# GATE | GATE CS 2013 | Question 65

• Difficulty Level : Hard
• Last Updated : 10 Sep, 2021

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?
(A) n/2
(B) n-1
(C) 2n-1
(D) 2n

Explanation: Given in the question, a grammar with no epsilon- and unit-production (i.e., of type A -> є and A -> a).

To get maximum number of Reduce moves, we should make sure than in each sentential form only one terminal is reduced. Since there is no unit production, so last 2 tokens will take only 1 move.

So To Reduce input string of n tokens, first Reduce n-2 tokens using n-2 reduce moves and then Reduce last 2 tokens using production which has . So total of n-2+1 = n-1 Reduce moves.

Suppose the string is abcd. ( n = 4 ).

We can write the grammar which accepts this string 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 that no production is for unit or epsilon. Hence 3 reductions here.

We can get less number of reductions with some other grammar which also does’t produce unit or epsilon productions,

```S->abA
A-> cd```

The Right Most Derivation for the above as:

```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.

Thus, Option B.

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