Bounded Verification with On-the-Fly Discrepancy Computation

Abstract

Simulation-based verification algorithms can provide formal safety guarantees for nonlinear and hybrid systems. The previous algorithms rely on user provided model annotations called discrepancy function, which are crucial for computing reachtubes from simulations. In this paper, we eliminate this requirement by presenting an algorithm for computing piece-wise exponential discrepancy functions. The algorithm relies on computing local convergence or divergence rates of trajectories along a simulation using a coarse over-approximation of the reach set and bounding the maximal eigenvalue of the Jacobian over this over-approximation. The resulting discrepancy function preserves the soundness and the relative completeness of the verification algorithm. We also provide a coordinate transformation method to improve the local estimates for the convergence or divergence rates in practical examples. We extend the method to get the input-to-state discrepancy of nonlinear dynamical systems which can be used for compositional analysis. Our experiments show that the approach is effective in terms of running time for several benchmark problems, scales reasonably to larger dimensional systems, and compares favorably with respect to available tools for nonlinear models.

Multi-agent Safety Verification Using Symmetry Transformations

“…Computing reachtubes is computationally expensive. For example, one of the approaches for such computation entails simulating the system using an ODE solver, solving nontrivial optimization problems numerically to compute sensitivity functions, and computing the Minkowski sum of the solution with the sensitivity function [13,15,16]. Another approach entails Taylor approximations of the dynamics, integrations, and optimizations of the time discretization of the solution [8,6].…”

Progress on Powertrain Verification Challenge with C2E2

“…One of the main developments that enabled this verification, is the implementation of a new algorithm in C2E2 (presented in detail in [9]) for automatic computation of local discrepancy along trajectories of the system. Using this improved C2E2, we were not only able to find counterexamples, but also verify the key STL requirements of the powertrain benchmark in the order of minutes.…”

“…An important technical detail that makes the implementation scale is the coordinate transformation proposed in [9]. For Jacobian matrices with complex eigenvalues the local discrepancy computed directly using the above algorithm can be a positive exponential even though the actual trajectories are not diverging.…”

“…Then it computes an exponential (possibly negative) bound on the divergence rate of trajectories over S(x 0 ) by finding a bound on the maximum eigenvalue of the symmetric part of the Jacobian J f over the region S(x 0 ). We refer the reader to the technical report [9] for the details of this algorithm.…”

“…In this paper, we use the algorithm presented in [9] for computing local discrepancy functions on-the-fly along validated simulations. This algorithm requires the Jacobian J f and a Lipschitz constant L f of the ODE.…”