Sampling Polynomial Trajectories for LTL Verification

This paper concerns the verification of continuous-time polynomial spline trajectories against linear temporal logic specifications (LTL without 'next'). Each atomic proposition is assumed to represent a state space region described by a multivariate polynomial inequality. The proposed approach is based on sampling the trajectories in a manner that captures all region transitions, to yield a discrete word called a trace, which is amenable to established formal methods for path checking. The continuous-time trajectory is shown to satisfy the specification if and only if the trace does. General topological conditions on the sample points are derived that ensure a trace is recorded for arbitrary continuous paths, given arbitrary region descriptions. Using techniques from computer algebra, a trace generation algorithm is developed to satisfy these conditions when the path and region boundaries are defined by polynomials. The proposed PolyTrace algorithm has polynomial complexity in the number of atomic propositions, and is guaranteed to produce a trace of any polynomial path. Its performance is demonstrated via numerical simulations.