Undecidable Long-term Behavior in Classical Physics: Foundations, Results, and Interpretation
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The behavior of some systems is non-computable in a precise new sense. One infamous problem is that of the stability of the solar system: Given the initial positions and velocities of several mutually gravitating bodies, will any eventually collide or be thrown off to infinity? Many have made vague suggestions that this and similar problems are undecidable: no finite procedure can reliably determine whether a given configuration will eventually prove unstable. But taken in the most natural way, this is trivial. The state of a system corresponds to a point in a continuous space, and virtually no set of points in space is strictly decidable. A new, more pragmatic concept is therefore introduced: a set is decidable up to measure zero (d.m.z.) if there is a procedure to decide whether a point is in that set and it only fails on some points that form a set of zero volume. This volume and probability: we can ignore a zero-volume set of states because the state of an arbitrary system almost certainly will not fall in that set. D.m.z. is also closer to the intuition of decidability than other notions in the literature, which are either less strict or apply only to special sets, like closed sets. Certain complicated sets are not d.m.z., most remarkably including the set of known stable orbits for planetary systems (the KAM tori). This suggests that the stability problem is indeed undecidable in the precise sense of d.m.z. Carefully extending decidability concepts from idealized models to actual systems, we see that even deterministic aspects of physical behavior can be undecidable in a clear and significant sense.