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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References

  1. 1.
    Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. A series of modern surveys in mathematics. Springer (2010)Google Scholar
  2. 2.
    Buss, J.F., Frandsen, G.S., Shallit, J.: The computational complexity of some problems of linear algebra. J. Comput. Syst. Sci. 58(3), 572–596 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer (2007)Google Scholar
  4. 4.
    Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. American Journal of Mathematics 79(3), 497–516 (1957)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Ghorbal, K., Platzer, A.: Characterizing algebraic invariants by differential radical invariants. Tech. Rep. CMU-CS-13-129, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213 (November 2013), http://reports-archive.adm.cs.cmu.edu/anon/2013/abstracts/13-129.html
  6. 6.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics. Springer (1977)Google Scholar
  7. 7.
    Hilbert, D.: Über die Theorie der algebraischen Formen. Mathematische Annalen 36(4), 473–534 (1890)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Hillar, C.J., Lim, L.H.: Most tensor problems are NP-hard. J. ACM 60(6), 45 (2013)CrossRefGoogle Scholar
  9. 9.
    Lafferriere, G., Pappas, G.J., Yovine, S.: Symbolic reachability computation for families of linear vector fields. J. Symb. Comput. 32(3), 231–253 (2001)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lanotte, R., Tini, S.: Taylor approximation for hybrid systems. In: Morari, Thiele (eds.) [13], pp. 402–416Google Scholar
  11. 11.
    Liu, J., Zhan, N., Zhao, H.: Computing semi-algebraic invariants for polynomial dynamical systems. In: Chakraborty, S., Jerraya, A., Baruah, S.K., Fischmeister, S. (eds.) EMSOFT, pp. 97–106. ACM (2011)Google Scholar
  12. 12.
    Matringe, N., Moura, A.V., Rebiha, R.: Generating invariants for non-linear hybrid systems by linear algebraic methods. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 373–389. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Morari, M., Thiele, L. (eds.): HSCC 2005. LNCS, vol. 3414. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Neuhaus, R.: Computation of real radicals of polynomial ideals II. Journal of Pure and Applied Algebra 124(13), 261–280 (1998)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reasoning 41(2), 143–189 (2008)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Platzer, A.: A differential operator approach to equational differential invariants - (invited paper). In: Beringer, L., Felty, A.P. (eds.) ITP. LNCS, vol. 7406, pp. 28–48. Springer (2012)CrossRefGoogle Scholar
  18. 18.
    Platzer, A.: Logics of dynamical systems. In: LICS, pp. 13–24. IEEE (2012)Google Scholar
  19. 19.
    Platzer, A.: The structure of differential invariants and differential cut elimination. Logical Methods in Computer Science 8(4), 1–38 (2012)Google Scholar
  20. 20.
    Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 176–189. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Rodríguez-Carbonell, E., Kapur, D.: An abstract interpretation approach for automatic generation of polynomial invariants. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 280–295. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  22. 22.
    Rodríguez-Carbonell, E., Tiwari, A.: Generating polynomial invariants for hybrid systems. In: Morari, Thiele (eds.) [13], pp. 590–605Google Scholar
  23. 23.
    Sankaranarayanan, S.: Automatic invariant generation for hybrid systems using ideal fixed points. In: Johansson, K.H., Yi, W. (eds.) HSCC, pp. 221–230. ACM (2010)Google Scholar
  24. 24.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. Formal Methods in System Design 32(1), 25–55 (2008)CrossRefMATHGoogle Scholar
  25. 25.
    Stengel, R.F.: Flight Dynamics. Princeton University Press (2004)Google Scholar
  26. 26.
    Tiwari, A.: Approximate reachability for linear systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 514–525. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  27. 27.
    Tiwari, A.: Abstractions for hybrid systems. Formal Methods in System Design 32(1), 57–83 (2008)CrossRefMATHGoogle Scholar
  28. 28.
    Tiwari, A., Khanna, G.: Nonlinear systems: Approximating reach sets. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 600–614. Springer, Heidelberg (2004)CrossRefGoogle Scholar

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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