A spatio-temporal algorithmic point of view on firing squad synchronisation problem

Abstract

Firing Squad Synchronization Problems are well known to be solvable by voluminous transition tables describing signals traveling and colliding. In this paper, we show that it is possible to solve it by expressing directly the fact that we want a recursive division of the space into two parts of equal size, and a notification when no further division is possible. Using fields - objects associating a value to every point in space and time - as primitive objects, the solution is designed algorithmically by a semantically-intuitive decomposition of the global evolution into simpler evolutions. The system we obtain has several interesting characteristics: it is understandable, time-optimal, tackles many initial configurations, and allows a new interpretation of the traditional signals and collisions point of view. We will quickly sketch how we can obtain a finite state automaton by reduction of the system using the Lipschitz-continuity of involved fields, and a kind of tail-recursivity property of the dependencies. © 2012 Springer-Verlag Berlin Heidelberg.