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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.

Example of DFA minimization:

Construct a minimum state automata equivalent to given automata?

(RGPV 2008)


Transition table for above automata.

StateInput = aInput = b
->q0 Initial stateq1q3
q4 Final stateq4q4

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:

StateInput = aInput = b
->{q0,q2} Initial state{q1,q3}{q1,q3}
{q4} Final state{q4}{q4}


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