Cellular automaton

A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. It is made up of a grid of small units called cells. Each cell can be in one of a few simple states, like on and off. The grid can have many dimensions, and each cell looks at its nearby neighbors to decide what state to change to next.

The idea of cellular automata began in the 1940s, created by scientists Stanislaw Ulam and John von Neumann while they worked together at Los Alamos National Laboratory. Interest grew much larger in the 1970s thanks to a famous example called Conway's Game of Life. This special kind of cellular automaton shows how simple rules can create complex patterns that change over time.

These systems are useful in many areas, such as physics, theoretical biology, and modeling tiny structures in materials called microstructure. Scientists like Stephen Wolfram have studied these patterns carefully. Some cellular automata can become very complicated and might even be able to perform any calculation a computer can do, which researchers call being computationally universal or able to simulate a Turing machine.

Overview

A cellular automaton is like a grid of squares, where each square can be black or white. The color of each square changes based on the colors of its neighboring squares. The neighbors are usually the squares directly next to it, either just the side squares or also the corner squares. There are many possible ways to decide how the colors change, leading to many different patterns over time.

When we make these grids on a computer, we often use a smaller, finite grid instead of an infinite one. We have to decide what to do with the squares on the edges. One common way is to connect the edges, so the grid wraps around like a tube or a doughnut. This helps avoid special rules just for the edge squares and makes the patterns easier to calculate.

History

Stanisław Ulam studied crystal growth using simple patterns in the 1940s at the Los Alamos National Laboratory. His colleague, John von Neumann, worked on systems that could copy themselves. Together, they developed the idea of cellular automata, a way to model how groups of small units change based on their neighbors.

In later years, many scientists built on this idea. One famous example is the Game of Life, created by John Conway and shared by Martin Gardner in Scientific American. With just a few simple rules, it shows surprising patterns and can even perform calculations. Other researchers, like Stephen Wolfram, explored how simple rules can create complex behaviors in nature.

Classification

Wolfram created a way to group cellular automata into four classes based on how they behave. These classes help us understand the different patterns and behaviors that can emerge from simple rules.

• Class 1: Patterns quickly become uniform and stable. Any randomness in the starting pattern disappears.
• Class 2: Patterns become stable or oscillating structures. Some randomness may remain, but changes stay local.
• Class 3: Patterns behave in a chaotic, unpredictable way. Any stable structures that appear are quickly destroyed.
• Class 4: Patterns develop complex, interacting structures that can last a long time. These systems might be capable of universal computation, meaning they could, in theory, perform any calculation.

These classes are guidelines, and some cellular automata might show features of more than one class. Researchers have tried to create more precise classifications, but some of these remain challenging to define clearly. The idea of grouping systems into four classes originated from studies in chemistry and thermodynamics.

Elementary cellular automata

Main article: Elementary cellular automaton

An elementary cellular automaton is a simple system where each cell can be in one of two states, like on or off. Each cell looks at its two neighbors to decide its next state. Because each cell and its neighbors can be in different combinations, there are 256 possible rules for how these cells change over time.

Some rules, like rule 30, rule 90, rule 110, and rule 184, are especially interesting. Rule 30 creates patterns that look random and chaotic. Rule 110, however, creates patterns that are neither fully random nor fully repeating, allowing complex structures to form and interact in surprising ways. This rule shows that even very simple systems can perform complex calculations.

Rule space

A cellular automaton rule is like a set of instructions that tells each cell what to do next based on its neighbors. These rules can be thought of as points inside a shape called a hypercube. For simple rules, this shape has 8 sides, and for more complex ones, it has 32 sides.

We can measure how different two rules are by counting the steps needed to move from one point to another in this hypercube. This helps us see if rules that behave similarly are close together. Scientists have noticed that some rules that stay mostly the same are found in one area, while others that change a lot are found in another. This idea is linked to something called the "edge of chaos," which is similar to how changes happen in the study of heat and energy.

Applications

Further information: Patterns in nature

Cellular automata can simulate many natural processes. For example, the colorful patterns on some seashells are created by natural cellular automata. Each pigment cell on a seashell follows simple rules based on its neighbors, producing beautiful designs as the shell grows. Plants also use a similar process to control tiny openings on their leaves that let gases in and out.

In chemistry, cellular automata can mimic reactions that create swirling patterns, similar to those seen in certain chemical mixtures. In physics, these models help scientists study how materials change under different conditions, like how magnets lose their magnetism when heated.

Cellular automata are also used in computer science for tasks like creating random numbers and solving certain types of problems. They have even been used to make music and generate landscapes in video games.

Specific rules

Some well-known cellular automata rules include Conway's game of life, Langton's ant, and Rule 90. These rules help create interesting patterns and simulations in a grid of cells. Other notable examples are Brian's Brain, Langton's loops, and Wireworld, each offering unique ways to explore how simple rules can lead to complex behavior.