Complete Algorithms for Algebraic Strongest Postconditions and Weakest Preconditions in Polynomial ODE'S

A system of polynomial ordinary differential equations (ode's) is specified via a vector of multivariate polynomials, or vector field, F. A safety assertion psi -> [F]phi means that the system's trajectory will lie in a subset phi (the postcondition) of the state-space, whenever the initial state belongs to a subset psi(the precondition). We consider the case when phi and psi are algebraic varieties, that is, zero sets of polynomials. In particular, polynomials specifying the postcondition can be seen as conservation laws implied by psi. Checking the validity of algebraic safety assertions is a fundamental problem in, for instance, hybrid systems. We consider generalized versions of this problem, and offer algorithms to: (1) given a user specified polynomial set P and a precondition psi, find the smallest algebraic postcondition phi including the variety determined by the valid conservation laws in P (relativized strongest postcondition); (2) given a user specified postcondition phi, find the largest algebraic precondition psi(weakest precondition). The first algorithm can also be used to find the weakest algebraic invariant of the system implying all conservation laws in P valid under psi. The effectiveness of these algorithms is demonstrated on a challenging case study from the literature.