ITP 2012: Interactive Theorem Proving pp 28-48

# A Differential Operator Approach to Equational Differential Invariants

(Invited Paper)
• André Platzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7406)

## Abstract

Hybrid systems, i.e., dynamical systems combining discrete and continuous dynamics, have a complete axiomatization in differential dynamic logic relative to differential equations. Differential invariants are a natural induction principle for proving properties of the remaining differential equations. We study the equational case of differential invariants using a differential operator view. We relate differential invariants to Lie’s seminal work and explain important structural properties resulting from this view. Finally, we study the connection of differential invariants with partial differential equations in the context of the inverse characteristic method for computing differential invariants.

## Keywords

Hybrid System Invariant Function Dynamic Logic Invariant Equation Differential Invariant
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(1), 3–34 (1995)
2. 2.
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)
3. 3.
Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer (1992)Google Scholar
4. 4.
Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput. 5(1/2), 29–35 (1988)
5. 5.
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 2nd edn., vol. 19. AMS (2010)Google Scholar
6. 6.
Gentzen, G.: Untersuchungen über das logische Schließen. II. Math. Zeit. 39(3), 405–431 (1935)
7. 7.
Grigor’ev, D.Y.: Complexity of Quantifier Elimination in the Theory of Ordinary Differential Equations. In: Davenport, J.H. (ed.) ISSAC 1987 and EUROCAL 1987. LNCS, vol. 378, pp. 11–25. Springer, Heidelberg (1989)
8. 8.
Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Gupta, Malik [9], pp. 190–203Google Scholar
9. 9.
Gupta, A., Malik, S. (eds.): CAV 2008. LNCS, vol. 5123. Springer, Heidelberg (2008)
10. 10.
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer (1977)Google Scholar
11. 11.
Henzinger, T.A.: The theory of hybrid automata. In: LICS, pp. 278–292. IEEE Computer Society, Los Alamitos (1996)Google Scholar
12. 12.
Hilbert, D.: Über die Theorie der algebraischen Formen. Math. Ann. 36(4), 473–534 (1890)
13. 13.
Proceedings of the 27th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2012, Dubrovnik, Croatia. IEEE Computer Society (2012)Google Scholar
14. 14.
Lie, S.: Über Differentialinvarianten, vol. 6. Teubner (1884); English Translation: Mr. Ackerman (1975); Sophus Lie’s 1884 Differential Invariant Paper. Math. Sci. Press, Brookline, Mass.: (1884)Google Scholar
15. 15.
Lie, S.: Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen. Teubner, Leipzig (1893)
16. 16.
Lie, S.: Über Integralinvarianten und ihre Verwertung für die Theorie der Differentialgleichungen. Leipz. Berichte 49, 369–410 (1897)Google Scholar
17. 17.
Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer (1993)Google Scholar
18. 18.
Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008)
19. 19.
Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)
20. 20.
Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer, Heidelberg (2010)
21. 21.
Platzer, A.: The complete proof theory of hybrid systems. In: LICS [13]Google Scholar
22. 22.
Platzer, A.: Logics of dynamical systems (invited tutorial). In: LICS [13]Google Scholar
23. 23.
Platzer, A.: The structure of differential invariants and differential cut elimination. In: Logical Methods in Computer Science (to appear, 2012)Google Scholar
24. 24.
Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. In: Gupta, Malik [9], pp. 176–189Google Scholar
25. 25.
Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. CAV 2008 35(1), 98–120 (2009); Special issue for selected papers from CAV 2008
26. 26.
Platzer, A., Clarke, E.M.: Formal Verification of Curved Flight Collision Avoidance Maneuvers: A Case Study. In: Cavalcanti, A., Dams, D.R. (eds.) FM 2009. LNCS, vol. 5850, pp. 547–562. Springer, Heidelberg (2009)
27. 27.
Platzer, A., Quesel, J.-D., Rümmer, P.: Real World Verification. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 485–501. Springer, Heidelberg (2009)
28. 28.
Prajna, S., Jadbabaie, A.: Safety Verification of Hybrid Systems Using Barrier Certificates. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 477–492. Springer, Heidelberg (2004)
29. 29.
Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE T. Automat. Contr. 52(8), 1415–1429 (2007)
30. 30.
Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. Form. Methods Syst. Des. 32(1), 25–55 (2008)
31. 31.
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)
32. 32.
Zeidler, E. (ed.): Teubner-Taschenbuch der Mathematik. Teubner (2003)Google Scholar