Cellular Automata in Automata Theory (2025)

Cellular Automata are used in various fields such as computer science, mathematics, physics, and biology. It was originally proposed by Stanislaw Ulam and John von Neumann in the 1940s.

Cellular automata provide a simple and also powerful way to simulate complex systems. In this chapter, we will see the basics of cellular automata, discuss their key characteristics, and walk through a detailed example for a better understanding.

Cellular Automata

Cellular Automata consist of a grid of cells, where each cell can be in one of a finite number of states, usually represented as 0 or 1. The grid can be one-dimensional or multi-dimensional.

In this chapter, we will focus on one-dimensional cellular automata first for simplicity. The state of each cell evolves over time according to a set of rules that consider the states of neighbouring cells.

Key Concepts of Cellular Automata

Here are some Key Concepts of Cellular Automata −

• Cells and Grid − The basic unit of a cellular automaton is the cell. Cells are arranged in a grid, which can be one-dimensional (a line), two-dimensional (a plane), or higher dimensions. Each cell has a state, which is updated based on the states of its neighbouring cells.
• Neighbourhood − The neighbourhood of a cell consists of the cell itself and its adjacent cells. In a one-dimensional cellular automaton, a cell typically has two neighbours: one on the left and one on the right. The neighbourhood can also include more distant cells depending on the rules.
• Rules − The evolution of the cellular automaton is controlled by a set of rules. These rules finds the next state of a cell based on the current state of the cell and the states of its neighbours. The rules are applied simultaneously to all cells, and gives a synchronous updates across the grid.
• Discrete Time − Cellular automata evolve in discrete time steps. At each time step, the state of every cell is updated according to the rules, and the entire grid transitions to a new configuration.

Characteristics of Cellular Automata

Cellular automata have several important characteristics that make them unique and useful for modelling complex systems −

• Synchronous Computation − All cells in the grid update their states simultaneously at each time step. This ensures that the evolution of the automaton is uniform and predictable.
• Finite States − Each cell can exist in a finite number of states, usually two (0 or 1), but more states are possible depending on the application.
• Local Interactions − The state of each cell depends only on its own state and the states of its immediate neighbours. This local interaction can lead to complex global behaviour in the automaton.
• Deterministic or Non-Deterministic − Cellular automata can be either deterministic, where the rules produce a single predictable outcome, or non-deterministic, where the rules may allow multiple possible outcomes.

Example of One-Dimensional Cellular Automata

Let us see with a basic example to understand how a one-dimensional cellular automaton works. We will use a simple rule-based system where each cell can be in one of two states: 0 or 1.

The Update Rule

In a one-dimensional cellular automaton, each cell has three neighbours: the cell itself, the cell to its left, and the cell to its right. The next state of a cell is determined by applying a rule to the current states of these three cells.

There are 256 possible rules for a one-dimensional CA with binary states (0 and 1). Each rule can be represented by a binary number, where each bit in the number specifies the next state of the cell for a particular neighbourhood configuration.

In null stuffing 0 is stuffed at both sides of the original sequence. This is described in the following table.

For example, consider the rule 210. The binary equivalent of 210 is 11010010. This binary number corresponds to the next states for each possible neighbourhood configuration, as shown below −

The cell representation of 111 and 010 is like the following −

Applying the Rule

Let us apply this rule to an initial configuration of cells. Suppose we start with the following sequence −

Initial Sequence (t0): 1001

Now, we will calculate the next three-time steps (t1, t2, t3) using the wrap-around technique. In this technique, the sequence is considered circular, meaning that the first and last cells are treated as neighbours.

• Time Step 1 (t1) − Neighbourhoods: 110, 100, 001, 011. Next State: 1, 1, 1, 0, Result: 1110
• Time Step 2 (t2) − Neighbourhoods: 011, 110, 101, 011, Next State: 0, 1, 1, 0, Result: 0110
• Time Step 3 (t3) − Neighbourhoods: 001, 010, 100, 000, Next State: 1, 0, 1, 1, Result: 1011

Thus, the evolution of the cellular automaton over three timesteps from the initial sequence 1001 is −

• t0: 1001
• t1: 1110
• t2: 0110
• t3: 1011

After null stuffing, it will be −

• t0: 010010
• t1: 001100
• t2: 010110
• t3: 000010

This example shows how a simple set of rules can produce a sequence of states over time in a cellular automaton.

Applications of Cellular Automata

Let us see some of the Cellular automata applications in various fields −

Computer Science

• Cryptography − Cellular automata are used to design secure cryptographic systems by exploiting their complex and unpredictable behaviour.
• Fault Tolerance − In digital circuits, cellular automata can help detect and correct errors by simulating the behaviour of faulty components.

Physics

• Simulation of Gas Behaviour − Cellular automata can model the behaviour of gases, allowing researchers to study properties like diffusion and pressure.
• Forest Fire Propagation − By simulating the spread of fire through a forest, cellular automata can help predict and manage wildfire outbreaks.

Biology

• Bone Erosion Simulation − Cellular automata can simulate biological processes like bone erosion, helping in medical research and treatment planning.

Conclusion

In this chapter, we covered the basic structure of cellular automata including cells, neighbourhoods, rules, and discrete time steps. We also seen a detailed example of a one-dimensional cellular automaton using rule 210. Finally, we discussed the various applications of cellular automata in fields such as computer science, physics, and biology.