Abstract Interpretation with Higher-Dimensional Ellipsoids and Conic Extrapolation

TL;DR

This paper introduces a static analysis framework using higher-dimensional ellipsoids to automatically discover quadratic invariants in programs, leveraging semidefinite programming for computation and proof validation.

Contribution

It extends abstract interpretation with ellipsoidal cones to identify symbolic quadratic invariants, incorporating conic extrapolation and semidefinite programming techniques.

Findings

Successfully discovers quadratic invariants in loops

Uses semidefinite programming for domain operations and proofs

Extends static analysis of digital filters with higher-dimensional ellipsoids

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 higher-dimensional 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.