What does the formula Cell state_t = f(neighborhood states_{t-1}) mean?

Conditions

• $t$ represents the current discrete time step or generation index.
• $t-1$ represents the previous discrete time step.
• $f$ is the transition rule mapping previous neighborhood states to the next cell state.

Reasoning, step by step

1. Identify $\text{Cell state}_t$ as the state of a cell at time $t$.
2. Identify $\text{neighborhood states}_{t-1}$ as the collection of states belonging to neighboring cells at time $t-1$.
3. Recognize $f$ as the update function or rule that maps the previous neighborhood states to the current cell state.
4. Combine these to understand that the next state is a function of the previous neighborhood states.

Example

The whiteboard shows the equation "Cell state_t = f(neighborhood states_{t-1})", and the speaker explains the current cell state is "equal to a function of the neighborhood states at time t minus one".

Common misconceptions

• Thinking that $f$ is a fixed arithmetic operation like addition, whereas it can be any rule that inspects patterns.
• Assuming the formula applies to continuous time, rather than discrete generations.

Watch the explanation

Explore next

• How do generations over time work in the cellular automaton described here?
• Why does each cell need a neighborhood in a cellular automaton?
• What is the definition of a cellular automaton given in this clip?