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

  • Khalil Ghorbal
  • Andrew Sogokon
  • André Platzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8931)


This paper presents a theoretical and experimental comparison of sound proof rules for proving invariance of algebraic sets, that is, sets satisfying polynomial equalities, under the flow of polynomial ordinary differential equations. Problems of this nature arise in formal verification of continuous and hybrid dynamical systems, where there is an increasing need for methods to expedite formal proofs. We study the trade-off between proof rule generality and practical performance and evaluate our theoretical observations on a set of heterogeneous benchmarks. The relationship between increased deductive power and running time performance of the proof rules is far from obvious; we discuss and illustrate certain classes of problems where this relationship is interesting.


Singular Locus Polynomial Vector Invariant Equation Differential Invariance Proof Rule 
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  1. 1.
    Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43(6), 1002–1045 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Collins, G.E.: Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  3. 3.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms - an introduction to computational algebraic geometry and commutative algebra, 2nd edn. Springer (1997)Google Scholar
  5. 5.
    Darboux, J.G.: Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. Bulletin des Sciences Mathématiques et Astronomiques 2(1), 151–200 (1878), Google Scholar
  6. 6.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput. 5(1/2), 29–35 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulas over ordered fields. Journal of Symbolic Computation 24, 209–231 (1995)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: ISSAC, pp. 75–83. ACM, New York (2002)Google Scholar
  9. 9.
    Ghorbal, K., Platzer, A.: Characterizing algebraic invariants by differential radical invariants. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 279–294. Springer, Heidelberg (2014)Google Scholar
  10. 10.
    Ghorbal, K., Sogokon, A., Platzer, A.: A hierarchy of proof rules for checking differential invariance of algebraic sets. Tech. Rep. CMU-CS-14-140, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA (November 2014),
  11. 11.
    Ghorbal, K., Sogokon, A., Platzer, A.: Invariance of conjunctions of polynomial equalities for algebraic differential equations. In: Müller-Olm, M., Seidl, H. (eds.) SAS 2014. LNCS, vol. 8723, pp. 151–167. Springer, Heidelberg (2014)Google Scholar
  12. 12.
    Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advanced series in nonlinear dynamics. World Scientific (2001)Google Scholar
  13. 13.
    Lie, S.: Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen. Teubner, Leipzig (1893)Google Scholar
  14. 14.
    Lindelöf, E.: Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre. Comptes rendus hebdomadaires des séances de l’Académie des sciences 116, 454–458 (1894)Google Scholar
  15. 15.
    Liu, J., Zhan, N., Zhao, H.: Computing semi-algebraic invariants for polynomial dynamical systems. In: EMSOFT, pp. 97–106. ACM (2011)Google Scholar
  16. 16.
    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
  17. 17.
    Mayr, E.W.: Membership in polynomial ideals over Q is exponential space complete. In: Cori, R., Monien, B. (eds.) STACS 1989. LNCS, vol. 349, pp. 400–406. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  18. 18.
    Nagumo, M.: Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen. Proceedings of the Physico-Mathematical Society of Japan 24, 551–559 (1942) (in German)Google Scholar
  19. 19.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer (2000)Google Scholar
  20. 20.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reasoning 41(2), 143–189 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Platzer, A.: The complete proof theory of hybrid systems. In: LICS, pp. 541–550. IEEE (2012)Google Scholar
  23. 23.
    Platzer, A.: A differential operator approach to equational differential invariants - (invited paper). In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 28–48. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Platzer, A.: The structure of differential invariants and differential cut elimination. Logical Methods in Computer Science 8(4), 1–38 (2012)Google Scholar
  25. 25.
    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
  26. 26.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. Form. Methods Syst. Des. 32(1), 25–55 (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Taly, A., Tiwari, A.: Deductive verification of continuous dynamical systems. In: FSTTCS. LIPIcs, vol. 4, pp. 383–394 (2009)Google Scholar
  28. 28.
    Tarski, A.: A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc. 59 (1951)Google Scholar
  29. 29.
    Tiwari, A.: Abstractions for hybrid systems. Form. Methods Syst. Des. 32(1), 57–83 (2008)CrossRefzbMATHGoogle Scholar
  30. 30.
    Walter, W.: Ordinary Differential Equations. Springer, New York (1998)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Khalil Ghorbal
    • 1
  • Andrew Sogokon
    • 2
  • André Platzer
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.LFCS, School of InformaticsUniversity of EdinburghEdinburghScotland, UK

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