Derivations

Question

Let G be a CFG , l be the number of left most derivations , r be the number of right most derivations and p be the number of parse trees.Assume l , p , r are computed for a particular string . For a given CFG 'G' and given string 'w'. What is the relation between l , p and r

a) l <=p<=r

b)l=p=r

c)l>=p<=r

d)none

Comments

For unambiguous grammar b)

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Please explain.

both lmd and rmd trees are parse trees. For any unambiguous grammar we have unique lmd and rmd. Hence l = r.

and parse trees are l + r.

Correct me if I’m wrong .

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@anil

L =r   means left most derivation and right most derivation is same

so which means there is only one parse tree possible

so l=p=r

p=L+r

is when its ambiguous and we have more than one left and right most derivations

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In any CFG $G,$ for every $w \in L(G);$ we have:

The number of parse trees of $w$ = The number of LMDs of $w$ = The number of RMDs of $w$.

Answer 1

For a particular string, there can be just one Leftmost derivation, and just one Rightmost derivation. So, l = r.

Now, both the derivations translate into the same parse tree. But in case of ambiguous grammars, there can be more than one parse trees. However, RMD and LMD for a particular string remain just one.

Hence, l = r $\leq$ p

So, none of the above.

Option D.

 

Assuming the grammar is unambiguous, Option B.

Answer 2

In any CFG $G,$ for every $w \in L(G);$ we have:

The number of parse trees of $w$ = The number of LMDs of $w$ = The number of RMDs of $w$.

i.e. For every string $w \in L(G)$, the number of parse trees is same as the number of leftmost derivations & that is same as the number of rightmost derivations.

For every parse tree, there is a unique leftmost, and a unique rightmost derivation. Similarly, for every leftmost derivation, there is a unique parse tree. Similarly, for every rightmost derivation, there is a unique parse tree.

So, answer is Option B, $l = r = p.$