Logical-Reasoning
April 20, 2024GATE 2017 [Set-1]
April 20, 2024Question 7886 – Theory-of-Computation
Consider the following languages:
I. {am bn cp dq ∣ m + p = n + q, where m, n, p, q ≥ 0}
II. {am bn cp dq ∣ m = n and p = q, where m, n, p, q ≥ 0}
III. {am bn cp dq ∣ m = n = p and p ≠ q, where m, n, p, q ≥ 0}
IV. {am bn cp dq ∣ mn = p + q, where m, n, p, q ≥ 0}
Which of the above languages are context-free?
Correct Answer: B
Question 9 Explanation:
i) am bn cp dq | m+p = n+q,
m-n = q-p
First we will push a’s in the stack then we will pop a’s after watching b’s.
Then some of a’s might left in the stack.
Now we will push c’s in the stack and then pop c’s by watching d’s.
And then the remaining a’s will be popped off the stack by watching some more d’s.
And if no d’s is remaining and the stack is empty then the language is accepted by CFG.
ii) am bn cp dq | m=n, p=q
Push a’s in stack. Pop a’s watching b’s.
Now push c’s in stack.
Pop c’s watching d’s.
So it is context free language.
iii) am bn cp dq | m=n=p & p≠q
Here three variables are dependent on each other. So not context free.
iv) Not context free because comparison in stack can’t be done through multiplication operation.
m-n = q-p
First we will push a’s in the stack then we will pop a’s after watching b’s.
Then some of a’s might left in the stack.
Now we will push c’s in the stack and then pop c’s by watching d’s.
And then the remaining a’s will be popped off the stack by watching some more d’s.
And if no d’s is remaining and the stack is empty then the language is accepted by CFG.
ii) am bn cp dq | m=n, p=q
Push a’s in stack. Pop a’s watching b’s.
Now push c’s in stack.
Pop c’s watching d’s.
So it is context free language.
iii) am bn cp dq | m=n=p & p≠q
Here three variables are dependent on each other. So not context free.
iv) Not context free because comparison in stack can’t be done through multiplication operation.
I and IV only
I and II only
II and III only
II and IV only