Duplicate rules in CFG

Question

Suppose I have the following CFG that I would like to bring to CNF

S -> TA | bA | Ab | b

A -> Aa|a

T -> Ab

Then I have two options to get rid of the nonsolitary terminal a in Aa

1 - Make a new nonterminal K and introduce the rule

The grammar in this case becomes

K->a

S -> TA | bA | Ab | b

A -> AK|a

T -> Ab

2 - Observe that A->a already and use it

The grammar in this case becomes

S -> TA | bA | Ab | b

A -> AA|a

T -> Ab

My intuition is that both should be correct since I don't see that changing a nonterminal's name would change the language the grammar generates but I am not entirely sure.

Thanks.

Answer 1

In this case, both are correct, but in general, you should use method 1.

For example, suppose that you had

$$A \to Aa|a|b.$$

You might be tempted to say that we have $A \to a$ already, so let's just use $A$ instead of creating a new non-terminal, and convert this to

$$A \to AA|a|b.$$

This would be an incorrect transformation, because originally $A$ generates the language $a,b,aa,ba,aaa,baa,\cdots$, and after the (incorrect) conversion, $A$ generates $a,b,aa,ab,ba,bb,aaa,aab,\cdots$

So, use method 1. Method 1 is always correct. It is just a "happy accident" that method 2 happens to work in this case, but in general, it often won't work.