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Communication Dans Un Congrès Année : 2014

Characterizing Algebraic Invariants by Differential Radical Invariants

Résumé

We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion.
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Dates et versions

hal-01660904 , version 1 (11-12-2017)

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Khalil Ghorbal, André Platzer. Characterizing Algebraic Invariants by Differential Radical Invariants. Tools and Algorithms for the Construction and Analysis of Systems - 20th International Conference, TACAS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2014, Grenoble, France, April 5-13, 2014. Proceedings, 2014, Grenoble, France. pp.279--294, ⟨10.1007/978-3-642-54862-8_19⟩. ⟨hal-01660904⟩

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