Abstract geometrical computation 7: geometrical accumulations and computably enumerable real numbers
Using rules to automatically extend a drawing on an Euclidean space might lead to accumulating drawings into a single point. Such points are characterized in the context of Abstract geometrical computation. Colored line segments (traces of signals) are drawn according to rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. Time and space are continuous and accumulations can happen. Constructions exist to unboundedly accelerate a computation and provide, in a finite duration, exact analog values as limits/accumulations. Starting with rational numbers for coordinates and speeds, the time of any isolated accumulation is a c.e. (computably enumerable) real number. There is a signal machine and an initial configuration that accumulates at any c.e. time. Similarly, the spatial positions of isolated accumulations are exactly the d-c.e. (difference of computably enumerable) numbers. Moreover, there is a signal machine that can accumulate at any c.e. time or d-c.e. position depending only on the initial configuration. These existence results rely on a two-level construction: an inner structure simulates a Turing machine that output orders to the outer structure which handles the accumulation.
abstract geometrical computations geometrical accumulations signal machine computable analysis c.e. and d-c.e. real numbers