Attractor-Based Obstructions to Growth in Homogeneous Cyclic Boolean Automata.

Kirk Dombrowski,Yuri Cantor,Bilal Khan

doi:10.4172/jcsb.1000209

Abstract

We consider a synchronous Boolean organism consisting of N cells arranged in a circle, where each cell initially takes on an independently chosen Boolean value. During the lifetime of the organism, each cell updates its own value by responding to the presence (or absence) of diversity amongst its two neighbours’ values. We show that if all cells eventually take a value of 0 (irrespective of their initial values) then the organism necessarily has a cell count that is a power of 2. In addition, the converse is also proved: if the number of cells in the organism is a proper power of 2, then no matter what the initial values of the cells are, eventually all cells take on a value of 0 and then cease to change further. We argue that such an absence of structure in the dynamical properties of the organism implies a lack of adaptiveness, and so is evolutionarily disadvantageous. It follows that as the organism doubles in size (say from m to 2m) it will necessarily encounter an intermediate size that is a proper power of 2, and suffers from low adaptiveness. Finally we show, through computational experiments, that one way an organism can grow to more than twice its size and still avoid passing through intermediate sizes that lack structural dynamics, is for the organism to depart from assumptions of homogeneity at the cellular level.

Highlights

The subject of cellular automata has received much attention sinceJohn Von Neumann’s seminal work [1] on the dynamics of a grid of cells which evolve in discrete time steps according to rules based on their neighbor’s values
Cellular automata are frequently studied by considering their collective dynamics
A random graph model for automata was introduced by Stuart

Summary

Introduction

John Von Neumann’s seminal work [1] on the dynamics of a grid of cells which evolve in discrete time steps according to rules based on their neighbor’s values (e.g., see the surveys in ref. [2,3]). States” (i.e., states that are unreachable from any other state), as well as determining whether a network can reach a state in which all cells have value 1 (i.e., a question that is known as the “All-Ones Problem”). Miller considered consensus in the standing ovation problem as a means to examine behavior in social networks using computational models [9]. We too (at least initially) consider only cellular automata that are homogenous at the cellular level, that is, cyclic networks in which all cells operate according to an identical update rule. The networks we will consider are so simple as to lack community substructures, and yet always reach consensus regardless of noise This is possible because (as we shall prove) their dynamics exhibit a single unique attractor.

Background

19. We begin with some definitions in Section

5: For each function x

Conclusions

E Stat Nonlin Soft Matter Phys 72