3 Search Results for "Magaud, Nicolas"

Document

DOI: 10.4230/LIPIcs.ITP.2024.33

Correctly Compiling Proofs About Programs Without Proving Compilers Correct

Authors: Audrey Seo, Christopher Lam, Dan Grossman, and Talia Ringer

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

Guaranteeing correct compilation is nearly synonymous with compiler verification. However, the correctness guarantees for certified compilers and translation validation can be stronger than we need. While many compilers do have incorrect behavior, even when a compiler bug occurs it may not change the program’s behavior meaningfully with respect to its specification. Many real-world specifications are necessarily partial in that they do not completely specify all of a program’s behavior. While compiler verification and formal methods have had great success for safety-critical systems, there are magnitudes more code, such as math libraries, compiled with incorrect compilers, that would benefit from a guarantee of its partial specification. This paper explores a technique to get guarantees about compiled programs even in the presence of an unverified, or even incorrect, compiler. Our workflow compiles programs, specifications, and proof objects, from an embedded source language and logic to an embedded target language and logic. We implement two simple imperative languages, each with its own Hoare-style program logic, and a system for instantiating proof compilers out of compilers between these two languages that fulfill certain equational conditions in Coq. We instantiate our system on four compilers: one that is incomplete, two that are incorrect, and one that is correct but unverified. We use these instances to compile Hoare proofs for several programs, and we are able to leverage compiled proofs to assist in proofs of larger programs. Our proof compiler system is formally proven sound in Coq. We demonstrate how our approach enables strong target program guarantees even in the presence of incorrect compilation, opening up new options for which proof burdens one might shoulder instead of, or in addition to, compiler correctness.

Cite as

Audrey Seo, Christopher Lam, Dan Grossman, and Talia Ringer. Correctly Compiling Proofs About Programs Without Proving Compilers Correct. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{seo_et_al:LIPIcs.ITP.2024.33,
  author =	{Seo, Audrey and Lam, Christopher and Grossman, Dan and Ringer, Talia},
  title =	{{Correctly Compiling Proofs About Programs Without Proving Compilers Correct}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{33:1--33:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.33},
  URN =		{urn:nbn:de:0030-drops-207612},
  doi =		{10.4230/LIPIcs.ITP.2024.33},
  annote =	{Keywords: proof transformations, compiler validation, program logics, proof engineering}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2022.11

Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families

Authors: Catherine Dubois, Nicolas Magaud, and Alain Giorgetti

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

Abstract

There are several ways to formally represent families of data, such as lambda terms, in a type theory such as the dependent type theory of Coq. Mathematical representations are very compact ones and usually rely on the use of dependent types, but they tend to be difficult to handle in practice. On the contrary, implementations based on a larger (and simpler) data structure combined with a restriction property are much easier to deal with. In this work, we study several families related to lambda terms, among which Motzkin trees, seen as lambda term skeletons, closable Motzkin trees, corresponding to closed lambda terms, and a parameterized family of open lambda terms. For each of these families, we define two different representations, show that they are isomorphic and provide tools to switch from one representation to another. All these datatypes and their associated transformations are implemented in the Coq proof assistant. Furthermore we implement random generators for each representation, using the QuickChick plugin.

Cite as

Catherine Dubois, Nicolas Magaud, and Alain Giorgetti. Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dubois_et_al:LIPIcs.TYPES.2022.11,
  author =	{Dubois, Catherine and Magaud, Nicolas and Giorgetti, Alain},
  title =	{{Pragmatic Isomorphism Proofs Between Coq Representations: Application to Lambda-Term Families}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.11},
  URN =		{urn:nbn:de:0030-drops-184548},
  doi =		{10.4230/LIPIcs.TYPES.2022.11},
  annote =	{Keywords: Data Representations, Isomorphisms, dependent Types, formal Proofs, random Generation, lambda Terms, Coq}
}

Document

DOI: 10.4230/LIPIcs.ITP.2022.25

Proof Pearl: Formalizing Spreads and Packings of the Smallest Projective Space PG(3,2) Using the Coq Proof Assistant

Authors: Nicolas Magaud

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract

We formally implement the smallest three-dimensional projective space PG(3,2) in the Coq proof assistant. This projective space features 15 points and 35 lines, related by an incidence relation. We define points and lines as two plain datatypes (one with 15 constructors for points, and one with 35 constructors for lines) and the incidence relation as a boolean function, instead of using the well-known coordinate-based approach relying on GF(2)⁴. We prove that this implementation actually verifies all the usual properties of three-dimensional projective spaces. We then use an oracle to compute some characteristic subsets of objects of PG(3,2), namely spreads and packings. We formally verify that these computed objects exactly correspond to the spreads and packings of PG(3,2). For spreads, this means identifying 56 specific sets of 5 lines among 360 360 (= 15× 14× 13× 12× 11) possible ones. We then classify them, showing that the 56 spreads of PG(3,2) are all isomorphic whereas the 240 packings of PG(3,2) can be classified into two distinct classes of 120 elements. Proving these results requires partially automating the generation of some large specification files as well as some even larger proof scripts. Overall, this work can be viewed as an example of a large-scale combination of interactive and automated specifications and proofs. It is also a first step towards formalizing projective spaces of higher dimension, e.g. PG(4,2), or larger order, e.g. PG(3,3).

Cite as

Nicolas Magaud. Proof Pearl: Formalizing Spreads and Packings of the Smallest Projective Space PG(3,2) Using the Coq Proof Assistant. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{magaud:LIPIcs.ITP.2022.25,
  author =	{Magaud, Nicolas},
  title =	{{Proof Pearl: Formalizing Spreads and Packings of the Smallest Projective Space PG(3,2) Using the Coq Proof Assistant}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.25},
  URN =		{urn:nbn:de:0030-drops-167349},
  doi =		{10.4230/LIPIcs.ITP.2022.25},
  annote =	{Keywords: Coq, projective geometry, finite models, spreads, packings, PG(3, 2)}
}

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2 Magaud, Nicolas
1 Dubois, Catherine
1 Giorgetti, Alain
1 Grossman, Dan
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3 Theory of computation → Logic and verification
1 Software and its engineering → Compilers
1 Theory of computation → Automated reasoning
1 Theory of computation → Hoare logic
1 Theory of computation → Lambda calculus
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2 Coq
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1 Data Representations
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