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.
O. Bournez—This author’s research was partially supported by a French National Research Agency’s grant (ANR-15-CE40-0016-02).
N. Dershowitz—This author’s research benefited from a fellowship at the Institut d’Études Avancées de Paris (France), with the financial support of the French National Research Agency’s “Investissements d’avenir” program (ANR-11-LABX-0027-01 Labex RFIEA+).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
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)
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)
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)
Bournez, O., Dershowitz, N., Néron, P.: Axiomatizing analog algorithms. ArXiv e-prints (2016). http://arxiv.org/abs/1604.04295
Bush, V.: The differential analyser. J. Franklin Inst. 212, 447–488 (1931)
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)
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)
Dershowitz, N., Gurevich, Y.: A natural axiomatization of computability and proof of Church’s Thesis. Bull. Symbolic Logic 14, 299–350 (2008)
Fu, M.Q., Zucker, J.: Models of computation for partial functions on the reals. J. Log. Algebraic Methods Program. 84, 218–237 (2015)
Graça, D.S., Buescu, J., Campagnolo, M.L.: Computational bounds on polynomial differential equations. Appl. Math. Comput. 215, 1375–1385 (2009)
Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. J. Complex. 19, 644–664 (2003)
Gurevich, Y.: Sequential abstract-state machines capture sequential algorithms. ACM Trans. Comput. Log. 1, 77–111 (2000)
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)
Nyce, J.M.: Guest editor’s introduction. IEEE Ann. Hist. Comput. 18, 3–4 (1996)
Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reasoning 41, 143–189 (2008)
Reisig, W.: On Gurevich’s theorem on sequential algorithms. Acta Informatica 39, 273–305 (2003)
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)
Shannon, C.E.: Mathematical theory of the differential analyser. J. Math. Phys. 20, 337–354 (1941)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Bournez, O., Dershowitz, N., Néron, P. (2016). Axiomatizing Analog Algorithms. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-40189-8_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40188-1
Online ISBN: 978-3-319-40189-8
eBook Packages: Computer ScienceComputer Science (R0)