Noise vs computational intractability in dynamics
Abstract
Computation plays a key role in predicting and analyzing natural phenomena. There are two fundamental barriers to our ability to computationally understand the longterm behavior of a dynamical system that describes a natural process. The first one is unaccountedfor 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 Turingcompleteness. By the undecidability of the Halting Problem, the longterm prospects of a system that can simulate a Turing Machine cannot be determined computationally. We investigate the interplay between these two forces  unaccountedfor errors and Turingcompleteness. We show that the introduction of even a small amount of noise into a dynamical system is sufficient to "destroy" Turingcompleteness, and to make the system's longterm behavior computationally predictable. On a more technical level, we deal with longterm 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 noncomputable, 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 arXiv:
 arXiv:1201.0488
 Bibcode:
 2012arXiv1201.0488B
 Keywords:

 Computer Science  Computational Complexity;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Chaotic Dynamics;
 37C20;
 37C40;
 H.1.1;
 F.1.1;
 F.1.3
 EPrint:
 ITCS 2012. 37 pages, 1 figure