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Since we love to prove things, we must first prove that we can restrict the structure of our grammars--without losing any expressiveness i. So, how do we do that? Bn for that B. So clearly, any derivation in G1 is also a derivation in G. Therefore, we have built a grammar in G1 for any grammar in G. All we said is that you can pick out an arbitrary A-production from a random grammar--and that if you apply this transformation to the grammar, you will get back an equivalent grammar.

The easiest thing to do is rename the variables in V for example, make their names now be A1, A NB Some orderings are easier to work with than others, but any ordering will work. Do this first for A1, then for A2, and so on, till you reach An. Note that after applying this algorithm for a particular variable, all of the productions for A1 up to Ai will be only be the forms shown by cases 1,2,3. Then again working your way up the numbering scheme apply the second algorithm to all of the productions of the form in case 3.

This leaves all productions to have RHS which either begin with a terminal, or with a higher numbered variable. Now An-1 will have the same property, do it for An-2, and continue down to A1. Finally we have to take care of the B-productions, but notice that, due to the structure of algorithm 2, the RHS of any B-production must start with an A-variable.

And, we just made it so that all of the A-productions have terminals as the leftmost symbol of their RHS. So, we can just substitute the A-productions in to the B-productions, and now every production will be of the appropriate form.


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