Indirect left recursion in my grammar?

Question

My grammar appears to have a case of indirect left recursion, looked at some of the other similar question, cannot quite make a mental connection between them and my grammar, I can't get my head around how to solve it.

A  ::= A' a
     | A
     | b

A' ::= c
     | A

A' is called from A but A' is c or A , this is causing the left recursion, how could this be rearranged to an equivalent grammar while eliminating the left recursion?

Answer 1

You have the following productions:

1:  A  -> A' a
2:  A  -> A
3:  A  -> b
4:  A' -> c
5:  A' -> A

First note that production #2 makes this grammar ambiguous, and is actually kind of pointless. Let's remove it.

1:  A  -> A' a
3:  A  -> b
4:  A' -> c
5:  A' -> A

The "Left recursion" article on Wikipedia contains an (unsourced) algorithm to eliminate all left recursion , including indirect left recursion. Let's ignore this specific algorithm and focus on the idea instead: first turn indirect recursion to direct recursion through substitution, then resolve direct recursion by adding a tail non-terminal.

For instance, we can substitute A' in production #1 by replacing it with

6:  A  -> c a    (see #1 and #4)
7:  A  -> A a    (see #1 and #5)

The grammar becomes as follows:

4:  A' -> c
5:  A' -> A
6:  A  -> c a
7:  A  -> A a

and we've already turned all indirect recursion into direct recursion. All that remains is to remove the direct recursion for A :

4:  A' -> c
5:  A' -> A
6:  A  -> c a T
8:  T  -> epsilon
9:  T  -> a T