Noise vs computational intractability in dynamics

TL;DR

This paper explores how small amounts of noise can simplify the long-term behavior prediction of complex dynamical systems, transforming intractable problems into computationally manageable ones.

Contribution

It demonstrates that introducing noise into Turing-complete systems can make their long-term statistical properties computable, revealing a trade-off between unpredictability and tractability.

Findings

Noise destroys Turing-completeness in dynamical systems.

Perturbing systems makes their invariant measures computable.

Noise can make long-term statistical behavior predictable.

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.