I am new to grammars and I want to learn context free grammars which are the base of programming languages. After solving some problems, I encountered the language
$$\{a^nb^nc^n\mid n\geq 1\}\,.$$
Can anybody tell me if this is a context free language? I can't make any context free grammar for it and I don't know any other proof.
This is my approach to prove that a given language is not a CFL.
Try hard to come up with a Context free grammar for the given language. If you can come up with such a grammar, then the language is indeed a CFL.
If you can't, then you can use the pumping lemma to show that a given language is not a CFL.
Assume L is context free. Then L satisfies P.L. Then there exists n by the P.L. Let z = a^n b^n c^n Because |z|>=n and z in L, by PL there exist uvwxy s.t.
z = uvwxy
|vwx| <= n
|vx| >= 1
forall i>=0, u v^i w x^i y in L
But if there exist u,v,w,x,y satisfying the first three constraints, we can show that there exists an i s.t. the fourth constraint fails.
Case uvw consists only of "a". Then when i=0, the string is not in L (fewer "a"s than "b"s or "c"s).
Case vwx contains only "b"s - similar.
Case vwx contains only "c"s - similar.
Case vwx contains only two types of characters from {a,b,c}. Then uwy has more of the remaining type of characters.
The string vw cannot contain "a"s, "b"s, and "c"s, since vwx is a substring of z and has length <=n (cannot contain the last "a" and the first "c" both).i.e this is a contradiction and our assumption is wrong that L is CFL.
CFL.
$\endgroup$
You are asking two different questions. First, is $\{a^nb^nc^n : n \geq 1 \}$ a context-free grammar? Second, is $\{a^nb^nc^n : n \geq 1 \}$ a context-free language?
The answer to the first question is No, since $\{a^nb^nc^n : n \geq 1 \}$ is not a grammar, let alone a context-free grammar; it is a language. What you probably meant to ask was: "Does $\{a^nb^nc^n : n \geq 1 \}$ have a context-free grammar?", which is the same as the second question.
The answer to the second question is also No. In fact, it is one of the standard examples of non-context-free languages, and as such most textbooks will have proofs to that effect.
