Order Independence in Asynchronous Cellular Automata

A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions F_Y, and a permutation pi of the vertex set (or more generally, a word w over the vertex set) that describes the order in which these local functions are to be applied. In this article we investigate the special case where Y is a circular graph with n vertices and all of the local functions are identical. The 256 possible local functions are known as Wolfram rules and the resulting sequential dynamical systems are called finite asynchronous elementary cellular automata, or ACAs, since they resemble classical elementary cellular automata, but with the important distinction that the vertex functions are applied sequentially rather than in parallel. An ACA is said to be pi-independent if the set of periodic states does not depend on the choice of pi, and our main result is that for all n>3 exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005 Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with this property. In addition to reproving and extending this earlier result, our proofs of pi-independence also provide significant insight into the dynamics of these systems.Comment: 18 pages. New version distinguishes between functions that are pi-independent but not w-independen

Computing Aggregate Properties of Preimages for 2D Cellular Automata

Computing properties of the set of precursors of a given configuration is a common problem underlying many important questions about cellular automata. Unfortunately, such computations quickly become intractable in dimension greater than one. This paper presents an algorithm --- incremental aggregation --- that can compute aggregate properties of the set of precursors exponentially faster than na{\"i}ve approaches. The incremental aggregation algorithm is demonstrated on two problems from the two-dimensional binary Game of Life cellular automaton: precursor count distributions and higher-order mean field theory coefficients. In both cases, incremental aggregation allows us to obtain new results that were previously beyond reach

Complex networks derived from cellular automata

We propose a method for deriving networks from one-dimensional binary cellular automata. The derived networks are usually directed and have structural properties corresponding to the dynamical behaviors of their cellular automata. Network parameters, particularly the efficiency and the degree distribution, show that the dependence of efficiency on the grid size is characteristic and can be used to classify cellular automata and that derived networks exhibit various degree distributions. In particular, a class IV rule of Wolfram's classification produces a network having a scale-free distribution.Comment: 10 pages, 8 figure

Lenia and Expanded Universe

We report experimental extensions of Lenia, a continuous cellular automata family capable of producing lifelike self-organizing autonomous patterns. The rule of Lenia was generalized into higher dimensions, multiple kernels, and multiple channels. The final architecture approaches what can be seen as a recurrent convolutional neural network. Using semi-automatic search e.g. genetic algorithm, we discovered new phenomena like polyhedral symmetries, individuality, self-replication, emission, growth by ingestion, and saw the emergence of "virtual eukaryotes" that possess internal division of labor and type differentiation. We discuss the results in the contexts of biology, artificial life, and artificial intelligence.Comment: 8 pages, 5 figures, 1 table; submitted to ALIFE 2020 conferenc