Iota-Delta Function for Generating Cellular Automata

Universality has always played a major role in every branch of science. In particular, universality in cellular automata (CA) and computational systems has been explored extensively, especially the elementary CA (ECA) rule 110 and its properties. Instead of considering universal rules, one may think of universal functions that, when applied to the cellular automaton net, generate every CA. This approach asks that one imagine the cellular automaton net as a functional space whose values are determined by means of the conjectured function and its parameters. Following this idea, different functional parameters should generate different CAs. This universal function, the iota-delta function, has been proved to exist and is defined as follows:

.

In the past, only integer arguments of the iota-delta function were analyzed. The function itself represents a filtering process in which the mod or remainder operator is applied with respect to prime numbers. In order to represent the CA, values are interpreted as colors. In an ECA, which takes only black and white colors, binary values (1 and 0) are used and , as the outputs of the function for integer arguments are governed by the last mod operator applied. It has been shown that the ECA can be easily generated as a linear combination of three cells from a past step plus a constant when is taken as 5 and as 2. Thus, the value of a given cell at time step and position can be readily given as

=.

The functional structure of this equation shows that it can be extended to cover not only the ECA described by Wolfram [3], but also other CA that depend on not only integer arguments of the iota-delta function, but also on real parameters. Besides that, one may build CA rules that do not depend only on the linear combination of the cells in the past step, but also on their products.