### 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 shall discuss 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 language | English (US) |
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Title of host publication | The Nature of Computation |

Subtitle of host publication | Logic, Algorithms, Applications - 9th Conference on Computability in Europe, CiE 2013, Proceedings |

Number of pages | 1 |

DOIs | |

State | Published - Jul 26 2013 |

Event | 9th Conference on Computability in Europe - The Nature of Computation: Logic, Algorithms, Applications, CiE 2013 - Milan, Italy Duration: Jul 1 2013 → Jul 5 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7921 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th Conference on Computability in Europe - The Nature of Computation: Logic, Algorithms, Applications, CiE 2013 |
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Country | Italy |

City | Milan |

Period | 7/1/13 → 7/5/13 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*The Nature of Computation: Logic, Algorithms, Applications - 9th Conference on Computability in Europe, CiE 2013, Proceedings*(Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7921 LNCS). https://doi.org/10.1007/978-3-642-39053-1_4