GATE 2014 [Set-3]
April 4, 2025GATE 2013
April 5, 2025GATE 2013
| Question 9 |
What is the maximum number of reduce moves that can be taken by a bottom-up parser for a grammar with no epsilon- and unit-production (i.e., of type A → є and A → a) to parse a string with n tokens?
| n/2 | |
| n-1 | |
| 2n-1 | |
| 2n |
Question 9 Explanation:
Since it is given that the grammar cannot have:
1) epsilon production
2) production of the form A → a
Consider the grammar:
S → Sa | a
If we were to derive the string “aaa” whose length is 3 then the number of reduce moves that would have been required are shown below:
S → Sa
→ Saa
→ aaa
This shows us that it has three reduce moves. The string length is 3 and the number of reduce moves is also 3. So presence of such kinds of production might give us the answer “n” for maximum number of reduce moves. But these productions are not allowed as per the question.
Also note that if a grammar does not have unit production then the maximum number of reduce moves can not exceed “n” where “n” denotes the length of the string.
3) No unit productions
Consider the grammar:
S → A
A → B
B → C
C → a
If we were to derive the string “a” whose length is 1 then the number of reduce moves that would have been required are shown below:
S → A
A → B
B → C
C → a
This shows us that it has four reduce moves. The string length is 1 and the number of reduce moves is 4. So presence of such kind of productions might give us the answer “n+1” or even more, for maximum number of reduce moves. But these productions are not allowed as per the question.
Now keeping in view the above points suppose we want to parse the string “abcd”. (n = 4) using bottom-up parsing where strings are parsed finding the rightmost derivation of a given string backwards. So here we are concentrating on deriving right most derivations only.
We can write the grammar which accepts this string which in accordance to the question, (i.e., with no epsilon- and unit-production (i.e., of type A → є and A → B) and no production of the form A → a) as follows:
S → aB
B → bC
C → cd
The Right Most Derivation for the above is:
S → aB (Reduction 3)
→ abC (Reduction 2)
→ abcd (Reduction 1)
We can see here the number of reductions present is 3.
We can get less number of reductions with some other grammar which also doesn’t produce unit or epsilon productions or production of the form A → a
S → abA
A → cd
The Right Most Derivation for the above is:
S → abA (Reduction 2)
→ abcd (Reduction 1)
Hence 2 reductions.
But we are interested in knowing the maximum number of reductions which comes from the 1st grammar. Hence total 3 reductions as maximum, which is (n – 1) as n = 4 here.
1) epsilon production
2) production of the form A → a
Consider the grammar:
S → Sa | a
If we were to derive the string “aaa” whose length is 3 then the number of reduce moves that would have been required are shown below:
S → Sa
→ Saa
→ aaa
This shows us that it has three reduce moves. The string length is 3 and the number of reduce moves is also 3. So presence of such kinds of production might give us the answer “n” for maximum number of reduce moves. But these productions are not allowed as per the question.
Also note that if a grammar does not have unit production then the maximum number of reduce moves can not exceed “n” where “n” denotes the length of the string.
3) No unit productions
Consider the grammar:
S → A
A → B
B → C
C → a
If we were to derive the string “a” whose length is 1 then the number of reduce moves that would have been required are shown below:
S → A
A → B
B → C
C → a
This shows us that it has four reduce moves. The string length is 1 and the number of reduce moves is 4. So presence of such kind of productions might give us the answer “n+1” or even more, for maximum number of reduce moves. But these productions are not allowed as per the question.
Now keeping in view the above points suppose we want to parse the string “abcd”. (n = 4) using bottom-up parsing where strings are parsed finding the rightmost derivation of a given string backwards. So here we are concentrating on deriving right most derivations only.
We can write the grammar which accepts this string which in accordance to the question, (i.e., with no epsilon- and unit-production (i.e., of type A → є and A → B) and no production of the form A → a) as follows:
S → aB
B → bC
C → cd
The Right Most Derivation for the above is:
S → aB (Reduction 3)
→ abC (Reduction 2)
→ abcd (Reduction 1)
We can see here the number of reductions present is 3.
We can get less number of reductions with some other grammar which also doesn’t produce unit or epsilon productions or production of the form A → a
S → abA
A → cd
The Right Most Derivation for the above is:
S → abA (Reduction 2)
→ abcd (Reduction 1)
Hence 2 reductions.
But we are interested in knowing the maximum number of reductions which comes from the 1st grammar. Hence total 3 reductions as maximum, which is (n – 1) as n = 4 here.
Correct Answer: B
Question 9 Explanation:
Since it is given that the grammar cannot have:
1) epsilon production
2) production of the form A → a
Consider the grammar:
S → Sa | a
If we were to derive the string “aaa” whose length is 3 then the number of reduce moves that would have been required are shown below:
S → Sa
→ Saa
→ aaa
This shows us that it has three reduce moves. The string length is 3 and the number of reduce moves is also 3. So presence of such kinds of production might give us the answer “n” for maximum number of reduce moves. But these productions are not allowed as per the question.
Also note that if a grammar does not have unit production then the maximum number of reduce moves can not exceed “n” where “n” denotes the length of the string.
3) No unit productions
Consider the grammar:
S → A
A → B
B → C
C → a
If we were to derive the string “a” whose length is 1 then the number of reduce moves that would have been required are shown below:
S → A
A → B
B → C
C → a
This shows us that it has four reduce moves. The string length is 1 and the number of reduce moves is 4. So presence of such kind of productions might give us the answer “n+1” or even more, for maximum number of reduce moves. But these productions are not allowed as per the question.
Now keeping in view the above points suppose we want to parse the string “abcd”. (n = 4) using bottom-up parsing where strings are parsed finding the rightmost derivation of a given string backwards. So here we are concentrating on deriving right most derivations only.
We can write the grammar which accepts this string which in accordance to the question, (i.e., with no epsilon- and unit-production (i.e., of type A → є and A → B) and no production of the form A → a) as follows:
S → aB
B → bC
C → cd
The Right Most Derivation for the above is:
S → aB (Reduction 3)
→ abC (Reduction 2)
→ abcd (Reduction 1)
We can see here the number of reductions present is 3.
We can get less number of reductions with some other grammar which also doesn’t produce unit or epsilon productions or production of the form A → a
S → abA
A → cd
The Right Most Derivation for the above is:
S → abA (Reduction 2)
→ abcd (Reduction 1)
Hence 2 reductions.
But we are interested in knowing the maximum number of reductions which comes from the 1st grammar. Hence total 3 reductions as maximum, which is (n – 1) as n = 4 here.
1) epsilon production
2) production of the form A → a
Consider the grammar:
S → Sa | a
If we were to derive the string “aaa” whose length is 3 then the number of reduce moves that would have been required are shown below:
S → Sa
→ Saa
→ aaa
This shows us that it has three reduce moves. The string length is 3 and the number of reduce moves is also 3. So presence of such kinds of production might give us the answer “n” for maximum number of reduce moves. But these productions are not allowed as per the question.
Also note that if a grammar does not have unit production then the maximum number of reduce moves can not exceed “n” where “n” denotes the length of the string.
3) No unit productions
Consider the grammar:
S → A
A → B
B → C
C → a
If we were to derive the string “a” whose length is 1 then the number of reduce moves that would have been required are shown below:
S → A
A → B
B → C
C → a
This shows us that it has four reduce moves. The string length is 1 and the number of reduce moves is 4. So presence of such kind of productions might give us the answer “n+1” or even more, for maximum number of reduce moves. But these productions are not allowed as per the question.
Now keeping in view the above points suppose we want to parse the string “abcd”. (n = 4) using bottom-up parsing where strings are parsed finding the rightmost derivation of a given string backwards. So here we are concentrating on deriving right most derivations only.
We can write the grammar which accepts this string which in accordance to the question, (i.e., with no epsilon- and unit-production (i.e., of type A → є and A → B) and no production of the form A → a) as follows:
S → aB
B → bC
C → cd
The Right Most Derivation for the above is:
S → aB (Reduction 3)
→ abC (Reduction 2)
→ abcd (Reduction 1)
We can see here the number of reductions present is 3.
We can get less number of reductions with some other grammar which also doesn’t produce unit or epsilon productions or production of the form A → a
S → abA
A → cd
The Right Most Derivation for the above is:
S → abA (Reduction 2)
→ abcd (Reduction 1)
Hence 2 reductions.
But we are interested in knowing the maximum number of reductions which comes from the 1st grammar. Hence total 3 reductions as maximum, which is (n – 1) as n = 4 here.