1.9k views

Consider the grammar

• S $\rightarrow Aa \mid b$
• A $\rightarrow Ac \mid Sd \mid \epsilon$

Construct an equivalent grammar with no left recursion and with minimum number of production rules.

edited | 1.9k views

(b) As it is the case of indirect recursion so let first make it as direct recursion then apply rules of removal of left recursion.

to make it as direct recursion first production remain unchanged while in second production substitute the right hand side of first production wherever it comes.In the question $S$ comes in middle of $A$ so substitute the right hand side of production $S$.Now after substituting it looks like:

• $A \rightarrow Ac\mid Aad \mid bd \mid \epsilon$

Now remove direct recursion from it

For removal of direct recursion rule:

• $A \rightarrow A\alpha_1 \mid \ldots \mid A\alpha_n \mid \beta_1 \mid \ldots \mid \beta_m$

Replace these with two sets of productions, one set for $A:$

• $A \rightarrow \beta_1A^\prime \mid \ldots \mid \beta_mA^\prime$

and another set for the fresh nonterminal $A^{\prime}$

• $A^\prime \rightarrow \alpha_1A^\prime \mid \ldots \mid \alpha_nA^\prime \mid \epsilon$

After applying these rule we'll get:

• $A \rightarrow bdA'\mid A'$
• $A' \rightarrow cA'\mid adA' \mid \epsilon$

Now complete production without left recursion is:

• $S \rightarrow Aa \mid b$
• $A \rightarrow bdA'\mid A'$
• $A' \rightarrow cA'\mid adA' \mid \epsilon$
by Junior (569 points)
selected
0

i have a doubt that in the production for S, if on rhs A is replaced by A'S production...so that is also an indirect recursion??

• S →Aa∣b
• A →Ac∣Sd∣ϵ

to

• S →Aca|Sda∣b

on removing left rec

S->aS'|bS'|Aca

S'->daS'|null

A->bdA'|A'

A'->cA'|adA'| +1
they are asking for minimum production rules
+1 vote
• S →Aa∣b
• A →Ac∣Sd∣ϵ

Remove indirect recursion first:-

A-> Ac | Aad | bd |ϵ

I have replace S production with RHS in above . Now we have got the grammer with direct left recursion.

Now let us remove left recursion

S →Aa∣b

A -> bd A' | A'

A' -> cA' | ad A' | ϵ

by Boss (24.9k points)
edited
0
Why are we not replacing the A in S production... Isn't it indirect recursion??

And also from the above grammar we are not able to generate some strings Eg 'b'.

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its not generating b. This is not correc.Please refer above answer.I will update this answer in some time
0
In above answer also the same confusion is there. Please update soon.
0
You can not leave S, the we will miss out S --- > b , Moreover keeping S---> Aa is necessary to reach A. And what we did is,after reaching A,we never go back to S, i.e. we eliminated the indirect recursion. We handle that with help of A' .
0
Yes it was incorrect. I have update it now.

@Garry:- I saw your above comment where you generated byu replacing A in S.I think you are getting 4 productions but with other way only 3 are possible.So,we will prefer 3
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i still didn't get it, that why u haven't replaced A in S  production??

how will we get to know in exam ,that 3 is the minimum no, we might get confused!!
0

@Gate Fever there is a rule of doing it , we number the productions then replace in an order, you can google how to remove indirect left recursion.

+1
ive seen in many examples that

in starting symbol,left recursion is not removed however in all non terminals,left recursion is removed