Wednesday, March 29, 2023

Cellular Automata and Turing Machines

MEXICO CITY (proceso.com.mx).-Cellular automata are a model that makes it possible to study some complex systems. They are basically about “cells” that have certain rules for reproducing over time. Their behavior is normally analyzed through computer simulations, which show how the automata’s generation changes at each tick of the clock.

The simplest automata to study are those in one dimension. For example, imagine a series of cells in a row that are next to each other. It may happen that some cells are empty while others contain cells. Once this occurs, the growth rules for this initial generation (called the zero generation) are defined. And then we can say: If there are two cells on each side of the cell of interest, this central cell dies due to overpopulation. Be careful, the term “overpopulation” here only applies in the context of these automata. Another rule is that if a cell exists next to the cell of interest, the cell of interest is kept alive for the next generation.

There are some 256 possible rules that can be applied and it is suddenly surprising that when drawing subsequent generations, one below the other, a complex system is observed, even with random planning, a A matter that is not easy to explain. Stephen Wolfram, the creator of the Mathematica software, has worked on this topic for more than 20 years. In his book (available for free at https://www.wolframscience.com/nks/), he discusses all kinds of automata in one dimension and shows interesting results. For example, adding cells of different colors (so that, for example, there are more possible combinations in reproduction), does not add complexity to the study of these automata.

Similarly, Wolfram decides to modify the behavior of automata by changing the way they are displayed on a computer screen and suddenly finds other behaviors that are complex and, again, resist analysis. In fact, there comes a time when automata are defined on the basis of substitution rules, that is, an automaton can become one or more cells according to some rules and then what we get is a formal automaton which is a Complies with the grammar, which is nothing more than one or more replacement rules.

Although Wolfram puts it in the context of a graphical cell simulation, it is clearly a grammar, eg: A is replaced with AB and B is replaced with BA. So if our automaton starts with A, we can do the following substitutions:

a to ab

AB to ABBA

ABBA to ABBABB

If we count the elements that are produced in each replacement, we will find that there are twice as many elements as in the previous generation. And this is, after all, a way to produce results twice from a very simple symbol substitution grammar. In fact, Wolfram makes another substitution of elements that allows us to calculate the Fibonacci series. And suddenly, we find ourselves with the following result: Cellular automata are Turing machines, which are defined as abstract machines that allow any computation. it speaks to the power of automata

In fact, thanks to automata we can model how some animals have pigmented skin, such as zebras and some types of snakes. Even in one dimension cellular automata can be seen pigmented in snail shells. And yes, in this case a pattern isn’t as clear as in the simulation, but we can’t deny the similarities in these dyes against what we see on a computer screen.

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