Abstract
The specification of cyber-physical systems usually relies on continuous functions over dense real numbers whereas their implementation is discrete. Proving the correctness of the discrete implementation with respect to the continuous specification remains a challenge in the presence of dense real numbers. In this paper, we propose a refinement-based formal method, relying on Event-B, for such developments. We illustrate our proposal with the development of a simple stability controller for a generic plant model. The continuous function that models the system behavior is refined as a discrete model of the same kind preserving stability expressed as a safety invariants of the continuous model. The obtained discrete model uses discrete time (instants modeled on \(\mathbb {N}\)), whereas the continuous model is based on dense time (on \(\mathbb {R}\)). The Rodin Platform, together with the Theory plug-in handling the Real datatype and its properties supported the whole developments and proofs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abrial, J.-R.: Modeling in Event-B: System and Software Engineering, 1st edn. Cambridge University Press, New York, NY, USA (2010)
Abrial, J.-R., Butler, M., Hallerstede, S., Hoang, T.S., Mehta, F., Voisin, L.: Rodin: an open toolset for modelling and reasoning in Event-B. Int. J. Softw. Tools Technol. Transfer 12(6), 447–466 (2010)
Abrial, J.-R., Butler, M., Hallerstede, S., Leuschel, M., Schmalz, M., Voisin, L.: Proposals for mathematical extensions for event-b. Technical report (2009)
Abrial, J.-R., Hallerstede, S.: Refinement, decomposition, and instantiation of discrete models: Application to Event-B. Fundamenta Informat. 77(1), 1–28 (2007)
Alur, R.: Formal verification of hybrid systems. In: Chakraborty, S., Jerraya, A., Baruah, S. K., Fischmeister, S. (eds.) Proceedings of the 11th International Conference on Embedded Software, EMSOFT - ESWeek, Taipei, Taiwan, October 9–14, 2011, pp. 273–278. ACM (2011)
Babin, G., At-Ameur, Y., Pantel, M.: Formal verification of runtime compensation of web service compositions: A refinement and proof based proposal with Event-B. In: International Conference on SCC 2015 IEEE, pp. 98–105, June
Banach, R.: Pliant modalities in Hybrid Event-B. In: Liu, Z., Woodcock, J., Zhu, H. (eds.) Theories of Programming and Formal Methods. LNCS, vol. 8051, pp. 37–53. Springer, Heidelberg (2013)
Banach, R., Butler, M., Qin, S., Verma, N., Zhu, H.: Core Hybrid Event-B I: Single Hybrid Event-B machines. Science of Computer Programming (2015)
Banach, R., Zhu, H., Su, W., Huang, R.: Formalising the continuous/discrete modeling step. In: Derrick, J., Boiten, E.A., Reeves, S. (eds.) Proceedings 15th International Refinement Workshop, Refine 2011, Limerick, Ireland, 20th June 2011, volume 55 of EPTCS, pp. 121–138 (2011)
Banach, R., Zhu, H., Su, W., Wu, X.: ASM and controller synthesis. In: Derrick, J., Fitzgerald, J., Gnesi, S., Khurshid, S., Leuschel, M., Reeves, S., Riccobene, E. (eds.) ABZ 2012. LNCS, vol. 7316, pp. 51–64. Springer, Heidelberg (2012)
Boldo, S., Lelay, C., Melquiond, G.: Coquelicot: A user-friendly library of real analysis for Coq. Math. Comput. Sci. 9(1), 41–62 (2015)
Butler, M., Abrial, J.-R., Banach, R.: From Action Systems to Distributed Systems: The Refinement Approach, chapter Modelling and Refining Hybrid Systems in Event-B and Rodin, p. 300. Taylor & Francis, February 2016
Butler, M., Maamria, I.: Practical theory extension in Event-B. In: Liu, Z., Woodcock, J., Zhu, H. (eds.) Theories of Programming and Formal Methods. LNCS, vol. 8051, pp. 67–81. Springer, Heidelberg (2013)
Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM POPL 1977, pp. 238–252, New York, NY, USA. ACM (1977)
Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Monniaux, D., Rival, X.: The ASTRÉE analyzer. In: Sagiv, M. (ed.) ESOP 2005. LNCS, vol. 3444, pp. 21–30. Springer, Heidelberg (2005)
Goubault, É.: Static analyses of the precision of floating-point operations. In: Cousot, P. (ed.) SAS 2001. LNCS, vol. 2126, p. 234. Springer, Heidelberg (2001)
Henzinger. T. A.: The theory of hybrid automata. In: Inan, M.K., Kurshan, R.P. (eds.) Verification of Digital and Hybrid Systems, volume 170 of NATO ASI Series, pp. 265–292. Springer-Verlag (2000)
Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Hytech: A model checker for hybrid systems. International Journal on STTT 1(1–2), 110–122 (1997)
Ishii, D., Melquiond, G., Nakajima, S.: Inductive verification of hybrid automata with strongest postcondition calculus. In: Johnsen, E.B., Petre, L. (eds.) IFM 2013. LNCS, vol. 7940, pp. 139–153. Springer, Heidelberg (2013)
Lee, E.A., Seshia, S.A.: Introduction to Embedded Systems - A Cyber-Physical Systems Approach. LeeSeshia.org, edition 1.5 edition (2014)
Muller, J.-M., Brisebarre, N., de Dinechin, F., Jeannerod, C.-P., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.: Handbook of Floating-Point Arithmetic. Birkhäuser (2010)
Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer-Verlag, Heidelberg (2010)
Su, W., Abrial, J.-R., Zhu, H.: Formalizing hybrid systems with Event-B and the Rodin platform. Science of Computer Programming, 94, Part 2:164–202 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Babin, G., Aït-Ameur, Y., Nakajima, S., Pantel, M. (2015). Refinement and Proof Based Development of Systems Characterized by Continuous Functions. In: Li, X., Liu, Z., Yi, W. (eds) Dependable Software Engineering: Theories, Tools, and Applications. SETTA 2015. Lecture Notes in Computer Science(), vol 9409. Springer, Cham. https://doi.org/10.1007/978-3-319-25942-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-25942-0_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25941-3
Online ISBN: 978-3-319-25942-0
eBook Packages: Computer ScienceComputer Science (R0)