# Proof any CFG can be converted to GNF

I studied algorithm for converting grammars to GNF and was wondering about a proof that any CFG can be converted into GNF. I found this paper - ENGELFRIET, Joost. An Elementary Proof of Double Greibach Normal Form, 1992, but I do not understand the proof at all.

Any CFG can be converted into CNF, so we try to prove that any grammar in CNF can be converted into GNF.

At first, a grammar H (not GNF, but can be easily converted into GNF by substitution and by removing rules containing empty string) is constructed from original CNF denoted G as follows:

1) for rules Y → a $$A \rightarrow a[Y,A]$$ 2) for rules C → DQ $$[D, A] \rightarrow Q[C, A]$$ 3) $$[A, A] \rightarrow \epsilon$$

Than there are mentioned two statements about this new grammar. These statements are essential for the proof, because they show that the grammar H is equivalent to G. And I do not know why they are right:

$$([Y,A] \rightarrow^*_H w) \iff (A \rightarrow ^*_G Y w)$$ and $$(A \rightarrow^*_H w) \iff (A \rightarrow ^*_G w)$$

There is also written that these statements can be shown by induction. I tried to prove them but it seems to me that to prove one of them, I also need to prove the second. So I am stuck at cycle.

Could somebody please explain it to me?

• Welcome to CS.SE! Can you edit the question to give a self-contained definition of the notation? Also, can you give a full reference for the paper (title, authors, where published), so that others who have a question about the same paper and search on it find this, and so that the question still makes sense even if the link stops working? Thanks!
– D.W.
Commented May 1, 2017 at 15:32
• These look like some central claims (you can derive $w$ in one grammar if and only of you can derive it in the other). Taking them out of context makes them impossible to make sense of. Be more specific: which step of the proof don't you understand? Commented May 1, 2017 at 15:50
• I just do not understand why should the two mentioned statements are right, everything other makes me sense. I tried to prove it as they said. The second says that the grammars G and H are equivalent (right?), so I would guess that I really need to prove the first statement to prove the second, but it is not obvious to me how to prove the first one (without the second of course) Commented May 1, 2017 at 16:12

The key is to understand what the nonterminal $[Y,A]$ actually represents. It means a derivation that has a leftward "backbone" which starts from $A$ and ends just before $Y$, as in the picture below.
Now the first production in $H$ can be explained as follows. If I start with $A$ and want to produce GrNF then I must generate a terminal. We choose to generate the leftmost terminal $a$ of the subtree under $A$. So, we produce $a$, but we still must find a derivation $[Y,A]$ of the form in the picture, where $Y$ is the nonterminal that actually produces $a$ using the available production $Y\to a$.