Hybrid automata combine finite automata and dynamical systems, and model the interaction of digital with physical systems. Formal analysis that can guarantee the safety of all behaviors or rigorously witness failures, while unsolvable in general, has been tackled algorithmically using, e.g., abstraction, bounded model-checking, assisted theorem proving.
Nevertheless, very few methods have addressed the time-unbounded reachability analysis of hybrid automata and, for current sound and automatic tools, scalability remains critical. We develop methods for the polyhedral abstraction of hybrid automata, which construct coarse overapproximations and tightens them incrementally, in a CEGAR fashion. We use template polyhedra, i.e., polyhedra whose facets are normal to a given set of directions.
While, previously, directions were given by the user, we introduce (1) the first method
for computing template directions from spurious counterexamples, so as to generalize and
eliminate them. The method applies naturally to convex hybrid automata, i.e., hybrid
automata with (possibly non-linear) convex constraints on derivatives only, while for linear
ODE requires further abstraction. Specifically, we introduce (2) the conic abstractions,
which, partitioning the state space into appropriate (possibly non-uniform) cones, divide
curvy trajectories into relatively straight sections, suitable for polyhedral abstractions.
Finally, we introduce (3) space-time interpolation, which, combining interval arithmetic
and template refinement, computes appropriate (possibly non-uniform) time partitioning
and template directions along spurious trajectories, so as to eliminate them.
We obtain sound and automatic methods for the reachability analysis over dense
and unbounded time of convex hybrid automata and hybrid automata with linear ODE.
We build prototype tools and compare—favorably—our methods against the respective
state-of-the-art tools, on several benchmarks
