Programming with Infinitesimals: A While-Language for Hybrid System Modeling

  • Kohei Suenaga
  • Ichiro Hasuo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)


We add, to the common combination of a \(\textsc{While}\)-language and a Hoare-style program logic, a constant \(\mathtt{dt}\) that represents an infinitesimal (i.e. infinitely small) value. The outcome is a framework for modeling and verification of hybrid systems: hybrid systems exhibit both continuous and discrete dynamics and getting them right is a pressing challenge. We rigorously define the semantics of programs in the language of nonstandard analysis, on the basis of which the program logic is shown to be sound and relatively complete.




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  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. Comp. Sci. 138(1), 3–34 (1995)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bliudze, S., Krob, D.: Modelling of complex systems: Systems as dataflow machines. Fundam. Inform. 91(2), 251–274 (2009)MATHMathSciNetGoogle Scholar
  3. 3.
    Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM Journ. Comput. 7(1), 70–90 (1978)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Hurd, A.E., Loeb, P.A.: An Introduction to Nonstandard Real Analysis. Academic Press, London (1985)MATHGoogle Scholar
  5. 5.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reasoning 41(2), 143–189 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Platzer, A., Quesel, J.-D.: KeYmaera: A hybrid theorem prover for hybrid systems (System description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Rust, H.: Operational Semantics for Timed Systems: A Non-standard Approach to Uniform Modeling of Timed and Hybrid Systems. LNCS, vol. 3456. Springer, Heidelberg (2005)MATHGoogle Scholar
  9. 9.
    Suenaga, K., Hasuo, I.: Programming with infinitesimals: A While-language for hybrid system modeling. Extended version with proofs (April 2011)Google Scholar
  10. 10.
    Winskel, G.: The Formal Semantics of Programming Languages. MIT Press, Cambridge (1993)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kohei Suenaga
    • 1
  • Ichiro Hasuo
    • 2
  1. 1.JSPS Research Fellow, Kyoto UniversityJapan
  2. 2.University of TokyoJapan

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