understanding basics of CFG

Question

Have just started the chapter about CFL in Sipser's book and already fail to understand the basics.

Let this be the grammar of some language:

S -> A0A 
A -> 00A | 11A | 10A | 01A | e

I am really confused about this A0A part. Does it mean that the left hand side from 0 should be always the same as the right hands side. Does this mean 00011 or 000 are not in this language then?

Answer 1

Any S is any option you have for any A , followed by a single literal 0 and then another option for A . Each A is independent.

The string 00011 is in the language because you can pick your two A s (for example) such that the first one is 00A and the second one is 11A . If you pick recursively the empty string ( e ) for both of the "remaining" A s, when you concatenate everything you'll end up with the string 00011 .

You can do a similar thing to obtain the string 000 .

Answer 2

A transforms to either empty, or two binary digits then A. It means that A transforms to any sequence of even number of binary digits.

S transforms to A, then 0, then A. This means that S is transforms to even number of binary digits, then 0, then even number of binary digits. That is, S is any sequence of odd number of binary digits with 0 in the middle.

Does it mean that the left hand side from 0 should be always the same as the right hands side.

No, why? Two different A's can transform to different sequences.