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# What is Minimization of DFA?

## Problem

Given a Deterministic Finite Automata (DFA), try to reduce the DFA by removing unreachable states and removing similar rows.

## Solution

### Step 1

Remove the unreachable states from q0

From the initial states, we are not able to reach q2 and q4. So, remove these two states as shown below −

After removing unreachable states, the partial minimized DFA is as follows −

### Step 2

The **transition table** is given below −

States | 0 | 1 |
---|---|---|

->q0 | q1 | q3 |

q1 | q0 | q3 |

*q3 | q5 | q5 |

*q5 | q5 | q5 |

### Step 3

Divide tables into 2 tables as shown below −

**Table 1** starts from the non-final states.

States | 0 | 1 |
---|---|---|

->q0 | q1 | q3 |

q1 | q0 | q3 |

**Table 2** starts from the final states.

States | 0 | 1 |
---|---|---|

*q3 | q5 | q5 |

*q5 | q5 | q5 |

### Step 4

Remove similar rows.

Table 1 has no similar rows

Table 2 has similar rows. So, skip q5 and replace q5 by q3

States | 0 | 1 |
---|---|---|

q3 | q5 | q3 |

### Step 5

Combine two tables as shown below −

States | 0 | 1 |
---|---|---|

->q0 | q1 | q3 |

q1 | q0 | q3 |

*q3 | q3 | q3 |

Thus, the **minimized DFA** will be as follows −

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