Abstract Interpretation with Higher-Dimensional Ellipsoids and Conic Extrapolation

Interpretation with Higher-Dimensional Ellipsoids and Conic Extrapolation ∗ Mendes Oulamara† École Normale Supérieure 45 rue d’Ulm 75005 Paris, France [email protected] Arnaud J. Venet Carnegie Mellon University NASA Ames Research Center Moffett Field, CA 94035 [email protected] CAV 2015, 18-24 July 2015 Abstract The inference and the verification of numerical relationships among variables of a program is one of the main goals of static analysis. In this paper, we propose an Abstract Interpretation framework based on higherdimensional ellipsoids to automatically discover symbolic quadratic invariants within loops, using loop counters as implicit parameters. In order to obtain non-trivial invariants, the diameter of the set of values taken by the numerical variables of the program has to evolve (sub-)linearly during loop iterations. These invariants are called ellipsoidal cones and can be seen as an extension of constructs used in the static analysis of digital filters. Semidefinite programming is used to both compute the numerical results of the domain operations and provide proofs (witnesses) of their correctness. keywords: static analysis, semidefinite programming, ellipsoids, conic extrapolationThe inference and the verification of numerical relationships among variables of a program is one of the main goals of static analysis. In this paper, we propose an Abstract Interpretation framework based on higherdimensional ellipsoids to automatically discover symbolic quadratic invariants within loops, using loop counters as implicit parameters. In order to obtain non-trivial invariants, the diameter of the set of values taken by the numerical variables of the program has to evolve (sub-)linearly during loop iterations. These invariants are called ellipsoidal cones and can be seen as an extension of constructs used in the static analysis of digital filters. Semidefinite programming is used to both compute the numerical results of the domain operations and provide proofs (witnesses) of their correctness. keywords: static analysis, semidefinite programming, ellipsoids, conic extrapolation