Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

Abstract

In this paper we seek to provide greater automation for formal deductive verification tools working with continuous and hybrid dynamical systems. We present an efficient procedure to check invariance of conjunctions of polynomial equalities under the flow of polynomial ordinary differential equations. The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities. We contrast this approach to an alternative one which combines fast and sufficient (but not necessary) conditions using differential cuts for soundly restricting the system evolution domain.

Differential Elimination and Algebraic Invariants of Polynomial Dynamical Systems

“…It may be possible to generalize this to the differential case and prove Lemma 94 while replacing Triangulate with a method of singly exponential complexity. The results of [100] and [91] imply that the invariant yielded by RGA o can be obtained by other techniques with doubly exponential complexity (Section 5.2). This further supports the idea that some modification of Triangulate might dispense with the nonelementary bounds.…”

“…In [91] Ghorbal, Sogokon, and Platzer give an invariant checking algorithm that is related to LZZ but is restricted to algebraic sets and improves efficiency in that case. Like LZZ, the method in [91] uses Gröbner bases and nonlinear real arithmetic.…”

“…In [91] Ghorbal, Sogokon, and Platzer give an invariant checking algorithm that is related to LZZ but is restricted to algebraic sets and improves efficiency in that case. Like LZZ, the method in [91] uses Gröbner bases and nonlinear real arithmetic. The algorithm computes successive Lie derivatives of the input A ⊆ R[x] and checks ideal membership (where Gröbner bases come in) until the first-order Lie derivatives of all preceding elements belong to the ideal generated by those elements.…”

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

“…In the examples featuring singularities and high integrals in the benchmarks we see DRI as the clear winner, simply because there was no other rule that was tailored to work on this class. Indeed, the structure of these invariant sets can be rather involved, making it difficult to characterize in a single proof rule; however, sometimes it is pos-sible to exploit the structure of high integrals inside a proof system and arrive at very efficient proofs that outperform DRI [11].…”

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

“…In the examples featuring singularities and high integrals in the benchmarks we see DRI as the clear winner, simply because there was no other rule that was tailored to work on this class. Indeed, the structure of these invariant sets can be rather involved, making it difficult to characterize in a single proof rule; however, sometimes it is possible to exploit the structure of high integrals inside a proof system and arrive at very efficient proofs that outperform DRI [12].…”