Abstract
In this paper, we consider the definition of a three-valued semantics for a μ-calculus on abstractions of hybrid automata. To this end, we first develop a framework that is general in the sense that it provides a preservation result for several possible semantics of the modal operators. In a second step, we instantiate our framework to two particular abstractions. To this end, a key issue is the consideration of both over- and underapproximated reachability, while classic simulation-based abstractions rely only on overapproximations, and therefore limit the preservation to the universal (μ-calculus’) fragment. To specialize our general result, we consider (1) modal abstractions, where the notions of ‘may’ and ‘must’ transitions are extended from the purely discrete to the hybrid time framework, and (2) so-called discrete bounded bisimulation abstractions.
Similar content being viewed by others
References
Alur R., Courcoubetis C., Henzinger T.: Computing accumulated delays in real-time systems. Formal Methods Syst. Design 11(2), 137–155 (1997)
Alur R., Dill D.: A theory of timed automata. Theoret. Comp. Sci. 126(2), 183–235 (1994)
Alur R., Henzinger T., Ho P.-H.: Automatic symbolic verification of embedded systems. IEEE Trans. Softw. Eng. 22(3), 181–201 (1996)
Alur R., Henzinger T., Lafferriere G., Pappas G.: Discrete abstractions of hybrid systems. Proc. IEEE 88(7), 971–984 (2000)
Bauer K.: Three-valued μ-calculus on hybrid automata. Master’s thesis, Department of Computer Science. University of Kaiserslautern, Germany (2007)
Bauer, K., Gentilini, R., Schneider, K. A uniform approach to three-valued semantics for μ-calculus on abstractions of hybrid automata. In: Chockler, H., Hu, A. (eds). International Haifa verification conference (HVC). LNCS, vol. 5394, pp. 38–52. Springer, Haifa (2009)
Bauer K., Schneider K.: From synchronous programs to symbolic representations of hybrid systems. In: Johansson, K., Yi, W. (eds) Hybrid systems: computation and control (HSCC), pp. 41–50. ACM, Stockholm (2010)
Bensalem, S., Bouajjani, A., Loiseaux, C., Sifakis, J.: Property preserving simulations. In: von Bochmann, G., Probst, D. (eds) Computer aided verification (CAV). LNCS, vol. 663, pp. 260–273. Springer, Montreal (1993)
Bruns G., Godefroid P.: Model checking partial state spaces with 3-valued temporal logics. In: Halbwachs, N., Peled, D. (eds) Computer Aided Verification (CAV), LNCS, vol. 1633, pp. 274–287. Springer, Trento (1999)
Davoren J.: On hybrid systems and the modal μ-calculus. In: Panos, J., Kohn, W., Lemmon, M., Nerode, A., Sastry, S. (eds) Hybrid Systems V. LNCS, vol. 1567, pp. 38–69. Springer, Berlin (1999)
Davoren J., Nerode A.: Logics for hybrid systems. Proc. IEEE 88(7), 985–1010 (2000)
Fitting M.: Kleene’s three valued logics and their children. Fundamenta Informaticae 20(1–3), 113–131 (1994)
Fränzle, M.: What will be eventually true of polynomial hybrid automata? In: Kobayashi, N., Pierce, B. (eds.) Theoretical aspects of computer software (TACS). LNCS, vol. 2215, pp. 340–359. Springer, Sendai (2001)
Gentilini R., Schneider K., Mishra B.: Successive abstractions of hybrid automata for monotonic CTL model checking. In: Artemov, S., Nerode, A. (eds) International Symposium on Logical Foundations of Computer Science (LFCS). LNCS, vol. 4514, pp. 224–240. Springer, New York (2007)
Ghosh R., Tiwari A., Tomlin C.: Automated symbolic reachability analysis with application to delta-notch signaling automata. In: Maler, O., Pnueli, A. (eds) Hybrid systems: computation and control (HSCC). LNCS, vol. 2623, pp. 233–248. Springer, Prague (2003)
Ghosh R., Tomlin C.: Lateral inhibition through Delta-Notch signaling: a piecewise affine hybrid model. In: Di Benedetto, M., Sangiovanni-Vincentelli, A. (eds) Hybrid systems: computation and control (HSCC), LNCS, vol. 2034, pp. 232–246. Springer, Rome (2001)
Godefroid P., Huth M., Jagadeesan R.: Abstraction-based model checking using modal transition systems. In: Larsen, K., Nielsen, M. (eds) Conference on Concurrency Theory (CONCUR). LNCS, vol. 2154, pp. 426–440. Springer, Aalborg (2001)
Grumberg O., Lange M., Leucker M., Shoham S.: Don’t know in the μ-calculus. In: Cousot, R. (eds) Verification, Model Checking, and Abstract Interpretation (VMCAI). LNCS, vol. 3385, pp. 233–249. Springer, Paris (2005)
Henzinger M., Henzinger T., Kopke P.: Computing simulations on finite and infinite graphs. In: Seberry, J., Pieprzyk, J. (eds) Annual symposium on foundations of computer science (FOCS), pp. 453. IEEE Computer Society, New Brunswick (1995)
Henzinger T.: The theory of hybrid automata. In: Symposium on Logic in Computer Science (LICS), pp. 278–292. IEEE Computer Society, New Brunswick (1996)
Henzinger T., Kopke P., Puri A., Varaiya P.: What’s decidable about hybrid automata?. J. Comp. Syst. Sci. 57(1), 94–124 (1998)
Kannellakis P., Smolka S.: CCS expressions, finite state processes, and three problems of equivalence. Inform. Comput. 86(1), 43–68 (1990)
Kleene S.: Introduction to Metamathematics. North Holland, Amsterdam (1952)
Lafferriere G., Pappas G., Sastry S.: o-Minimal hybrid systems. Math. Control Signals Syst. 13(1), 1–21 (2000)
Lafferriere G., Pappas G., Yovine S.: A new class of decidable hybrid systems. In: Vaandrager, F., van Schuppen, J. (eds) Hybrid Systems: Computation and Control (HSCC). LNCS, vol. 1569, pp. 137–151. Springer, Berg en Dal (1999)
Miller J.: Decidability and complexity results for timed automata and semi-linear hybrid automata. In: Lynch, N., Krogh, B. (eds) Hybrid Systems: Computation and Control (HSCC). LNCS, vol. 1790, pp. 296–309. Springer, Pittsburgh (2000)
Piazza C., Antoniotti M., Mysore V., Policriti A., Winkler F., Mishra B.: Algorithmic algebraic model checking I: Challenges from systems biology. In: Etessami, K., Rajamani, S. (eds) Computer Aided Verification (CAV). LNCS, vol. 3576, pp. 5–19. Springer, Edinburgh (2005)
Ratschan S., She Z.: Safety verification of hybrid systems by constraint propagation based abstraction refinement. In: Morari, M., Thiele, L. (eds) Hybrid Systems: Computation and Control (HSCC). LNCS, vol. 3414, pp. 573–589. Springer, Zurich (2005)
Shoham S., Grumberg O.: Monotonic abstraction-refinement for CTL. In: Jensen, K., Podelski, A. (eds) Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 2988, pp. 546–560. Springer, Barcelona (2004)
Shoham S., Grumberg O.: Multi-valued model checking games. In: Peled, D., Tsay, Y.-K. (eds) Automated Technology for Verification and Analysis (ATVA). LNCS, vol. 3707, pp. 354–369. Springer, Taipei (2005)
Shoham S., Grumberg O.: 3-valued abstraction: more precision at less cost. In: Symposium on Logic in Computer Science (LICS), pp. 399–410. IEEE Computer Society, Seattle (2006)
Tiwari A., Khanna G.: Series of abstractions for hybrid automata. In: Tomlin, C., Greenstreet, M. (eds) Hybrid Systems: Computation and Control (HSCC). LNCS, vol. 2289, pp. 465–478. Springer, Stanford (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bauer, K., Gentilini, R. & Schneider, K. A uniform approach to three-valued semantics for μ-calculus on abstractions of hybrid automata. Int J Softw Tools Technol Transfer 13, 273–287 (2011). https://doi.org/10.1007/s10009-010-0161-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10009-010-0161-y