constructing CFG

Question

How can I construct a Context-Free Grammer for the language x^ay^bz^2(a+b) where a>=0, b>=0. Thanks for helps...

Answer 1

Think of it this way

x^a y^b z^2(a+b) = x^a y^b z^2a z^2b = x^a y^b (z^2)^b (z^2)^a 

Therefore

S -> xSzz | S1
S1 -> yS1zz | e

Answer 2

Observe that for each x and for each y , you need to generate two z 's because of the 2( a + b ). Also, observe that each string can be viewed as an "inside" part of y 's and z 's, and an "outside" part of x 's and z 's.

Since for each y you need two z 's, the inside part can be described by (using capitals to denote non-terminal symbols and [] for the empty string):

I --> []
I --> y I z z

Now write a grammar for the outside part in the same way, but referring to I in the base case.

Answer 3

There are essentially two cases that you need to treat:

• You can either add an x at the beginning of the string, in which case you need to add two z 's at the end.
• Or you can add a y in the middle, in which case you also need to add two z 's at the end.

Try reducing either of these descriptions to a simpler grammer (eg a^nb^n ) for which you know the solution.

This hint should be enough to deduce the generative grammer.