GATE | GATE-CS-2006 | Question 85

In the correct grammar of above question, what is the length of the derivation (number of steps starring from S) to generate the string a^lb^m with l ≠ m?

(A)

max(l,m) + 2

(B)

l + m + 2

(C)

l + m + 3

(D)

max(l, m) + 3

Answer: (A)

Explanation:

Correct grammar of the last question was (D), which is:

S -> AC|CB
C -> aCb|epsilon
A -> aA|a
B -> Bb|b

Now, the most optimal and intuitive way to generate a string of the form a^lb^m would be to first use \”C -> aCb|epsilon\” production rule to get as many a and b as we can, which would be min(l,m). To get the rest of the string, we could just use latter two production rules accordingly. Formally deriving the string of the general format a^lb^m from the above grammar –

1.  S -> AC 
2.      -> A(aCb) 
3.      -> …. 
4.      -> ….
5.      -> A(a^m C b^m) 
6.      -> A(a^m b^m) 
7.       -> aA(a^m b^m) 
8.      -> …. 
9.      -> ….
10.    -> a^(l-m-1)A(a^m b^m) 
11.     -> a^l b^m

From above set of derivation steps we can count the total steps as follows:

Production 1 took 1 step          :           1                      [using S->AC]
Production 2-5 took steps      :      min(l,m)              [using C->aCb] 
Production 6 took 1 step         :           1                      [using C->epsilon]
Production 7-11 took steps      : max(l,m)-min(l,m)   [using A -> aA|a or B -> Bb|b]

Total steps   : max(l,m) + 2     

Hence, answer should be (A): max(l,m) + 2 

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