Complete algorithms for algebraic strongest postconditions and weakest preconditions in polynomial ODEs

A system of polynomial ordinary differential equations (ODEs) is specified via a vector of multivariate polynomials, or vector field, $F$. A safety assertion $\psi\rightarrow[F]\phi$ means that the trajectory of the system 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 a system's conservation laws implied by $\psi$. Checking the validity of algebraic safety assertions is a fundamental problem in, for instance, hybrid systems. We consider a generalized version of this problem, and offer an algorithm that, given a user specified polynomial set $P$ and an algebraic precondition $\psi$, finds the largest subset of polynomials in $P$ implied by $\psi$ (relativized strongest postcondition). Under certain assumptions on $\phi$, this algorithm can also be used to find the largest algebraic invariant included in $\phi$ and the weakest algebraic precondition for $\phi$. Applications to continuous semialgebraic systems are also considered. The effectiveness of the proposed algorithm is demonstrated on several case studies from the literature.