How to remove left recursion in a grammar with both left recursion and right recursion existing?

Question

The grammar is as follows:

E->a
E->E+E
E->S,E
E->(E)

S-> bS'
S'-> ;bS'
S'->

I have no idea how to remove the left recursion cause E contain terminals and non-terminals. And there're both left recursion and right recursion.

Answer 1

First let me explain what left recursion is: Any grammar of the form

A → Aα1 | Aα2 |...|Aαn | β1 |β2|...|βn

is said to contain left recursion because the non terminal A appears on the left side of the production and as the first symbol on the right side of the production as well. Such a grammar can be converted to an equivalent form without left recursion by applying the following transformation:

A → β1A' | β2A' |...| βnA'

A' → α1A' | α2A' |...|αnA'

Putting your grammar in the same form, we get

E → E+E | a | S,E | (E)

S → bS'

S' → ;bS' | ε

We can see that left recursion occurs only in the case of the E productions, and that too only when the right side is E+E So we can make the following matchings here:

α is equivalent to +E

β1 is equivalent to a

β2 is equivalent to S,E

β3 is equivalent to (E)

Now we can directly substitute to get the following

E → aE' | S,EE' | (E)E'

E' → +EE' | ε

Note that the productions related to S and S' remain unchanged. This grammar has no left recursion.