A Logical Product Approach to Zonotope Intersection

TL;DR

This paper introduces a new abstract domain combining zonotopes with other polyhedral domains, enabling efficient and precise interpretation of non-linear computations and intersections, addressing a key limitation of traditional zonotopic methods.

Contribution

It proposes a logical product approach that allows accurate and efficient intersection operations in zonotope-based abstract domains, enhancing reachability analysis and invariant generation.

Findings

Efficient interpretation of tests and intersections in combined domains.

Implementation in the APRON library demonstrating practical applicability.

Addresses a major limitation of zonotopic methods in hybrid systems analysis.

Abstract

We define and study a new abstract domain which is a fine-grained combination of zonotopes with polyhedric domains such as the interval, octagon, linear templates or polyhedron domain. While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e. intersections) efficiently. This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid sys- tems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic over-approximation of the intersection. We describe some examples and an im- plementation of our method in the APRON library, and discuss some further in- teresting combinations of zonotopes with non-linear or non-convex domains such as quadratic templates and maxplus polyhedra.