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

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

Symbolic and analytic techniques for resource analysis of Java bytecode

Recent work in resource analysis has translated the idea of amortised resource analysis to imperative languages using a program logic that allows mixing of assertions about heap shapes, in the tradition of separation logic, and assertions about consumable resources. Separately, polyhedral methods have been used to calculate bounds on numbers of iterations in loop-based programs. We are attempting to combine these ideas to deal with Java programs involving both data structures and loops, focusing on the bytecode level rather than on source code

Proof Generation from Delta-Decisions

We show how to generate and validate logical proofs of unsatisfiability from delta-complete decision procedures that rely on error-prone numerical algorithms. Solving this problem is important for ensuring correctness of the decision procedures. At the same time, it is a new approach for automated theorem proving over real numbers. We design a first-order calculus, and transform the computational steps of constraint solving into logic proofs, which are then validated using proof-checking algorithms. As an application, we demonstrate how proofs generated from our solver can establish many nonlinear lemmas in the the formal proof of the Kepler Conjecture.Comment: Appeared in SYNASC'1