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Ques:-  Let ∑= {0, 1} What will be the number of states in minimal DFA, if the Binary number string is congruent to (mod 8)?

*[ Can anybody explain this as I am getting 8 states for this since remainders will be 8 (0,1,2,3,4,5,6,7). But the answer is 4].
asked in Theory of Computation by (41 points) | 65 views
when you draw transaction table you get 8 states out of which 4 states are equal states, so that MDFA contain 4 states.
what does binary number string mean here?
Binary number string means strings can form using only the input 0,1.(strings having all combination of 0,1)

2 Answers

+2 votes


$4$ states.

answered by Junior (507 points)
0 votes

MDFA, for binary strings congruent to (mod 8) for input $\sum$(0,1) as follow:

transcation table:

                   states                      0                     1
                     Q0                      Q0                    Q1
                     Q1                      Q2                    Q3
                     Q2                      Q4                    Q5
                     Q3                      Q6                    Q7
                     Q4                      Q0                    Q1
                      Q5                      Q2                    Q3
                      Q6                      Q4                    Q5
                      Q7                      Q6                    Q7

In here, binary string r(mod n),number of states is depends on remainder & final state is Qr.

from table 4 equal states such as Q0 =Q4 ,Q1 =Q5 ,Q2 =Q6,Q3 =Q7.

So MDFA contain 4 states.

answered by Boss (12.6k points)
Thanks for the answer :)

Hi Thanks for the answer but recently I had discovered a flaw in your answer.


If you use 'Minimization of DFA' technique where we make 0equivalence,1equivalence states. You will not get q0 = q4, q1=q5,... as you have mentioned.

Your answer is correct but not the method.

The minimized DFA states after 'Minimization of DFA' are [q0],[q4],[q1,q3,q5,q7],[q2,q6].
where q0 is initial and final state and according to these minimized states we can say that "q0 and q4 are independent" and "q1,q3,q5,q7 are equal" and also "q2 and q6 are equal".

Please correct me if I am wrong.



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