Extending Coq with Imperative Features and its Application to SAT Verification

This work was supported in part by the french ANR DECERT initiativeInternational audienceCoq has within its logic a programming language that can be used to replace many deduction steps into a single computation, this is the so-called reflection. In this paper, we present two extensions of the evaluation mechanism that preserve its correctness and make it possible to deal with cpu-intensive tasks such as proof checking of SAT traces

Formal Proofs for Nonlinear Optimization

We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of $f$ over $K$. This method allows to bound in a modular way some of the constituents of $f$ by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table

Type classes for efficient exact real arithmetic in Coq

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275

Implementing and reasoning about hash-consed data structures in Coq

We report on four different approaches to implementing hash-consing in Coq programs. The use cases include execution inside Coq, or execution of the extracted OCaml code. We explore the different trade-offs between faithful use of pristine extracted code, and code that is fine-tuned to make use of OCaml programming constructs not available in Coq. We discuss the possible consequences in terms of performances and guarantees. We use the running example of binary decision diagrams and then demonstrate the generality of our solutions by applying them to other examples of hash-consed data structures

Décimales distantes de ππ: Preuves formelles de certains algorithmes pour les calculer et des garanties d'exactitude

International audienceWe describe how to compute very far decimals of $π$ and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of $π$ and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way. Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic-geometric means. All proofs and computations can be made inside the Coq system. We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation

Verification of program computations

Formal verification of complex algorithms is challenging. Verifying their implementations in reasonable time is infeasible using current verification tools and usually involves intricate mathematical theorems. Certifying algorithms compute in addition to each output a witness certifying that the output is correct. A checker for such a witness is usually much simpler than the original algorithm -- yet it is all the user has to trust. The verification of checkers is feasible with current tools and leads to computations that can be completely trusted. We describe a framework to seamlessly verify certifying computations. We demonstrate the effectiveness of our approach by presenting the verification of typical examples of the industrial-level and widespread algorithmic library LEDA. We present and compare two alternative methods for verifying the C implementation of the checkers. Moreover, we present work that was done during an internship at NICTA, Australia\u27s Information and Communications Technology Research Centre of Excellence. This work contributes to building a feasible framework for verifying efficient file systems code. As opposed to the algorithmic problems we address in this thesis, file systems code is mostly straightforward and hence a good candidate for automation.Die formale Verifikation der Implementierung komplexer Algorithmen ist schwierig. Sie übersteigt die Möglichkeiten der heutigen Verifikationswerkzeuge und erfordert für gewöhnlich komplexe mathematische Theoreme. Zertifizierende Algorithmen berechnen zu jeder Ausgabe ein Zerfitikat, das die Korrektheit der Antwort bestätigt. Ein Checker für ein solches Zertifikat ist normalerweise ein viel einfacheres Programm und doch muss ein Nutzer nur dem Checker vertrauen. Die Verifizierung von Checkern ist mit den heutigen Werkzeugen möglich und führt zu Berechnungen, denen völlig vertraut werden kann. Wir beschreiben eine Rahmenstruktur zur Verifikation zertifizierender Berechnungen und demonstrieren die Effektivität unseres Ansatzes an Hand typischer Beispiele aus der hochqualitätiven und oft eingesetzten LEDA Algorithmenbibliothek. We präsentieren und bewerten zwei alternative Methoden zur Verifikation von Checkerimplementierungen in C. Desweiteren beschreiben wir Ergebnisse, die während eines Praktikums am NICTA, dem Australischen Forschungszentrum für Informations- und Kommunikationstechnik, erzielt wurden. Diese Arbeit trägt zum Aufbau einer praktisch einsetzbaren Rahmenstruktur zur Verifizierung von Code für effiziente Dateisysteme bei. Im Gegensatz zu den algorithmischen Problemen, die wir in dieser Arbeiten behandeln, ist der Code für Dateisysteme weitgehend unkompliziert unddaher ein guter Kandidat zur Automatisierung