Undecidability inRⁿ: Riddled Basins, the KAM Tori, and the Stability of the Solar System

Abstract

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in l (or d-l) for any measure l, which is particularly appropriate for physics and in some ways more intuitive than Ko's (1991) recursive approximability (r.a.). For Lebesgue measure k, d-k implies r.a. Sets with positive kmeasure that are sufficiently "riddled" with holes are never d-k but are often r.a. This explicates Sommerer and Ott's (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-k.

Computing the uncomputable; or, The discrete charm of second-order simulacra

“…In particular, no riddled set with positive measure is d.m.z. (Parker 2003(Parker , 2005. Every decision procedure for such a riddled, positive-measure set will fail in a significant portion of cases.…”

Three concepts of decidability for general subsets of uncountable spaces

“…Grzegorczyk-computable functions are continuous [6,7], and the only subsets of R n that have continuous characteristic functions are R n and ∅. However, some sets of reals are intuitively more computable than others, and accordingly, various relaxed notions of a decidable or recursive set of reals have been introduced [4,9,10,[12][13][14][15]21,24] (see [24] for other references). Unlike the several concepts of decidability over N introduced in the 1930s, most of these concepts are far from equivalent.…”

“…However, in applications one may encounter sets that are not open, closed, convex, or regular, or one may simply not know whether a given set has such properties. Hence it also seems useful to discuss the computability of arbitrary sets in R n , as do Ko and others [2,3,9,10,[13][14][15].…”

“…[13], Ker-I Ko's recursive approximability (r.a.) [10], and decidability up to measure zero, or decidability "mod zero" (d.m.z.) [14]. 3 As defined here, these notions apply to subsets of any separable metric space with a dense computable sequence and equipped with an outer measure (a generalized notion of length, area, or n-dimensional volume).…”

Three topics in the theory of computing: Multi-resolution cellular automata, the Kolmogorov complexity characterization of regular languages, and hidden variables in Bayesian networks

“…We note that Myrvold [36] and Parker [40] "decidability up to measure zero" requires the set of points not correctly "decided" to be small in a measure-theoretic sense, while we require it to be small in a topological sense.…”