[Solved]: Constructing Context Free Grammar

Problem Detail: 

I am stuck and having a hard time with this question. I want to construct a CFG for the language $$L = \{{a^lb^mc^n | l,m\in N, n=|l-m|\}}$$ I know that the language consists of strings where:
1. number of a's = number of b's, so c=0
2. number of a's more than number of b's, c=l-m
3. number of a's less than number of b's, c=-(l-m)
I started with $$S->ab$$ $$S->aSb$$ This generates all of case one, where number of a's = number of b's and c=0. I know that I could increment a's and c's by having aSc but I cant put that in the second line because it could generate a(aSc)b which is not in the language.

Asked By : Data

Answered By : Denis

Hint: try to do each language separately, and then your final grammar is S->S1|S2|S3.

To do your case 2, you can write

$S->aSc|X$

$X->aXb|\epsilon$.

I.e. you start by generating the c's and the exceeding a's, and then you generate the b's and the remaining a.

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Question Source : http://cs.stackexchange.com/questions/16678