How do you do the last step when converting a CFG to a CNF?

Question

Given:

    S->AcA|BcB
    A->ccBc|ABA|cc
    B->c

step1

    S0->S
    S->AcA|BcB
    A->ccBc|ABA|cc
    B->c


step2             // change symbol to terminals?

    S0->S
    S->ABA|BBB
    A->BBBB|ABA|BB
    B->c

step3             // split?

    S0->S
    S->ABA
    S->BBB
    A->BBBB
    A->ABA
    A->BB
    B->c

step4             // what to do when A->AXA?

    S0->S
    S->ABA
    S->BBB
    A->BBBB   //??
    A->ABA    //??
    A->BB     //??
    B->c

I'm not sure how to continue.

Answer 1

There is no step 2 for you. Step 2 is removing epsilon rules, and you have no epsilon rules.

You don't have a step 3 either, because B->c has a terminal - not a nonterminal - on its right hand side. There are no unit rules of the form:

Terminal -> Terminal.

This leaves us after your step 1:

S0 -> S
S -> AcA|BcB
A -> ccBc|ABA|cc
B -> c

You need to get the rest of your rules in the form of:

X -> YZ //where X,Y,Z are all nonterminals

To convert a string of nonterminals and terminals into the above form, you take the first nonterminal or terminal from the front, and then you turn the rest of the string into a new rule, and use that rule on the end. I didn't explain that very well, so let's look at an example.

//To convert
S -> AcA
//we split it into A and cA, and define a new rule C -> cA, giving:
S -> AC
C -> cA
//Then, C -> cA needs to be converted to the same form, so just replace c with B
S -> AC
C -> BA

However, throw out the above, because first I'm going to just change all the cs to B, since that will happen anyway:

S0 -> S
S -> ABA|BBB
A -> BBBB|ABA|BB
B -> c

Now that I look at your question again, this is where you were at. You'd gone one step further to:

S0 -> S
S -> ABA
S -> BBB
A -> BBBB
A -> ABA
A -> BB
B -> c

If we take the first S, and use the method described above, we get:

S -> ABA
//goes to
S -> AC
C -> BA

Doing the rest of the rules, we get:

//second S
S -> BD
D -> BB

//first A
A -> DD

//second A:
A -> AC

Combining this all, we get:

S0 -> S
S -> AC
S -> BD
A -> DD
A -> AC
A -> BB
B -> c
C -> BA
D -> BB