Irrationality is needed to compute with signal machines with only three speeds

International audienceSpace-time diagrams of signal machines on finite configurations are composed of interconnected line segments in the Euclidean plane. As the system runs, a network emerges. If segments extend only in one or two directions, the dynamics is finite and simplistic. With four directions, it is known that fractal generation, accumulation and any Turing computation are possible. This communication deals with the three directions/sp eeds case. If there is no irrational ratio (between initial distances between signals or between speeds) then the network follows a mesh preventing accumulation and forcing a cyclic behavior. With an irrational ratio (here, the Golden ratio) between initial distances, it becomes possible to provoke an accumulation that generates infinitely many interacting signals in a bounded portion of the Euclidean plane. This b ehavior is then controlled and used in order to simulate a Turing machine and generate a 25-state 3-speed Turing-universal signal machin

The Euclid Abstract Machine

He took the golden Compasses, prepar’d In Gods Eternal store, to circumscribe This Universe, and all created things: One foot he center’d, and the other turn’d Round through the vast profunditie obscure, And said, thus farr extend, thus farr thy bounds, This be thy just Circumference, O World.

The euclid abstract machine: Trisection of the angle and the halting problem

Concrete non-computable functions are usually re- lated to the halting function. Is it possible to present examples of non-computability, which are unrelated to the halting prob- lem and its derivatives? We built an abstract machine based on the historic concept of compass and ruler constructions (a com- pass construction would suffice) which reveals the existence of non-computable functions not related with the halting problem. These natural, and the same time, non-computable functions can help to understand the nature of the uncomputable and the pur- pose, the goal, and the meaning of computing beyond Turing