GATE | GATE-CS-2014-(Set-2) | Question 65

Let L1 = {w ∈ {0,1} | w has at least as many occurrences
                                  of (110)’s as (011)’s}. 

Let L2 = { ∈ {0,1} | w has at least as many occurrences
                                 of (000)’s as (111)’s}. 

Which one of the following is TRUE?
(A) L1 is regular but not L2
(B) L2 is regular but not L!
(C) Both L2 and L1 are regular
(D) Neither L1 nor L2 are regular


Answer: (A)

Explanation: L1 is regular
let us consider the string 011011011011
In this string, number of occurrences of 011 are 4 but when we see here 110 is also occurred and the number of occurrence of 110 is 3.

Note that if i add a 0 at the last of string we can have same number of occurrences of 011 and 110 so this string is accepted. We can say if the string is ending with 011 so by appending a 0 we can make 110 also.

Now string2: 110110110110 in this number of occurrences of 110 is 4 and 011 is 3 which already satisfy the condition

So we can observe here that whenever 110 will be there string will be accepted

So with this idea we can build an automata for this. Therefore, it is regular.

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