Abstract Geometrical Computation 8: Small Machines, Accumulations and Rationality

TL;DR

This paper investigates the conditions under which small signal machines in abstract geometrical computation can produce accumulations, revealing a dichotomy based on rationality of speeds and distances, and linking it to Euclid's algorithm.

Contribution

It characterizes when accumulations can occur with small-speed machines, especially highlighting the role of rational and irrational ratios in their behavior.

Findings

No accumulation with 2 speeds.

Accumulation possible with 4 speeds.

Accumulation depends on rationality of ratios.

Abstract

In the context of abstract geometrical computation, computing with colored line segments, we study the possibility of having an accumulation with small signal machines, ie, signal machines having only a very limited number of distinct speeds. The cases of 2 and 4 speeds are trivial: we provide a proof that no machine can produce an accumulation in the case of 2 speeds and exhibit an accumulation with 4 speeds. The main result is the twofold case of 3 speeds. On the one hand, we prove that accumulations cannot happen when all ratios between speeds and all ratios between initial distances are rational. On the other hand, we provide examples of an accumulation in the case of an irrational ratio between 2 speeds and in the case of an irrational ratio between two distances in the initial configuration. This dichotomy is explained by the presence of a phenomenon computing Euclid's algorithm (gcd): it stops if and only if its input is commensurate (ie, of rational ratio).