Finite-State Machine Model for Cell Movement with Color Constraints (MATLAB Example)

Finite-State Machine Model for Cell Movement

Suppose a game starts in cell 1 and the movement rules are as follows:

• If the cell's color is 'red', the new state can be any cell adjacent to the current state that is also 'red'.* If the cell's color is 'green', the new state can be any cell adjacent to the current state that is also 'green'.

To capture these rules, we can define a finite-state machine (FSM) model with the following components:

• State Set (Q): This represents the possible states of the system.* Input Alphabet (∑): This represents the possible inputs to the system.* Transition Function (δ): This maps a current state and an input to a new state.

Defining the FSM Components

• Q = {1, 2, 3, 4, 5, 6} (representing the cells)* ∑ = {'red', 'green'} (representing the cell colors)* Transition Function (δ):

The transition function can be represented as a matrix in MATLAB where:

• Rows represent the current state.* Columns represent the input.* Matrix elements represent the next state.

**MATLAB Code for Transition Function (δ)**matlabdelta = [ [2, 4; 3, 5]; [1, 4; 3, NaN]; [2, NaN; 1, 5]; [1, 6; 2, 5]; [3, 6; 1, 4]; [4, 5; 3, NaN] ];

Explanation of the Transition Function Matrix:

• delta(1, 'red') = [2, 4] means that if the current state is 1 and the cell color is 'red', the next state can be either 2 or 4 (both adjacent and 'red').* delta(2, 'green') = 3 means that if the current state is 2 and the cell color is 'green', the next state can only be 3 (adjacent and 'green').* delta(3, 'red') = 2 means that if the current state is 3 and the cell color is 'red', the next state can only be 2 (adjacent and 'red').* delta(6, 'green') = 3 means that if the current state is 6 and the cell color is 'green', the next state can only be 3 (adjacent and 'green').

Note: NaN (Not a Number) values in the matrix indicate that there is no valid next state for that specific state and input combination.

This FSM model effectively captures the cell movement rules based on color, and the transition function represented as a matrix in MATLAB makes it easy to implement and analyze the system's behavior.