Abstract Geometrical Computation 11: Slanted Firing Squad Synchronisation on Signal Machines

TL;DR

This paper introduces a signal machine that achieves synchronization on any non-infinite slope by encoding the slope in the initial configuration, advancing the study of continuous geometrical computation.

Contribution

It designs a novel signal machine capable of synchronizing on arbitrary slopes, expanding the understanding of accumulation lines in continuous space.

Findings

Successfully encodes slope in initial configuration

Constructs an infinite tree for expansion computation

Provides tools for studying accumulation lines in signal machines

Abstract

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.