A Verified Algorithm for Geometric Zonotope/Hyperplane Intersection

Abstract

To perform rigorous numerical computations, one can use a generalization of interval arithmetic, namely affine arithmetic (AA), which works with zonotopes instead of intervals. Zonotopes are also widely used for reachability analysis of continuous or hybrid systems, where an important operation is the geometric intersection of zonotopes with hyperplanes. We have implemented a functional algorithm to compute the zonotope/hyperplane intersection and verified it in Isabelle/HOL. The algorithm is similar to convex hull computations, our verification is therefore inspired by Knuth's axioms for an orientation predicate of points in the plane, which have been successfully used to verify convex hull algorithms. The interesting fact is that we combine a mixture of different fields: a discrete geometrical algorithm to perform operations on the continuous sets represented by zonotopes.