Pattern synchronization in two-dimensional cellular spaces

This paper presents an algorithm for synchronizing (firing) an arbitrary, finite, connected pattern of cells in a potentially infinite two-dimensional grid of identical finite state cells. Earlier solution times have been quadratic in m, where m is the edge length of the smallest square enclosing the pattern. A linear solution is formulated

The Firing Squad Problem Revisited

In the classical firing squad problem, an unknown number of nodes represented by identical finite state machines is arranged on a line and in each time unit each node may change its state according to its neighbors\u27 states. Initially all nodes are passive, except one specific node located at an end of the line, which issues a fire command. This command needs to be propagated to all other nodes, so that eventually all nodes simultaneously enter some designated ``firing" state. A natural extension of the firing squad problem, introduced in this paper, allows each node to postpone its participation in the squad for an arbitrary time, possibly forever, and firing is allowed only after all nodes decided to participate. This variant is highly relevant in the context of decentralized distributed computing, where processes have to coordinate for initiating various tasks simultaneously. The main goal of this paper is to study the above variant of the firing squad problem under the assumptions that the nodes are infinite state machines, and that the inter-node communication links can be changed arbitrarily in each time unit, i.e., are defined by a dynamic graph. In this setting, we study the following fundamental question: what connectivity requirements enable a solution to the firing squad problem? Our main result is an exact characterization of the dynamic graphs for which the firing squad problem can be solved. When restricted to static directed graphs, this characterization implies that the problem can be solved if and only if the graph is strongly connected. We also discuss how information on the number of nodes or on the diameter of the network, and the use of randomization, can improve the solutions to the problem

Simulación de fenómenos promotores del desarrollo del pensamiento matemático

Actualmente existen investigaciones que estudian modelos que exhiben dinámicas descentralizadas; por ejemplo, la propagación de una epidemia, el comportamiento de las variables económicas, fenómenos ecológicos y la dinámica de ciudades, entre otros. Este taller pretende que los profesores de nivel secundaria, medio superior y superior utilicen el software NetLogo para modificar los parámetros de un modelo referente a la dinámica de interacción de un grupo de organismos con la finalidad de fomentar la discusión de nociones, que se encuentran intrínsecas en la dinámica, como son la autoorganización, el caos y la estabilidad y son promotoras del principio de la estabilidad del cambio que bien puede ser un modo de pensar y hacer matemáticas que se encuentra en ámbitos intra y extra escolares

A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata

The firing squad synchronization problem (FSSP) on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional arrays but two-dimensional arrays. In the present paper, we propose a simple recursive-halving based optimum-time synchronization algorithm that can synchronize any rectangle arrays of size m*n with a general at one corner in m+n+max(m, n)-3 steps. The algorithm is a natural expansion of the well-known FSSP algorithm proposed by Balzer [1967], Gerken [1987], and Waksman [1966] and it can be easily expanded to three-dimensional arrays, even to multi-dimensional arrays with a general at any position of the array.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249