New properties of 2D Cellular Automata found through Polynomial Cellular Neural Networks

Abstract

In this paper we show how Polynomial Cellular Neural Networks can be used to find new properties of twodimensional binary Cellular Automata (CA). In particular, we define formally a complexity index for totalistic and semitotalistic CA, and we discuss on the intrinsic complexity of universal CA finding a surprising result: universal rules are slightly more complex than linearly separable ones.

Learning in Polynomial Cellular neural networks using quadratic programming

“…Rules of totalistic CA can be conveniently represented in a Cartesian coordinate system. This original representation was proposed in [6], in which the sum of the nine neighbors of the automaton is on the horizontal axis, and the corresponding output on the vertical axis. Given a rule (or equivalently, a truth table), for each of the 10 input patterns we need to depict a red, in case of a firing pattern; blue, in case of a quenching pattern.…”

“…Considering the previous work [6], it is possible to find through a rigorous method the parameters -degree of the polynomial and the weights of the network -of a one-layer space-invariant Polynomial CelIular Neural Network implementing a totalistic CA rule. When implementing a totalistic CA, the mathematical representation can be simplified thanks to two preliminary considerations about the nature of the problem.…”