A-=>AaA/a then how can S1 will be true

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+41 votes

Which of the following statements are true?

- Every left-recursive grammar can be converted to a right-recursive grammar and vice-versa
- All $\epsilon$-productions can be removed from any context-free grammar by suitable transformations
- The language generated by a context-free grammar all of whose productions are of the form $X \rightarrow w$ or $X \rightarrow wY$ (where, $w$ is a string of terminals and $Y$ is a non-terminal), is always regular
- The derivation trees of strings generated by a context-free grammar in Chomsky Normal Form are always binary trees

- I, II, III and IV
- II, III and IV only
- I, III and IV only
- I, II and IV only

+50 votes

Best answer

Answer is C:

Statement 1 is **true**: Using GNF we can convert Left recursive grammar to right recursive and by using reversal of CFG and GNF we can convert right recursive to left recursive.

Statement 2 is **false**: because if $\epsilon$ is in the language then we can't remove $\epsilon$ production from Start symbol. (For example $L = a^*$)

Statement 3 is **true** because right linear grammar generates regular set

Statement 4 is **true**, only two non-terminals are there in each production in CNF. So it always form a binary tree.

0

2nd statement says *ϵ* can be removed from Grammar.

Suppose we have a grammar like

S -> AbaC

A -> BC

B -> b/ *ϵ*

C -> D/ *ϵ*

D -> d

In this case *ϵ* productions can be removed. Correct me if i am wrong..

+1

@Shamim Ahmed option b says all epsilon can be removed. If language can generate epsilon, how can we remove it? start symbol will always have it to generate epsilon.

+16 votes

1.Every left recursive grammar can be converted to a right recursive grammar and vice versa

yes their is algo.

2. All ∈ production can be removed from any CFG by suitable transformation

NO when language itself contain null then u can't remove null.

3. The language generated by a CFG all whose production are to the form X→ w or X→ wY

yup their is formula for it X-> aT* then regular since right linear form

4.The derivation trees of strings generated by a CFG in CNF are alays binary trees.

yes b/c cfn in form of S->AB or S->a so max two child at a time.

Answer is (C)

+6 votes

2nd is false

S->∈ if we have this production in our cfg we can remove it if we do so it changes language

All other options are true so ans is c

S->∈ if we have this production in our cfg we can remove it if we do so it changes language

All other options are true so ans is c

0

but,we have a procedure to remove null production in cfg isnt it???and the qs asked that all epsilon production can be removed or not...

+28

Yes, we can remove any Null production except in one case if we can derive $\epsilon$ from grammar, we cannot remove that.

eg. $S\rightarrow aS|\epsilon$ generating language $L = a^*$, and if we remove $\epsilon$ from it,

we will get $S\rightarrow aS|a$ generating a different language ,$L = a^+$

+1 vote

Explanation for I.

- Reverse "LLG for L" to get "RLG for L
^{R}" by changing A → Ba to A → aB - Convert "RLG for L
^{R}" directly to "FA for L^{R}" - Reverse "FA for L
^{R}" to get "FA for L" by- Change starting state to final state
- Reverse direction of each transition
- Create a new start state with
*ϵ*transitions to all accepting states - Convert "FA for L" directly to "RLG for L"

Similar approach can be derived for converting "RLG for L" to "LLG of L".

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