doi =		{10.4230/LIPIcs.ITP.2019.28},
  annote =	{Keywords: computable analysis, Coq, contionuous functionals, discontinuity, closed choice on the naturals}
}
Document
A Certificate-Based Approach to Formally Verified Approximations

Authors: Florent Bréhard, Assia Mahboubi, and Damien Pous

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)


Abstract
We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant. Based on interval arithmetic, this library also implements a technique of validation a posteriori based on the Banach fixed-point theorem. We illustrate this technique on the case of operations of division and square root. This library features a collection of abstract structures that organise the specfication of rigorous approximations, and modularise the related proofs. Finally, we provide an implementation of verified Chebyshev approximations, and we discuss a few examples of computations.

Cite as

Florent Bréhard, Assia Mahboubi, and Damien Pous. A Certificate-Based Approach to Formally Verified Approximations. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{brehard_et_al:LIPIcs.ITP.2019.8,
  author =	{Br\'{e}hard, Florent and Mahboubi, Assia and Pous, Damien},
  title =	{{A Certificate-Based Approach to Formally Verified Approximations}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.8},
  URN =		{urn:nbn:de:0030-drops-110638},
  doi =		{10.4230/LIPIcs.ITP.2019.8},
  annote =	{Keywords: approximation theory, Chebyshev polynomials, Banach fixed-point theorem, interval arithmetic, Coq}
}
Document
Primitive Floats in Coq

Authors: Guillaume Bertholon, Érik Martin-Dorel, and Pierre Roux

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)


Abstract
Some mathematical proofs involve intensive computations, for instance: the four-color theorem, Hales' theorem on sphere packing (formerly known as the Kepler conjecture) or interval arithmetic. For numerical computations, floating-point arithmetic enjoys widespread usage thanks to its efficiency, despite the introduction of rounding errors. Formal guarantees can be obtained on floating-point algorithms based on the IEEE 754 standard, which precisely specifies floating-point arithmetic and its rounding modes, and a proof assistant such as Coq, that enjoys efficient computation capabilities. Coq offers machine integers, however floating-point arithmetic still needed to be emulated using these integers. A modified version of Coq is presented that enables using the machine floating-point operators. The main obstacles to such an implementation and its soundness are discussed. Benchmarks show potential performance gains of two orders of magnitude.

Cite as

Guillaume Bertholon, Érik Martin-Dorel, and Pierre Roux. Primitive Floats in Coq. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bertholon_et_al:LIPIcs.ITP.2019.7,
  author =	{Bertholon, Guillaume and Martin-Dorel, \'{E}rik and Roux, Pierre},
  title =	{{Primitive Floats in Coq}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.7},
  URN =		{urn:nbn:de:0030-drops-110629},
  doi =		{10.4230/LIPIcs.ITP.2019.7},
  annote =	{Keywords: Coq formal proofs, floating-point arithmetic, reflexive tactics, Cholesky decomposition}
}
Document
A Proposal to add Interval Arithmetic to the C++ Standard Library

Authors: Sylvain Pion, Hervé Brönnimann, and Guillaume Melquiond

Published in: Dagstuhl Seminar Proceedings, Volume 6021, Reliable Implementation of Real Number Algorithms: Theory and Practice (2006)


Abstract
I will report on a recent effort by Guillaume Melquiond, Hervé Br"onnimann and myself to push forward a proposal to include interval arithmetic in the next C++ ISO standard. The goals of the standardization are to produce a unified specification which will serve as many uses of intervals as possible, together with hoping for very efficient implementations, closer to the compilers. I will describe how the standardization process works, explain some of the design choices made, and list some of the other questions arising in the process. We welcome any comment on the proposal.

Cite as

Sylvain Pion, Hervé Brönnimann, and Guillaume Melquiond. A Proposal to add Interval Arithmetic to the C++ Standard Library. In Reliable Implementation of Real Number Algorithms: Theory and Practice. Dagstuhl Seminar Proceedings, Volume 6021, pp. 1-25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006)


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@InProceedings{pion_et_al:DagSemProc.06021.4,
  author =	{Pion, Sylvain and Br\"{o}nnimann, Herv\'{e} and Melquiond, Guillaume},
  title =	{{A Proposal to add Interval Arithmetic to the C++ Standard Library}},
  booktitle =	{Reliable Implementation of Real Number Algorithms: Theory and Practice},
  pages =	{1--25},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6021},
  editor =	{Peter Hertling and Christoph M. Hoffmann and Wolfram Luther and Nathalie Revol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06021.4},
  URN =		{urn:nbn:de:0030-drops-7189},
  doi =		{10.4230/DagSemProc.06021.4},
  annote =	{Keywords: Interval arithmetic, C++, ISO standard}
}
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