Characterizing Algebraic Invariants by Differential Radical Invariants

  • Khalil Ghorbal
  • André Platzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8413)

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.

Keywords

invariant algebraic sets polynomial vector fields real algebraic geometry Zariski topology higher-order Lie derivation automated generation and checking symbolic linear algebra rank minimization formal verification 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Khalil Ghorbal
    • 1
  • André Platzer
    • 1
  1. 1.Carnegie Mellon UniversityPittsburghUSA

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