Combining Zonotope Abstraction and Constraint Programming for Synthesizing Inductive Invariants

Abstract

We propose to extend an existing framework combining abstract interpretation and continuous constraint programming for numerical invariant synthesis, by using more expressive underlying abstract domains, such as zonotopes. The original method, which relies on iterative refinement, splitting and tightening a collection of abstract elements until reaching an inductive set, was initially presented in combination with simple underlying abstract elements: boxes and octagons. The novelty of our work is to use zonotopes, a sub-polyhedric domain that shows a good compromise between cost and precision. As zonotopes are not closed under intersection, we had to extend the existing framework, in addition to designing new operations on zonotopes, such as a novel splitting algorithm based on paving zonotopes by sub-zonotopes and parallelotopes. We implemented this method on top of the APRON library, and tested it on programs with non-linear loops that present complex, possibly non-convex, invariants. We present some results demonstrating both the interest of this splitting-based algorithm to synthesize invariants on such programs, and the good compromise presented by its use in combination with zonotopes with respect to its use with both simpler domains such as boxes and octagons, and more expressive domains such as polyhedra.

Zonotope-based Controller Synthesis for LTL Specifications

“…Remark 5: Besides the proposed approach, other approaches like tiling and zonotopal subdivision techniques [41], [42] can be applied to the approximation of zonotopes. These techniques take centrally symmetric advantages of zonotopes to partition zonotopes into sub-zonotopes.…”