The coordinates of isolated accumulations are exactly computable real numbers
Abstract
In Abstract geometrical computation, Turing computability is provided by simples machines involving drawing colored line segments, called signals, accordin g to simple rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. These signal machines also provide a very powerful model of analog computation following both the approaches of computable analysis and of Blum, Shub and S male. The key is that accumulations can be devised to accelerate the computation and provide an exact analog values as limits in finite time. In the present paper, we show that starting with rational numbers for coordinates and speeds, the collections of positions of accumulations in both space and time are exactly the computable real numbers (as defined by computable analysis). Moreover, there is a signal machine that can provide an accumulation at any computable place and date.