Document Open Access Logo

Quantifiying the Robustness of Dynamical Systems. Relating Time and Space to Length and Precision

Authors Manon Blanc , Olivier Bournez

PDF
Thumbnail PDF

File

LIPIcs.CSL.2024.17.pdf
  • Filesize: 0.77 MB
  • 20 pages

Document Identifiers

Author Details

Manon Blanc
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX, Palaiseau, France
  • Université Paris-Saclay, LISN, Orsay, France
Olivier Bournez
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX, Palaiseau, France

Funding

This work was partially supported by Labex Digicosme and ANR δIFFERENCE.

Cite AsGet BibTex

Manon Blanc and Olivier Bournez. Quantifiying the Robustness of Dynamical Systems. Relating Time and Space to Length and Precision. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CSL.2024.17

Abstract

Reasoning about dynamical systems evolving over the reals is well-known to lead to undecidability. In particular, it is known that there cannot be reachability decision procedures for first-order theories over the reals extended with even very basic functions, or for logical theories that reason about real-valued functions, or decision procedures for state reachability. This mostly comes from the fact that reachability for dynamical systems over the reals is fundamentally undecidable, as Turing machines can be embedded into (even very simple) dynamical systems. However, various results in the literature have shown that decision procedures exist when restricting to robust systems, with a suitably-chosen notion of robustness. In particular, it has been established in the field of verification that if the state reachability is not sensitive to infinitesimal perturbations, then decision procedures for state reachability exist. In the context of logical theories over the reals, it has been established that decision procedures exist if we focus on properties not sensitive to arbitrarily small perturbations. For example by considering properties that are either true or δ-far from being true, for some δ > 0. In this article, we first propose a unified theory explaining in a uniform framework these statements, that were established in different contexts. More fundamentally, while all these statements are only about computability issues, we also consider complexity theory aspects. We prove that robustness to some precision is inherently related to the complexity of the decision procedure. When a system is robust, it makes sense to quantify at which level of perturbation it is. We prove that assuming robustness to a polynomial perturbation on precision leads to a characterisation of PSPACE. We prove that assuming robustness to polynomial perturbation on time or length leads to similar statements for PTIME. In other words, precision on computations is inherently related to space complexity, while length or time of trajectories, is intrinsically related to time complexity. These statements can also be interpreted in relation to several recent results about the computational power of analogue models of computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Computability
  • Theory of computation → Complexity classes
  • Theory of computation → Complexity theory and logic
  • Computer systems organization → Analog computers
Keywords
  • Computability
  • Complexity theory
  • Computable analysis
  • Verification
  • Decision
  • Robustness
  • Dynamical Systems
  • Models of computation
  • Analogue Computations

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Eugene Asarin and Ahmed Bouajjani. Perturbed Turing machines and hybrid systems. In Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS-01), pages 269-278, Los Alamitos, CA, June 16-19 2001. IEEE Computer Society Press. Google Scholar
  2. Eugene Asarin, Oded Maler, and Amir Pnueli. Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science, 138(1):35-65, February 1995. Google Scholar
  3. Manon Blanc and Olivier Bournez. A characterisation of functions computable in polynomial time and space over the reals with discrete ordinary differential equations: Simulation of Turing machines with analytic discrete odes. In Mathematical Foundations of Computer Science (MFCS'2023), 2023. Google Scholar
  4. Vincent Blondel and John Tsitsiklis. A survey of computational complexity results in systems and control. Automatica, 36(9):1249-1274, 2000. Google Scholar
  5. Olivier Bournez, Johanne Cohen, and Valentin Dardilhac. On the δ-decidability of decision problems for neural network questions. In Computability, Continuity, Constructivity - from Logic to Algorithms CCC'23, 2023. Google Scholar
  6. Olivier Bournez, Felipe Cucker, Paulin Jacobé de Naurois, and Jean-Yves Marion. Computability over an arbitrary structure. sequential and parallel polynomial time. In Andrew D. Gordon, editor, Foundations of Software Science and Computational Structures, 6th International Conference (FOSSACS'2003), volume 2620 of Lecture Notes in Computer Science, pages 185-199, Warsaw, 2003. Springer. Google Scholar
  7. Olivier Bournez, Riccardo Gozzi, Daniel S Graça, and Amaury Pouly. A continuous characterization of PSPACE using polynomial ordinary differential equations. Journal of Complexity, 77:101755, August 2023. URL: https://www.sciencedirect.com/science/article/pii/S0885064X23000249?dgcid)=author.
  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: https://doi.org/10.1145/3127496.
  9. Olivier Bournez, Daniel S. Graça, and Emmanuel Hainry. Robust computations with dynamical systems. In Mathematical Foundations of Computer Science, MFCS'2010, volume 6281 of Lecture Notes in Computer Science, pages 198-208. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15155-2_19.
  10. Olivier Bournez and Emmanuel Hainry. An analog characterization of elementary computable functions over the real numbers. In Josep Díaz, Juhani Karhumäki, Arto Lepistö, and Donald Sannella, editors, International Colloquium on Automata, Languages and Programming (ICALP 2004), volume 3142 of Lecture Notes in Computer Science, pages 269-280, 2004. Google Scholar
  11. Olivier Bournez and Amaury Pouly. A survey on analog models of computation. In Handbook of Computability and Complexity in Analysis, pages 173-226. Springer, 2021. Google Scholar
  12. Vasco Brattka, Peter Hertling, and Klaus Weihrauch. A tutorial on computable analysis. In New computational paradigms, pages 425-491. Springer, 2008. Google Scholar
  13. Mark Braverman. Computational complexity of Euclidean sets: Hyperbolic Julia sets are poly-time computable. Master’s thesis, University of Toronto, 2004. Google Scholar
  14. Martin Fränzle. Analysis of hybrid systems: An ounce of realism can save an infinity of states. In Jörg Flum and Mario Rodríguez-Artalejo, editors, Computer Science Logic, 13th International Workshop, CSL '99, 8th Annual Conference of the EACSL, Madrid, Spain, September 20-25, 1999, Proceedings, volume 1683 of Lecture Notes in Computer Science, pages 126-140. Springer, 1999. Google Scholar
  15. Sicun Gao, Jeremy Avigad, and Edmund M Clarke. Delta-decidability over the reals. In Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on, pages 305-314. IEEE, 2012. Google Scholar
  16. Daniel S. Graça, N. Zhong, and J. Buescu. Computability, noncomputability and undecidability of maximal intervals of IVPs. Transactions of the American Mathematical Society, 2006. To appear. Google Scholar
  17. Philip Hartman. Ordinary Differential Equations. John Wiley and Sons, 1964. Google Scholar
  18. Thomas A. Henzinger, Peter W. Kopke, Anuj Puri, and Pravin Varaiya. What’s decidable about hybrid automata? Journal of Computer and System Sciences, 57(1):94-124, August 1998. Google Scholar
  19. Thomas A. Henzinger and Jean-François Raskin. Robust undecidability of timed and hybrid systems. In Nancy A. Lynch and Bruce H. Krogh, editors, Hybrid Systems: Computation and Control, Third International Workshop, HSCC 2000, Pittsburgh, PA, USA, March 23-25, 2000, Proceedings, volume 1790 of Lecture Notes in Computer Science, pages 145-159. Springer, 2000. Google Scholar
  20. N. Immerman. Nondeterministic space is closed under complementation. In Structure in Complexity Theory Conference, 1988. Proceedings., Third Annual, pages 112-115. IEEE, 1988. Google Scholar
  21. Ker-I Ko. Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhaüser, Boston, 1991. Google Scholar
  22. Pascal Koiran, Michel Cosnard, and Max Garzon. Computability with low-dimensional dynamical systems. Theoretical Computer Science, 132(1-2):113-128, September 1994. Google Scholar
  23. Cristopher Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity, 4(3):199-230, 1991. Google Scholar
  24. Eike Neumann. Decision problems for linear recurrences involving arbitrary real numbers. Logical Methods in Computer Science, 17, 2021. Google Scholar
  25. Eike Neumann. On the complexity of robust eventual inequality testing for C-finite functions. In International Conference on Reachability Problems, pages 98-112. Springer, 2023. Google Scholar
  26. A. Puri, V. Borkar, and P. Varaiya. Epsilon-approximation of differential inclusions. In Proceedings of the 34th IEEE Conference on Decision and Control, pages 2892-2897, 1995. Google Scholar
  27. Anuj Puri. Dynamical properties of timed automata. Discrete Event Dynamic Systems, 10:87-113, 2000. Google Scholar
  28. Stefan Ratschan. Deciding predicate logical theories of real-valued functions. In Symposium on Mathematical Foundations of Computer Science (MFCS'2023), 2023. Google Scholar
  29. Claude E. Shannon. Mathematical theory of the differential analyser. Journal of Mathematics and Physics MIT, 20:337-354, 1941. Google Scholar
  30. Michael Sipser. Introduction to the Theory of Computation. PWS Publishing Company, 1997. Google Scholar
  31. R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta informatica, 26(3):279-284, November 1988. URL: https://doi.org/10.1007/BF00299636.
  32. Klaus Weihrauch. Computable Analysis: an Introduction. Springer, 2000. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact