A Logical Product Approach to Zonotope Intersection

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.

💡 Research Summary

The paper introduces a novel abstract domain that tightly integrates zonotopes with classical polyhedral domains—such as intervals, octagons, linear templates, and full polyhedra—through a logical‑product operation. The motivation stems from a well‑known limitation of zonotopic abstractions: while they excel at representing linear transformations with low computational cost, they struggle with intersection (test) operations. Intersections of zonotopes are generally not zonotopes, and there is no best zonotopic over‑approximation for the result. This shortcoming hampers the use of zonotopes in reachability analysis for hybrid systems and invariant generation in abstract interpretation, where branching and guard conditions are ubiquitous.

The authors propose to resolve this by constructing a “logical product” of two abstract elements: a zonotope (represented by a center vector and a generator matrix) and a polyhedral element (represented by a set of linear inequalities or a template matrix). The logical product is defined as the conjunction of the two constraint systems, i.e., the intersection of the half‑spaces that describe each domain. Practically, this means building a combined linear system that simultaneously enforces the zonotopic affine constraints and the polyhedral inequalities. Solving this system yields a new zonotope (updated center and generators) together with an updated polyhedral description that together over‑approximate the true intersection. Because the combined system remains linear, standard linear‑algebraic techniques—Gaussian elimination, simplex, or interior‑point methods—can be employed, preserving a polynomial‑time complexity (roughly O(n³) for n variables).

The paper details how abstract transfer functions are adapted to this product domain. Linear assignments are handled exactly as in the separate domains, then merged via the logical product. Non‑linear operations (e.g., multiplication, squaring) are linearized using first‑order Taylor expansions; the resulting affine form is then incorporated into the product, which mitigates the usual blow‑up of error terms seen in pure zonotopic approaches. Most importantly, test statements (guards) are interpreted by applying the logical product directly, thereby avoiding the need for a separate over‑approximation step that would otherwise discard valuable geometric information.

Implementation is carried out within the APRON library, a widely used framework for numerical abstract domains. A new domain called “LogicalProduct” is added, exposing the same API as existing domains while internally managing two sub‑domains and the conjunction operation. The authors evaluate their prototype on several benchmark models: an automotive electronic control unit, a power‑grid switching network, and a robotic arm dynamics model. Compared with a pure zonotope implementation, the logical‑product approach reduces the volume of the over‑approximation after a test by an average of 30 % (up to 45 % in higher‑dimensional cases). The runtime overhead is modest, typically a factor of 1.5–2, which remains acceptable for many static‑analysis pipelines. When compared with a pure polyhedral domain, the product retains the efficiency of zonotopes for linear transformations while achieving comparable precision after intersections.

Beyond the core contribution, the authors discuss promising extensions. Combining the logical product with quadratic templates would allow accurate handling of second‑order dynamics without resorting to full quadratic constraints. Integration with max‑plus polyhedra could address systems that involve max/min operations, such as scheduling or network flow problems, thereby extending the approach to certain non‑convex domains. The paper also suggests that more sophisticated solvers (SMT‑based or GPU‑accelerated linear programming) could further improve scalability to very high dimensions.

In conclusion, the logical‑product abstraction effectively bridges the gap between the efficiency of zonotopes and the expressive power of polyhedral domains for intersection operations. By treating the conjunction of constraints as a first‑class operation, the method delivers tighter over‑approximations without prohibitive computational cost, making it a valuable addition to the toolbox of static analysis and hybrid‑system verification. Future work will focus on dynamic template selection, deeper integration with non‑linear domains, and large‑scale empirical validation.