Question

Prove if the following languages are CFL or not.

If L is a CFL, give its CFG. Otherwise, prove it by Pumping Lemma.
If any closure property of CFL is applicable, apply them to simplify it before its proof.
L = {wwRw | w Î {a, b}*}

Answer 1

To prove if a language is a context-free language (CFL) or not, we can provide a context-free grammar (CFG) or use the Pumping Lemma. For the language L = {wwRw | w Î {a, b}*}, we can provide a CFG, but using the Pumping Lemma, we can prove that L is not a CFL.

Step-by-step explanation:

This question is asking whether the language L = {wwRw | w Î {a, b}*} is a context-free language (CFL) or not. To prove if a language is a CFL, we can provide a context-free grammar (CFG) for the language. However, if a language is not a CFL, we can prove it using the Pumping Lemma.

In this case, let's first try to find a CFG for L. We can define the CFG as follows:

S -> aSa | bSb | ε

Where S is the start symbol and ε represents the empty string.

Now, let's apply the Pumping Lemma to prove that L is not a CFL. Suppose L is a CFL. Let p be the pumping length of L. Consider the string s = a^p b^p a^p b^p a^p b^p, which is in L and has length greater than p. By the pumping lemma, s can be divided into five parts: uvxyz, where the length of vxy is less than or equal to p and the length of vy is at least 1.

By pumping down or up, we can show that the pumped string is not in L, which contradicts the assumption that L is a CFL. Therefore, the language L = {wwRw | w Î {a, b}*} is not a CFL.