Lecture Notes on Differential Invariants & Proof Theory

Lecture 10 on Differential Equations & Differential Invariants and Lecture 11 on Differential Equations & Proofs equipped us with powerful tools for proving properties of differential equations without having to solve them. Differential invariants (DI) [Pla10a] prove properties of differential equations by induction based on the right-hand side of the differential equation, rather than its much more complicated global solution. Differential cuts (DC) [Pla10a] made it possible to prove another property C of a differential equation and then change the dynamics of the system around so that it can never leave region C. Differential cuts turned out to be very useful when stacking inductive properties of differential equations on top of each other, so that easier properties are proved first and then assumed during the proof of the more complicated properties. Differential weakening (DW) [Pla10a] proves simple properties that are entailed by the evolution domain, which becomes especially useful after the evolution domain constraint has been augmented sufficiently by way of a differential cut. Just like in the case of loops, where the search for invariants is nontrivial, differential invariants also require some smarts (or good automatic procedures) to be found. Once a differential invariant has been identified, the proof follows easily, which is a computationally attractive property. Finding invariants of loops is very challenging. It can be shown to be the only fundamental challenge in proving safety properties of conventional discrete programs [HMP77]. Likewise, finding invariants and differential invariants is the only fundamental challenge in proving safety properties of hybrid systems [Pla08, Pla10b, Pla12a]. A more careful analysis even shows that just finding differential invariants is the only fundamental challenge for hybrid systems safety verification [Pla12a].