Is this grammar left recursive or factor?

Question

I am new to this topic of left recursion and left factoring, please help me in determining whether this grammar is left recursive or left factored, if it is then why ?

S-> aAd | bBd | aBe | bAe | cA | cB

Answer 1

Is it Left recursive? Answer: No.

By definition, "A grammar is left-recursive if we can find some non-terminal A which will eventually derive a sentential form with itself as the left-symbol."

example: Immediate left recursion occurs in rules of the form

A -> Aa | b

where a and b are sequences of nonterminals and terminals, and b doesn't start with A.

Clearly not the case in:

S-> aAd | bBd | aBe | bAe | cA | cB

Is it Left factored? Answer: Yes.

By definition, in left factoring, it is not clear which two alternative production to choose, to expand a non-terminal.

This ambiguity occurs, when you have two alternative production that starts with same terminal/non-terminal. In your case, I can see that thrice, two alternative paths:

S-> aAd | aBe 

S-> bBd |  bAe 

S-> cA | cB

If I remove the left factoring then the grammar becomes:

S-> aA'

A'-> Ad | Be

S-> bB'

B'-> Bd | Ae

S-> cC'

C'-> A | B

This slide explains the same in simpler words