Minimization of DFA

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,
  1. {q4}, 
  2. {q0,q2}, 
  3. {q1,q3}

State

Input = a

Input = b

->{q0,q2} Initial state

{q1,q3}

{q1,q3}

{q1,q3}

{q0,q2}

{q4}

{q4} Final state

{q4}

{q4}



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