A Computational Approach to Pocklington Certificates in Type Theory

Grégoire, Benjamin; Théry, Laurent; Werner, Benjamin

doi:10.1007/11737414_8

Cited by 18 publications

(15 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The underlying technique for these proofs via computations in Coq is also called proof by reflection (cf. [1,15,19,20] on this). In our case we may formalize a computational evaluator E in an executable way that checks whether the property formalized by D holds.…”

Section: Using Computational Evaluators To Speed Up the Proving Processmentioning

confidence: 87%

Certifying compilers using higher-order theorem provers as certificate checkers

Blech

Grégoire

2010

Form Methods Syst Des

Self Cite

View full text Add to dashboard Cite

Correct software requires compilers to work correctly. Especially code generation can be an error prone task, since it potentially uses sophisticated algorithms to produce efficient code.In this paper we present an approach to guarantee the correctness of compiler transformations with respect to a formal notion of correctness. We certify the results of each compilation run. With the help of a compiler generated certificate and a certificate checker, we verify the results of each compilation run automatically. Thereby we ensure the correctness of the compilation run without having to look at concrete compilation algorithms.We use higher-order theorem provers to check the certificates and to formally define syntax, and semantics of the involved languages as well as a criterion under which we regard a compilation as correct. The use of higher-order theorem provers ensures a small and well understood trusted computing base. The task of efficient certificate checking is especially crucial for the acceptance of certifying compilation. We present methods to facilitate this task, most notably by using computational reflection: We present small-in an executable way specified-evaluators that solve certain properties appearing in our certificates and are used to speed up certain subtasks in the checking process.We discuss an implemented prototype performing code generation. Using Coq and Isabelle/HOL as certificate checkers we highlight typical challenges and their solutions

show abstract

Section: Using Computational Evaluators To Speed Up the Proving Processmentioning

confidence: 87%

Certifying compilers using higher-order theorem provers as certificate checkers

Blech

Grégoire

2010

Form Methods Syst Des

Self Cite

View full text Add to dashboard Cite

show abstract

“…But also the conversion rule allows the computation steps not to appear in the proof; for instance 2 + 2 = 4 is simply proved by one reflexivity step, since this proposition is identified with 4 = 4 by conversion. In some cases this can lead to a dramatic space gain, using the result of certified computations inside a proof; spectacular recent applications include the formal proof of the four-color theorem [15] or formal primality proofs [18]. (3) Finally, type theories are naturally constructive.…”

Section: Introductionmentioning

confidence: 99%

On the strength of proof-irrelevant type theories

Werner¹

2008

Logical Methods in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.

show abstract

“…First, all our computations are done inside COQ with user-defined data-structures, i.e. we use the modular arithmetic defined in [10] based on a datatype composed of 256 constructors that simulates an 8-bit arithmetic. We would need a direct access to the machine 32-bit arithmetic instead.…”

Section: Some Benchmarksmentioning

confidence: 99%

“…We would need a direct access to the machine 32-bit arithmetic instead. In [10], some tests for Pocklington certificates with a native 32-bits arithmetic exhibit a speed-up of 80. Such speed-up would make it possible, in our case, to verify a 250-digit number in less than a minute.…”

Section: Some Benchmarksmentioning

confidence: 99%

See 1 more Smart Citation

Primality Proving with Elliptic Curves

Théry

Hanrot

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Elliptic curves are fascinating mathematical objects. In this paper, we present the way they have been represented inside the COQ system, and how we have proved that the classical composition law on the points is internal and gives them a group structure. We then describe how having elliptic curves inside a prover makes it possible to derive a checker for proving the primality of natural numbers.Key-words: elliptic curve, formalising mathematics, primality proving Prouver la primalité avec des courbes elliptiques Résumé : Les courbes elliptiques sont des objets mathématiques fascinants. Dans ce travail, nous présentons la façon dont elles ont été représentées dans le système COQ, et comment nous avons prouvé le fait que la loi de composition classique est bien interne et munit l'ensemble des points de la courbe d'une structure de groupe. Nous décrivons alors comment cela permet d'obtenir un vérificateur permettant de prouver la primalité de nombres naturels.

show abstract

A Computational Approach to Pocklington Certificates in Type Theory

Cited by 18 publications

References 6 publications

Certifying compilers using higher-order theorem provers as certificate checkers

Certifying compilers using higher-order theorem provers as certificate checkers

On the strength of proof-irrelevant type theories

Primality Proving with Elliptic Curves

Contact Info

Product

Resources

About