Axiomatizing Analog Algorithms

  • Olivier Bournez
  • Nachum Dershowitz
  • Pierre Néron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

Abstract

We propose a formalization of generic algorithms that includes analog algorithms. This is achieved by reformulating and extending the framework of abstract state machines to include continuous-time models of computation. We prove that every hybrid algorithm satisfying some reasonable postulates may be expressed precisely by a program in a simple and expressive language.

References

  1. 1.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers; NP completeness, recursive functions and universal machines. Bull. Am. Math. Soc. 21, 1–46 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boker, U., Dershowitz, N.: The Church-Turing thesis over arbitrary domains. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 199–229. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Boker, U., Dershowitz, N.: Three paths to effectiveness. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Fields of Logic and Computation. LNCS, vol. 6300, pp. 135–146. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Bournez, O., Campagnolo, M.L.: A survey on continuous time computations. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) Changing Conceptions of What is Computable, pp. 383–423. Springer, New York (2008)CrossRefGoogle Scholar
  5. 5.
    Bournez, O., Dershowitz, N., Falkovich, E.: Towards an axiomatization of simple analog algorithms. In: Agrawal, M., Cooper, S.B., Li, A. (eds.) TAMC 2012. LNCS, vol. 7287, pp. 525–536. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Bournez, O., Dershowitz, N., Néron, P.: Axiomatizing analog algorithms. ArXiv e-prints (2016). http://arxiv.org/abs/1604.04295
  7. 7.
    Bush, V.: The differential analyser. J. Franklin Inst. 212, 447–488 (1931)CrossRefMATHGoogle Scholar
  8. 8.
    Cohen, J., Slissenko, A.: On implementations of instantaneous actions real-time ASM by ASM with delays. In: Proceedings of 12th International Workshop on Abstract State Machines, Université de Paris, vol. 12, pp. 387–396 (2005)Google Scholar
  9. 9.
    Cohen, J., Slissenko, A.: Implementation of sturdy real-time abstract state machines by machines with delays. In: Proceedings of 6th International Conference on Computer Science and Information Technology, National Academy of Science of Armenia (2007)Google Scholar
  10. 10.
    Dershowitz, N., Gurevich, Y.: A natural axiomatization of computability and proof of Church’s Thesis. Bull. Symbolic Logic 14, 299–350 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fu, M.Q., Zucker, J.: Models of computation for partial functions on the reals. J. Log. Algebraic Methods Program. 84, 218–237 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Graça, D.S., Buescu, J., Campagnolo, M.L.: Computational bounds on polynomial differential equations. Appl. Math. Comput. 215, 1375–1385 (2009)MathSciNetMATHGoogle Scholar
  13. 13.
    Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. J. Complex. 19, 644–664 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gurevich, Y.: Sequential abstract-state machines capture sequential algorithms. ACM Trans. Comput. Log. 1, 77–111 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hasuo, I., Suenaga, K.: Exercises in nonstandard static analysis of hybrid systems. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 462–478. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. 16.
    Nyce, J.M.: Guest editor’s introduction. IEEE Ann. Hist. Comput. 18, 3–4 (1996)CrossRefGoogle Scholar
  17. 17.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reasoning 41, 143–189 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Reisig, W.: On Gurevich’s theorem on sequential algorithms. Acta Informatica 39, 273–305 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Rust, H.: Hybrid abstract state machines: using the hyperreals for describing continuous changes in a discrete notation. In: International Workshop on Abstract State Machines, Swiss Federal Institute of Technology, pp. 341–356 (2000)Google Scholar
  20. 20.
    Shannon, C.E.: Mathematical theory of the differential analyser. J. Math. Phys. 20, 337–354 (1941)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Suenaga, K., Hasuo, I.: Programming with infinitesimals: a While-language for hybrid system modeling. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 392–403. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Tucker, J.V., Zucker, J.I.: A network model of analogue computation over metric algebras. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 515–529. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Olivier Bournez
    • 1
  • Nachum Dershowitz
    • 2
  • Pierre Néron
    • 3
  1. 1.Laboratoire d’Informatique de l’X (LIX)École PolytechniquePalaiseauFrance
  2. 2.School of Computer ScienceTel Aviv UniversityRamat AvivIsrael
  3. 3.French Network and Information Security Agency (ANSSI)ParisFrance

Personalised recommendations