No of states in Minimal DFA

Question

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].

Comments

when you draw transaction table you get 8 states out of which 4 states are equal states, so that MDFA contain 4 states.

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what does binary number string mean here?

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Binary number string means strings can form using only the input 0,1.(strings having all combination of 0,1)

Answer 1

$4$ states.

Comments

there is a flaw in ur state ur states also accepts the binary numbers divisible by 16 {ends with 4 0's}

to make it correct remove the loop on final state with 0 and make the transaction to initial state like {state-4 ----0------>initial state}

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No, It is perfectly right because the numbers divisible by 16 will also be divisible by 8. If we remove the loop on the final state and change the transition to initial state on 0 then "10000" will not be accepted but it should be accepted because it is divisible by 8

Answer 2

it is in the form of 2^n

then minimal DFA contain n+1 states

here 4 states

suppose if the number is in odd like 3 ,5,7,9..

minimal DFA contain n states

it is valid only for the binary string

Answer 3

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 Q_r.

from table 4 equal states such as Q₀ =Q₄ ,Q₁ =Q₅ ,Q₂ =Q₆,Q₃ =Q_7.

So MDFA contain 4 states.

Comments

Thanks for the answer :)

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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.

Thanks