What language does this mean?

Question

this is the language:

L = { w belong {a,b,c}* | |w|= 3 * number(a) (w) }

Then, what does that mean?

Answer 1

It means that L is the language of strings w consisting of symbols 'a' , 'b' ' and 'c' , where the length of the string w equals to 3 times the number of symbol 'a' present in the string w .

The productions for this grammars should be such that if it add one 'a' then it also adds two 'b' , or two 'c' , or one 'b' ; one 'c' . Check below grammar:

S → ^ | SaSMSM |  SMSaSM | SMSMSa   
M → b | c

here ^ means epsilon.

To generate aabbcc use Right most derivation

1. S → SaSMSM
2. replace first S in rhs by ^ using S → ^
   S → SaSMSM → aSMSM
3. replace S → SaSMSM
   S → SaSMSM → aSaSMSMMSM
4. use S → ^
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM
5. use S → ^
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM
6. M → b
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM
7. use S → ^
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM
8. M → b
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM
9. M → c
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM → aabbcSM
10. use S → ^
    S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM → aabbcSM → aabbcM
11. M → c
    S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM → aabbcSM → aabbcM → aabbcc