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

Abstract

A system of polynomial ordinary differential equations (ode's) is specified via a vector of multivariate polynomials, or vector field, F. A safety assertion ψ −→ [F] φ means that the system's trajectory will lie in a subset φ (the postcondition) of the state-space, whenever the initial state belongs to a subset ψ (the precondition). We consider the case when φ and ψ are algebraic varieties, that is, zero sets of polynomials. In particular, polynomials specifying the postcondition can be seen as conservation laws implied by ψ. 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 ψ, find the smallest algebraic postcondition φ including the variety determined by the valid conservation laws in P (relativized strongest postcondition); (2) given a user specified postcondition φ, find the largest algebraic precondition ψ (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 ψ. The effectiveness of these algorithms is demonstrated on a few case studies from the literature.

Algorithms for exact and approximate linear abstractions of polynomial continuous systems

“…See for instance Ghorbal and Platzer' technique based on differential radical invariants [10], and references therein. More recently, Boreale [7] has introduced a (in a precise sense, complete) method to obtain a small algebraic invariant that includes a given algebraic X 0 . The abstraction method in the present paper, therefore, can be seen as building on all such approaches.…”

“…In order to prove unreachability of a given unsafe set X u , it is therefore sufficient to find an abstraction φ such that X 0 ⊆ V(φ) and V(φ) ∩ X u = ∅. We refer the reader to [23,7] for discussions concerning this aspect. We point out again that our definition of linear abstraction is more flexible than the one considered in [23,24].…”

“…It is easy to prove that A is an invariant variety if and only if there exists an invariant ideal J such that A = V(J); see e.g. [7]. Finally, we say a polynomial p is invariant w.r.t.…”

“…Note that the reduced equation system (12) does not depend on x 0 . Let v ∈ R M represent the coefficients of g with respect to φ, as in (7). We are interested in studying the approximation…”

Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

“…This is important, e.g., in the treatment of hybrid systems (see below). After the short version of the present paper [9] appeared, some preliminary progress towards this goal has been made, see [10].…”