Document Open Access Logo

Computability, Complexity and Programming with Ordinary Differential Equations (Invited Talk)

Author Olivier Bournez

Thumbnail PDF


  • Filesize: 0.56 MB
  • 13 pages

Document Identifiers

Author Details

Olivier Bournez
  • Laboratoire d'Informatique de l'X (LIX), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France


This material is based upon work supported by the {RACAF Project from Agence National de la Recherche}.

Cite AsGet BibTex

Olivier Bournez. Computability, Complexity and Programming with Ordinary Differential Equations (Invited Talk). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 3:1-3:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


Ordinary Differential Equations (ODEs) appear to be a universally adopted and very natural way for expressing properties for continuous time dynamical systems. They are intensively used, in particular in applied sciences. There exists an abundant literature about the hardness of solving ODEs with numerical methods. We adopt a dual view: we consider ODEs as a way to program or to describe our mathematical/computer science world. We survey several results considering ODEs under this computational perspective, with a computability and complexity theory point of view. In particular, we provide various reasons why polynomial ODEs should be considered as the continuous time analog of Turing machines for continuous-time computations, or should be used as a way to talk about mathematical logic. This has already applications in various fields: determining whether analog models of computation can compute faster than classical digital models of computation; solving complexity issues for computations with biochemical reactions in bioinformatics; machine independent characterizations of various computability and complexity classes such as PTIME or NPTIME, or proof of the existence of a universal polynomial ordinary differential equation whose solutions can approximate any continuous function if provided with a suitable well-chosen initial condition.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Models of computation
  • Theory of computation → Computability
  • Theory of computation → Recursive functions
  • Computer systems organization → Analog computers
  • Theory of computation → Complexity classes
  • Theory of computation → Complexity theory and logic
  • Mathematics of computing → Differential equations
  • Mathematics of computing → Ordinary differential equations
  • Ordinary differential equations
  • Models of computation
  • Analog Computations
  • discrete ordinary differential equations
  • Implicit complexity
  • recursion scheme


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Michael Boshernitzan. Universal formulae and universal differential equations. Annals of mathematics, 124(2):273-291, 1986. Google Scholar
  2. Daniel Bournez, Olivierand Graça and Amaury Pouly. On the Functions Generated by the General Purpose Analog Computer. Information and Computation, 257:34-57, 2017. URL:
  3. Olivier Bournez. Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theoretical Computer Science, 210(1):21-71, 1999. Google Scholar
  4. Olivier Bournez and Manuel L. Campagnolo. New Computational Paradigms. Changing Conceptions of What is Computable, chapter A Survey on Continuous Time Computations, pages 383-423. Springer-Verlag, New York, 2008. Google Scholar
  5. Olivier Bournez, Manuel L. Campagnolo, Daniel Graça, and Emmanuel S. Hainry. Polynomial differential equations compute all real computable functions on computable compact intervals. Journal of Complexity, 23(3):317-335, 2007. Google Scholar
  6. Olivier Bournez, Manuel L. Campagnolo, Daniel S. Graça, and Emmanuel Hainry. Polynomial differential equations compute all real computable functions on computable compact intervals. jcomp, 23(3):317-335, 2007. Google Scholar
  7. Olivier Bournez and Arnaud Durand. Recursion schemes, discrete differential equations and characterization of polynomial time computation. In MFCS 2019: 44th International Symposium on Mathematical Foundations of Computer Science, 2019. Google Scholar
  8. Olivier Bournez, Daniel S. Graça, and Amaury Pouly. Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. Journal of the ACM, 64(6):38:1-38:76, 2017. URL:
  9. Olivier Bournez, Daniel S. Graça, and Amaury Pouly. Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. The General Purpose Analog Computer and Computable Analysis are two efficiently equivalent models of computations. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 109:1-109:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  10. Olivier Bournez and Amaury Pouly. A universal ordinary differential equation. In International Colloquium on Automata Language Programming, ICALP'2017, 2017. Google Scholar
  11. Olivier Bournez and Amaury Pouly. A universal ordinary differential equation. To appear in Logical Methods in Computer Science, abs/1702.08328, 2020. URL:
  12. H. J. Buisman, H. M. M. ten Eikelder, P. A. J. Hilbers, and A. M. L. Liekens. Computing algebraic functions with biochemical reaction networks. Artificial Life, 15(1):5-19, 2009. URL:
  13. Vannevar Bush. The differential analyzer. A new machine for solving differential equations. J. Franklin Inst., 212:447-488, 1931. Google Scholar
  14. C. S. Calude and B. Pavlov. Coins, quantum measurements, and Turing’s barrier. Quantum Information Processing, 1(1-2):107-127, 2002. Google Scholar
  15. David C. Carothers, G. Edgar Parker, James S. Sochacki, and Paul G. Warne. Some properties of solutions to polynomial systems of differential equations. Electronic Journal of Differential Equations, 2005(40):1-17, 2005. Google Scholar
  16. Ho-Lin Chen, David Doty, and David Soloveichik. Rate-independent computation in continuous chemical reaction networks. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS '14, pages 313-326, New York, NY, USA, 2014. ACM. URL:
  17. Matthew Cook, David Soloveichik, Erik Winfree, and Jehoshua Bruck. Programmability of chemical reaction networks. In Algorithmic Bioprocesses, pages 543-584. Springer, 2009. Google Scholar
  18. B. Jack Copeland. Even Turing machines can compute uncomputable functions. In C.S. Calude, J. Casti, and M.J. Dinneen, editors, Unconventional Models of Computations. Springer-Verlag, 1998. Google Scholar
  19. B. Jack Copeland. Accelerating Turing machines. Minds and Machines, 12:281-301, 2002. Google Scholar
  20. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms (third edition). MIT press, 2009. Google Scholar
  21. E. B. Davies. Building infinite machines. The British Journal for the Philosophy of Science, 52:671-682, 2001. Google Scholar
  22. François Fages, Steven Gay, and Sylvain Soliman. Inferring reaction systems from ordinary differential equations. Theoretical Computer Science, 599:64-78, 2015. URL:
  23. Francois Fages, Guillaume Le Guludec, Olivier Bournez, and Amaury Pouly. Strong turing completeness of continuous chemical reaction networks and compilation of mixed analog-digital programs. In Computational Methods in Systems Biology-CMSB 2017, 2017. Google Scholar
  24. Andrei Gabrielov and Nicolai Vorobjov. Complexity of computations with pfaffian and noetherian functions. Normal forms, bifurcations and finiteness problems in differential equations, pages 211-250, 2004. Google Scholar
  25. Aleksandr Gelfond. Calculus of finite differences. Hindustan Publ. Corp., 1971. Google Scholar
  26. David Gleich. Finite calculus: A tutorial for solving nasty sums. Stanford University, 2005. Google Scholar
  27. Daniel S. Graça. Some recent developments on Shannon’s General Purpose Analog Computer. Mathematical Logic Quaterly, 50(4-5):473-485, 2004. Google Scholar
  28. Daniel S. Graça, Manuel L. Campagnolo, and J. Buescu. Computability with polynomial differential equations. Advances in Applied Mathematics, 2007. To appear. Google Scholar
  29. Daniel S. Graça and José Félix Costa. Analog computers and recursive functions over the reals. Journal of Complexity, 19(5):644-664, 2003. Google Scholar
  30. Ronald L Graham, Donald E Knuth, Oren Patashnik, and Stanley Liu. Concrete mathematics: a foundation for computer science. Computers in Physics, 3(5):106-107, 1989. Google Scholar
  31. Otto Hölder. Über die eigenschaft der gamma funktion keiner algebraische differentialgleichung zu genügen. Math. Ann., 28:1-13, 1887. Google Scholar
  32. FA Izadi, N Aliev, and G Bagirov. Discrete Calculus by Analogy. Bentham Science Publishers, 2009. Google Scholar
  33. Gustavo Lau. Discrete calculus. URL:
  34. Lenoard. Lipshitz and Lee A. Rubel. A differentially algebraic replacement theorem, and analog computability. Proceedings of the American Mathematical Society, 99(2):367-372, 1987. Google Scholar
  35. Cristopher Moore. Recursion theory on the reals and continuous-time computation. Theoretical Computer Science, 162(1):23-44, 1996. Google Scholar
  36. André Platzer and Yong Kiam Tan. Differential equation axiomatization: The impressive power of differential ghosts. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pages 819-828. ACM, 2018. Google Scholar
  37. Marian B. Pour-El. Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers). Transactions of the American Mathematical Society, 199:1-28, 1974. Google Scholar
  38. L. A. Rubel. A survey of transcendentally transcendental functions. American Mathematical Monthly, 96(9):777-788, 1989. Google Scholar
  39. Lee A. Rubel. A universal differential equation. Bulletin of the American Mathematical Society, 4(3):345-349, 1981. Google Scholar
  40. Keijo Ruohonen. Undecidability of event detection for ODEs. Journal of Information Processing and Cybernetics, 29:101-113, 1993. Google Scholar
  41. Keijo Ruohonen. Event detection for ODEs and nonrecursive hierarchies. In Proceedings of the Colloquium in Honor of Arto Salomaa. Results and Trends in Theoretical Computer Science (Graz, Austria, June 10-11, 1994), volume 812 of Lecture Notes in Computer Science, pages 358-371. Springer-Verlag, Berlin, 1994. URL:
  42. Claude E. Shannon. Mathematical theory of the differential analyser. Journal of Mathematics and Physics MIT, 20:337-354, 1941. Google Scholar
  43. Klaus Weihrauch. Computable Analysis: an Introduction. Springer, 2000. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail