Abstract
Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.
Change history
15 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s10817-021-09608-w
References
Ábrahám-Mumm, E., Steffen, M., Hannemann, U.: Verification of hybrid systems: formalization and proof rules in PVS. In: ICECCS, pp. 48–57. IEEE Computer Society, Los Alamitos (2001). doi:10.1109/ICECCS.2001.930163
Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking for real-time systems. In: LICS, pp. 414–425. IEEE Computer Society, Los Alamitos (1990)
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). doi:10.1016/0304-3975(94)00202-T
Anai, H., Weispfenning, V.: Reach set computations using real quantifier elimination. In: Benedetto, M.D.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC, LNCS, vol. 2034, pp. 63–76. Springer, Berlin (2001). doi:10.1007/3-540-45351-2_9
Asarin, E., Dang, T., Girard, A.: Reachability analysis of nonlinear systems using conservative approximation. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control, 6th International Workshop, HSCC 2003 Prague, Czech Republic, April 3–5, 2003, Proceedings, LNCS, vol. 2623, pp. 20–35. Springer, Berlin (2003). doi:10.1007/3-540-36580-X_5
Beckert, B.: Equality and other theories. In: D’Agostino, M., Gabbay, D., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods. Kluwer, Deventer (1999)
Beckert, B., Hähnle, R., Schmitt, P.H. (eds.): Verification of Object-Oriented Software: The KeY Approach, LNCS, vol. 4334. Springer, Berlin (2007)
Beckert, B., Platzer, A.: Dynamic logic with non-rigid functions: a basis for object-oriented program verification. In: Furbach, U., Shankar, N. (eds.) IJCAR, LNCS, vol. 4130, pp. 266–280. Springer, Berlin (2006)
Branicky, M.S.: Studies in hybrid systems: modeling, analysis, and control. Ph.D. thesis, Dept. Elec. Eng. and Computer Sci., Massachusetts Inst. Technol., Cambridge, MA (1995)
Branicky, M.S.: Universal computation and other capabilities of hybrid and continuous dynamical systems. Theor. Comput. Sci. 138(1), 67–100 (1995). doi:10.1016/0304-3975(94)00147-B
Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Automat. Contr. 43(1), 31–45 (1998). doi:10.1109/9.654885
Chaochen, Z., Ji, W., Ravn, A.P.: A formal description of hybrid systems. In: Alur, R., Henzinger, T.A., Sontag, E.D. (eds.) Hybrid Systems, LNCS, vol. 1066, pp. 511–530. Springer, Berlin (1995)
Chutinan, A., Krogh, B.H.: Computational techniques for hybrid system verification. IEEE Trans. Automat. Contr. 48(1), 64–75 (2003). doi:10.1109/TAC.2002.806655
Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int. J. Found. Comput. Sci. 14(4), 583–604 (2003)
Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT, Cambridge (1999)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)
Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM J. Comput. 7(1), 70–90 (1978). doi:10.1137/0207005
Damm, W., Hungar, H., Olderog, E.R.: Verification of cooperating travel agents. Int. J. Control 79(5), 395–421 (2006)
Damm, W., Mikschl, A., Oehlerking, J., Olderog, E.R., Pang, J., Platzer, A., Segelken, M., Wirtz, B.: Automating verification of cooperation, control, and design in traffic applications. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) Formal Methods and Hybrid Real-Time Systems, LNCS, vol. 4700, pp. 115–169. Springer, Berlin (2007)
Davoren, J.M.: On hybrid systems and the modal μ-calculus. In: Antsaklis, P.J., Kohn, W., Lemmon, M.D., Nerode, A., Sastry, S. (eds.) Hybrid Systems, LNCS, vol. 1567, pp. 38–69. Springer, Berlin (1997). doi:10.1007/3-540-49163-5_3
Davoren, J.M., Nerode, A.: Logics for hybrid systems. Proc. IEEE 88(7), 985–1010 (2000). doi:10.1109/5.871305
Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22(8), 465–476 (1979). doi:10.1145/359138.359142
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reason. 31(1), 33–72 (2003)
Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Sci. Comput. Program. 2(3), 241–266 (1982)
Emerson, E.A., Halpern, J.Y.: “Sometimes” and “Not Never” revisited: on branching versus linear time temporal logic. J. Assoc. Comput. Mach. 33(1), 151–178 (1986)
Fitting, M.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, New York (1996)
Fitting, M., Mendelsohn, R.L.: First-Order Modal Logic. Kluwer, Norwell (1999)
Fränzle, M.: Analysis of hybrid systems: an ounce of realism can save an infinity of states. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL, LNCS, vol. 1683, pp. 126–140. Springer, Berlin (1999)
Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC, LNCS, vol. 3414, pp. 258–273. Springer, Berlin (2005). doi:10.1007/b106766
Giese, M.: Incremental closure of free variable tableaux. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR, LNCS, vol. 2083, pp. 545–560. Springer, Berlin (2001). doi:10.1007/3-540-45744-5_46
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Mon.hefte Math. Phys. 38, 173–198 (1931). doi:10.1007/BF01700692
Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40, 330–349 (2007)
Hähnle, R., Schmitt, P.H.: The liberalized δ-rule in free variable semantic tableaux. J. Autom. Reason. 13(2), 211–221 (1994). doi:10.1007/BF00881956
Harel, D.: First-Order Dynamic Logic. Springer, New York (1979)
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT, Cambridge (2000)
Henzinger, T.A.: The theory of hybrid automata. In: LICS, pp. 278–292. IEEE Computer Society, Los Alamitos (1996)
Henzinger, T.A., Nicollin, X., Sifakis, J., Yovine, S.: Symbolic model checking for real-time systems. In: LICS, pp. 394–406. IEEE Computer Society, Los Alamitos (1992)
Hutter, D., Langenstein, B., Sengler, C., Siekmann, J.H., Stephan, W., Wolpers, A.: Deduction in the verification support environment (VSE). In: Gaudel, M.C., Woodcock, J. (eds.) FME, LNCS, vol. 1051, pp. 268–286. Springer, Berlin (1996)
Jifeng, H.: From CSP to hybrid systems. In: Roscoe, A.W. (ed.) A Classical Mind: Essays in Honour of C. A. R. Hoare, pp. 171–189. Prentice Hall, Hertfordshire (1994)
Kesten, Y., Manna, Z., Pnueli, A.: Verification of clocked and hybrid systems. Acta Inf. 36(11), 837–912 (2000). doi:10.1007/s002360050177
Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC, LNCS, vol. 1569, pp. 137–151. Springer, Berlin (1999)
Manna, Z., Sipma, H.: Deductive verification of hybrid systems using STeP. In: Henzinger, T.A., Sastry, S. (eds.) HSCC, LNCS, vol. 1386, pp. 305–318. Springer, Berlin (1998). doi:10.1007/3-540-64358-3_47
Morayne, M.: On differentiability of Peano type functions. Colloq. Math. LIII, 129–132 (1987)
Mysore, V., Piazza, C., Mishra, B.: Algorithmic algebraic model checking II: Decidability of semi-algebraic model checking and its applications to systems biology. In: Peled, D., Tsay, Y.K. (eds.) ATVA, LNCS, vol. 3707, pp. 217–233. Springer, Berlin (2005)
Perko, L.: Differential equations and dynamical systems. Springer, New York (1991)
Platzer, A.: Combining deduction and algebraic constraints for hybrid system analysis. In: Beckert, B. (ed.) VERIFY’07 at CADE, Bremen, Germany, CEUR Workshop Proceedings, vol. 259, pp. 164–178. CEUR-WS.org (2007)
Platzer, A.: Differential dynamic logic for verifying parametric hybrid systems. In: Olivetti, N. (ed.) TABLEAUX, LNCS, vol. 4548, pp. 216–232. Springer, Berlin (2007)
Platzer, A.: A temporal dynamic logic for verifying hybrid system invariants. In: Artëmov, S.N., Nerode, A. (eds.) LFCS, LNCS, vol. 4514, pp. 457–471. Springer, Berlin (2007)
Platzer, A., Clarke, E.M.: The image computation problem in hybrid systems model checking. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC, LNCS, vol. 4416, pp. 473–486. Springer, Berlin (2007)
Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57. IEEE, Piscataway (1977)
Pratt, V.R.: Semantical considerations on Floyd-Hoare logic. In: FOCS, pp. 109–121. IEEE, Piscataway (1976)
Rönkkö, M., Ravn, A.P., Sere, K.: Hybrid action systems. Theor. Comput. Sci. 290(1), 937–973 (2003)
Sibirsky, K.S.: Introduction to Topological Dynamics. Noordhoff, Leyden (1975)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)
Tavernini, L.: Differential automata and their discrete simulators. Nonlinear Anal. 11(6), 665–683 (1987). doi:10.1016/0362-546X(87)90034-4
Tinelli, C.: Cooperation of background reasoners in theory reasoning by residue sharing. J. Autom. Reason. 30(1), 1–31 (2003)
Tiwari, A.: Approximate reachability for linear systems. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control, 6th International Workshop, HSCC 2003 Prague, Czech Republic, April 3–5, 2003, Proceedings, LNCS, vol 2623, pp. 514–525. Springer, Berlin (2003). doi:10.1007/3-540-36580-X_37
Walter, W.: Ordinary Differential Equations. Springer, Berlin (1998)
Zhou, C., Ravn, A.P., Hansen, M.R.: An extended duration calculus for hybrid real-time systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) Hybrid Systems, LNCS, vol. 736, pp. 36–59. Springer, Berlin (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
The original online version of this article was revised due to retrospective open access order
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Platzer, A. Differential Dynamic Logic for Hybrid Systems. J Autom Reasoning 41, 143–189 (2008). https://doi.org/10.1007/s10817-008-9103-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10817-008-9103-8
Keywords
- Dynamic logic
- Differential equations
- Sequent calculus
- Axiomatisation
- Automated theorem proving
- Verification of hybrid systems