.Left recursion may also be introduced by two or more grammar rules. For example

S Aa | b

A Ac | Sd | ε

there is a left recursion because

S Aa Sda

. In such cases, left recursion is removed systematically

- Starting from the first rule and replacing all the occurrences of the first non terminal symbol

- Removing left recursion from the modified grammar

What we saw earlier was an example of immediate left recursion but there may be subtle cases where left recursion occurs involving two or more productions. For example in the grammar

S A a | b

A A c | S d | e

there is a left recursion because

S A a S d a

More generally, for the non-terminals A 0, A 1,..., An , indirect left recursion can be defined as being of the form:

A_n A₁ a ₁ | .

A₁ A₂ a₂ |

. .

A _n A_n a_(n+1) | .

Where a ₁ , a₂ ,..., a _n are sequences of non-terminals and terminals.

Following algorithm may be used for removal of left recursion in general case:

Input : Grammar G with no cycles or e -productions.

Output: An equivalent grammar with no left recursion .

Algorithm:

Arrange the non-terminals in some order A₁ , A₂ , A₃ ..... A_n .

for i := 1 to n do begin

replace each production of the form A_i A_jγ

by the productions A i d_{1 γ} | d_{2 γ} | ...........| d_kγ

where A_j d₁ | d₂ | .........| d_k are all the current Aj -productions;

end

eliminate the immediate left recursion among the Ai-productions.

end.