Converse Barrier Certificates for Set-Based Stochastic Reach-Avoid Verification

What the math gives to ML

The paper provides a constructive converse mechanism: for a continuous stochastic discrete-time system, a uniform strict reach-avoid probability margin over a compact initial set implies the existence of a barrier-like certificate satisfying a supermartingale drift condition and boundary inequalities. This is transferable to neural networks as a learned stochastic safety barrier or value certificate, rather than merely as a post-hoc verifier. The key engineering asset is the quantitative uniform margin: training should seek a certificate with slack, and empirical falsification should search for violations of the drift and boundary inequalities. A successful implementation predicts that certificate feasibility changes sharply when the requested probability threshold crosses the worst-case reach-avoid probability over the initial set.