How to come up with a solution of finite or infinite language using context free grammar?

Question

I am trying to come up with a solution of finite or infinite language using context free grammar.

I have these grammars to find if it's a solution of finite or infinite language

S -> XY|bb  Step 1
X -> XY|SS  Step 2
Y -> XY|SS  Step 3

So I would do

S -> XY            From step 1
S -> YYY           From step 2
S -> SSYY          From step 3
S -> SSSSY         From step 3
S -> SSSSSS        From step 3
S -> bbSSSSS       From step 1
S -> bbbbSSS       From step 1
S -> bbbbbbSSS     From step 1
S -> bbbbbbbbSS    From step 1
S -> bbbbbbbbbbS   From step 1
S -> bbbbbbbbbbbb  From step 1

bbbbbbbbbbbb 

So I know here how to generate a word like this, but how do I find if its finite or infinite language?

Answer 1

If you can show that S goes to something and itself ex: S->bS then your language is infinite. If someone attempted to make a finite list of everything the language could represent, you could always legally create something one 'b' character longer than anything in his finite list. Therefore any finite list isn't actually a complete list of everything the language could produce and the language is infinite.