Abstract
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.
This material is based upon work supported by the National Science Foundation by NSF CAREER Award CNS-1054246, NSF EXPEDITION CNS-0926181 and grant no. CNS-0931985. This research is also partially supported by the Defense Advanced Research Agency under contract no. DARPA FA8750-12-2-0291.
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Ghorbal, K., Platzer, A. (2014). Characterizing Algebraic Invariants by Differential Radical Invariants. In: Ábrahám, E., Havelund, K. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2014. Lecture Notes in Computer Science, vol 8413. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54862-8_19
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