Abstract Geometrical Computation 11: Slanted Firing Squad Synchronisation on Signal Machines
This work provides basic tools for studying computable accumulation lines in the signal machine model, addressing challenges in continuous space for researchers in abstract geometrical computation.
The paper tackles the problem of synchronizing finitely many cells in continuous space using signal machines, achieving accumulation on any non-infinite slope encoded in the initial configuration. It constructs an infinite tree where each node computes the tree's expansion to enable this synchronization.
Firing Squad Synchronisation on Cellular Automata is the dynamical synchronisation of finitely many cells without any prior knowledge of their range. This can be conceived as a signal with an infinite speed. Most of the proposed constructions naturally translate to the continuous setting of signal machines and generate fractal figures with an accumulation on a horizontal line, i.e. synchronously, in the space-time diagram. Signal machines are studied in a series of articles named Abstract Geometrical Computation. In the present article, we design a signal machine that is able to synchronise/accumulate on any non-infinite slope. The slope is encoded in the initial configuration. This is done by constructing an infinite tree such that each node computes the way the tree expands. The interest of Abstract Geometrical computation is to do away with the constraint of discrete space, while tackling new difficulties from continuous space. The interest of this paper in particular is to provide basic tools for further study of computable accumulation lines in the signal machine model.