Sampling Polynomial Trajectories for LTL Verification

TL;DR

This paper introduces a polynomial sampling method for verifying continuous-time polynomial trajectories against linear temporal logic specifications, ensuring accurate trace generation for formal verification.

Contribution

It presents a novel polynomial sampling algorithm that guarantees trace completeness for polynomial paths and LTL verification, with polynomial complexity and practical demonstrations.

Findings

The PolyTrace algorithm guarantees trace completeness for polynomial trajectories.

The method ensures verification accuracy by capturing all region transitions.

Numerical examples and a robotics case study validate the approach.

Abstract

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 samples a trajectory strategically, to capture every one of its region transitions. This yields a discrete word called a trace, which is amenable to established formal methods for path checking. The original continuous-time trajectory is shown to satisfy the specification if and only if its 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 examples and a case study from robotics.