Inaccessibility and undecidability in computation, geometry, and dynamical systems

Abstract

Non-self-similar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to (ideal) decision procedure, that is connected with undecidability. A halting set of a universal Turing machine (UTM), the Mandelbrot set and a riddled basin are mainly investigated as non-self-similar sets with a decision procedure. By encoding a symbol sequence to a point in a Euclidean space, a halting set of a UTM is shown to be geometrically represented as a non-self-similar set, having different patterns and different fine structures on arbitrarily small scales. The boundary dimension of this set is shown to be equal to the space dimension, implying that the ideal decision procedure is inaccessible in the presence of error. This property is shown to be invariant under application of “fractal” code transformations. Thus, a characterization of undecidability is given by the inaccessibility to the ideal decision procedure and its invariance against the code transformations. It is also shown that the distribution of halting time of the UTM, decays with a power law (or slower), and that this characteristic is also unchanged under code transformation. The Mandelbrot set is shown to have these features including the invariance against the code transformation, in common, and is connected with undecidable sets. In contrast, although a riddled basin, as a geometric representation of a certain context-free language, has the boundary dimension equal to the space dimension and a power law halting time distribution, these properties are not invariant against the code transformation. Thus, the riddled basin is ranked as middle between an ordinary fractal and a halting set of a UTM or the Mandelbrot set. Last, we propose undecidability for analog computation, and discuss the appropriate condition for coding.