Minimization of DFA

The DFA minimization is the process of reducing states in a deterministic finite automaton (DFA) and maintaining its language recognition abilities.

That means, DFA minimization is aimed at finding a DFA with the least number of states that can recognize the same language as the original DFA.

Some of the benefits of minimizing DFA:

• Reduced memory usage: DFAs with fewer states require less memory to store. This can be important for applications where memory usage is a constraint.
• Improved computational efficiency: DFAs with fewer states can process strings more quickly. This can be important for applications where processing speed is a concern.
• Enhanced understanding: DFAs with fewer states are generally easier to understand and analyze. This can be helpful for debugging and maintaining DFAs.
• Simplified hardware implementation: DFAs with fewer states are more amenable to hardware implementation. This can be important for applications where performance is critical.

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Example of DFA minimization:

Construct a minimum state automata equivalent to given automata?

(RGPV 2008)

Solution:

Transition table for above automata.

State	Input = a	Input = b
->q0 Initial state	q1	q3
q1	q2	q4
q2	q1	q1
q3	q2	q4
q4 Final state	q4	q4

Step 01: Remove steps which are unreachable from initial states.

Step 02: Split final states and non final states.

• A0 = {q4}
• A1 = {q0,q1,q2,q3}
• π0 = {q4}, {q0,q1,q2,q3}
• A0 cannot be partition further.

In A1,

• q0 is 1 equivalent to q2 for input a, but not equivalent to q1 and q3.
• q1 is 1 equivalent to q3 for input a and b, but not to q0 and q2.

So, A1 can be partitioned as,

• B0 = {q0, q2}
• B1 = {q1, q3}
• π1 = {q4}, {q0,q2}, {q1,q3}

Now, B0 and B1 can not be partitioned further.

• π2 = {q4}, {q0,q2}, {q1,q3}
• π2 = π1

In minimized DFA, we have three states,

• {q4},
• {q0,q2},
• {q1,q3}

Minimized DFA:

State	Input = a	Input = b
->{q0,q2} Initial state	{q1,q3}	{q1,q3}
{q1,q3}	{q0,q2}	{q4}
{q4} Final state	{q4}	{q4}

Reference:

• Introduction to the Theory of Computation” by Michael Sipser.