Noise vs computational intractability in dynamics

Mark Braverman, Alexander Grigo, Cristobal Rojas

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

Computation plays a key role in predicting and analyzing natural phenomena. There are two fundamental barriers to our ability to computationally understand the long-term behavior of a dynamical system that describes a natural process. The first one is unaccounted-for errors, which may make the system unpredictable beyond a very limited time horizon. This is especially true for chaotic systems, where a small change in the initial conditions may cause a dramatic shift in the trajectories. The second one is Turing-completeness. By the undecidability of the Halting Problem, the long-term prospects of a system that can simulate a Turing Machine cannot be determined computationally. We investigate the interplay between these two forces - unaccounted-for errors and Turing-completeness. We show that the introduction of even a small amount of noise into a dynamical system is sufficient to "destroy" Turing-completeness, and to make the system's long-term behavior computationally predictable. On a more technical level, we deal with long-term statistical properties of dynamical systems, as described by invariant measures. We show that while there are simple dynamical systems for which the invariant measures are non-computable, perturbing such systems makes the invariant measures efficiently computable. Thus, noise that makes the short term behavior of the system harder to predict, may make its long term statistical behavior computationally tractable. We also obtain some insight into the computational complexity of predicting systems affected by random noise.

Original languageEnglish (US)
Title of host publicationITCS 2012 - Innovations in Theoretical Computer Science Conference
Pages128-141
Number of pages14
DOIs
StatePublished - Feb 6 2012
Event3rd Conference on Innovations in Theoretical Computer Science, ITCS 2012 - Cambridge, MA, United States
Duration: Jan 8 2012Jan 10 2012

Publication series

NameITCS 2012 - Innovations in Theoretical Computer Science Conference

Other

Other3rd Conference on Innovations in Theoretical Computer Science, ITCS 2012
CountryUnited States
CityCambridge, MA
Period1/8/121/10/12

All Science Journal Classification (ASJC) codes

  • Computational Theory and Mathematics

Keywords

  • invariant distributions
  • perturbation
  • predictability
  • robustness
  • uncomputability

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    Braverman, M., Grigo, A., & Rojas, C. (2012). Noise vs computational intractability in dynamics. In ITCS 2012 - Innovations in Theoretical Computer Science Conference (pp. 128-141). (ITCS 2012 - Innovations in Theoretical Computer Science Conference). https://doi.org/10.1145/2090236.2090247