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Consider a DFA over $\Sigma=\{a,b\}$ accepting all strings which have number of a's divisible by $6$ and number of $b$'s divisible by $8$. What is the minimum number of states that the DFA will have?

1. $8$
2. $14$
3. $15$
4. $48$
edited | 5k views

Answer is (D). It can be proved using Myhill Nerode theorem. We need a separate state for $\#a \ mod \ 6 = 0..5$ and $\#b \ mod \ 8 = 0..7$. Each combination of them must also be a new state giving $6*8 = 48$ minimum states required in the DFA.

Reading Myhill-Nerode theorem might be confusing though it is actually very simple.

http://courses.cs.washington.edu/courses/cse322/05wi/handouts/MyhillNerode.pdf

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what would be an answer if I change the given condition to "number of a's divisible by 6 or number of b's divisible by 8"
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same 48 states ..no of final states increased and .. if they asked for same symbol then we have to take lcm
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if they asked for same symbol then we have to take lcm

explain this  sid1221

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We can take Cartesian product of given two dfas M1 & M2, accepting the languages L1 and L2 respectively where:

L1={w over Σ={a,b} accepting all strings which have number of a's divisible by 6}, say has 'm' states

L2={w over Σ={a,b} accepting all strings which have number of b's divisible by 8}, say has 'n' states

Step 1: Take Cartesian product and create a dfa M with m*n # of states. (total # of states)

Step 2: The difference of AND/OR comes on basis of selection of final states in M..

If OR(means UNION) so in M choose all states as final states wherever final states of EITHER M1 or M2 occurs..

If AND(means INTERSECTION) so in M choose all states as final states wherever final states of BOTH M1 and M2 occurs..

So by question, M1 has 6 states, M2 has 8 states. Final dfa has 6*8=48 states.(only total # of states is asked here and it will be minimum).

Please correct me if I am wrong.
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Can't this be done in 24 states as mod 8 dfa can be done in 4 states(since minimum is asked) and mod 6 in 6 states. So, 6x4 -> 24 states. Is 24 not considered because it's not in option .Is it the case ?
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@Spandan, how mod 8 dfa can be done in 4 states?
Answer is D: 6 states for a's and 8 states for b's, and condition is AND so number of states will be 6X8 = 48 states.
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if the condition would have been OR, then the min dfa will contain 14 or 15 states ?
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what about OR case?? can anybody give the right solution?
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OR does not make a difference in this question - just some more states become final.
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For 'OR' case, we have either a is divisible by 6 or b is by 8. Total #states = 48 as in 'AND' case.

Only final states will be more. We will count final states like (0,x) and (x, 0) where x E [0,5] for 'a' and [0,7] for 'b'.

For 'a' we have 6 such final states, for 'b' we have 8.

Total 6 + 8 - 1 = 13 final states as we have counted state (0,0) twice.

I will give a short trick for divisibility: |w| i.e length of string is divisible by a,b.

If a doesn't divide b and b doesn't divide a then Min. states = a*b

Case 1 : Divisible by a and b ----> If a divides b or b divides a then Min. No .of states = LCM(a,b)
Case 2: Divisible by a or b  ------> If a divides b or b divides a then Min. states =  GCD(a,b)

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@amitabh, you are wrong in case of Divisible by a or b , for example just check divisible by 2 or 3 and we  require atleast 6 states in min. dfa but your answer gives GCD(2,3) = 1

Pls check for your multiplication part too.as i am not sure though.

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Edited
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anyone verify this"

I will give a short trick for divisibility: |w| i.e length of string is divisible by a,b.

If a doesn't divide b and b doesn't divide a then Min. states = a*b

Case 1 : Divisible by a and b ----> If a divides b or b divides a then Min. No .of states = LCM(a,b)
Case 2: Divisible by a or b  ------> If a divides b or b divides a then Min. states =  GCD(a,b)

"

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Can you please mention the source or the authenticity of this, my friend.
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never apply these rule...fjust ollow properties
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Option D is correct as LCM of 6 and 8 is 24, also 48 is a multiple of 24 .
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how LCM helps?
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No need to even thnk about lcm..its simply multiplication refer the example in the image

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Please tell me the book name
You are referring to
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someone plz atleast mention the book or source for these type of questions, i can't find this concept in peter linz

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