Constructing a CFG

Question

I literally can't find the answer to this:

L = {vw | v element of {a,b) , w element of {b,c} , number of a's <= number of c's}

V --> aV | bV | e

W --> bW | cW | e

But I cannot think of how to combine the construction of the words v and w after one another and keeping in mind the mentioned restriction.. Anyone who could lend me a hand?

Answer 1

As I explained in my answers tips for writing cfg , correct approach is first understand all possible patterns of strings in language then write rules.

What is this language L? In strings of language L:

1. All 'a' s comes before any 'c' s
2. Number of 'a' are less than or equals to number of 'c' , So in grammar of L, if a production rule adds one symbol 'a' then it must also add one or more 'c' .
3. There is no restriction on occurrence of symbol 'b' , it can appear anywhere any number of times. 4. Empty (null) string is also belongs to L as number of 'a' = number if 'c' == 0. And for same reason a string consists of only symbol 'b' is also acceptable.
5. Any string without 'a' also belongs L (in other words (c + b)* is subset of L).

Now writing grammar rules are easy (read comments to understand each production rules):

S →  BaBSBCB | ^    // add `a` add `C` also, B can be any where so added B   
Z →  CZ | BZ | ^    // to create `(c + b)*`
C →  cC | c         // C always generates one or more `c`s  
B →  bB | ^         // there is no restriction on B it generates `b`s or ^