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Question 4 1. Here is a question on regular languages just to get you in shape for the final exam. Suppose that $L$ is a regular language and $w$ is any word, not necessarily in $L$. We define the set $L/w = \{ x \in \Sigma^* | xw \in L \}$ . Show that $L/w$ is regular. 2. Suppose that $G$ is a context-free grammar. Show that the question \"Is $L(G)$ regular?\" is undecidable. Here is a possible approach. Let $N$ be some language that is known to be context-free but not regular (for example, $\{a^nb^n | n \ge 0\}$). Now consider the language $L = (N#\Sigma^) \cup (\Sigma^#L(G))$, where # is some symbol that is not in $L(G)$ or $N$. Prove that $L$ is always context-free but is regular if and only if $L(G) = \Sigma^*$. This, by itself, does not complete the question, so you have to complete all the remaining steps as well as proving the claim. Also, you should think about why I put part (1) together with this question? You are free to ignore this hint, but if you do so, I will mark you just as rigorously as the people who used the hint.

          Question 4
1. Here is a question on regular languages just to get you in shape for the final exam.
Suppose that $L$ is a regular language and $w$ is any word, not necessarily in $L$. We
define the set
$L/w = \{ x \in \Sigma^* | xw \in L \}$ .
Show that $L/w$ is regular.
2. Suppose that $G$ is a context-free grammar. Show that the question \"Is $L(G)$ regular?\" 
is undecidable. Here is a possible approach. Let $N$ be some language that is known to
be context-free but not regular (for example, $\{a^nb^n | n \ge 0\}$). Now consider the language
$L = (N#\Sigma^*) \cup (\Sigma^*#L(G))$, where # is some symbol that is not in $L(G)$ or $N$. Prove
that $L$ is always context-free but is regular if and only if $L(G) = \Sigma^*$. This, by itself,
does not complete the question, so you have to complete all the remaining steps as well
as proving the claim. Also, you should think about why I put part (1) together with
this question? You are free to ignore this hint, but if you do so, I will mark you just
as rigorously as the people who used the hint.

$Question 4 1. Here is a question on regular languages just to get you in shape for the final exam. Suppose that L is a regular language and w is any word, not necessarily in L. We define the set L/w = { x ∈Σ^* | xw ∈ L } . Show that L/w is regular. 2. Suppose that G is a context-free grammar. Show that the question Ïs L(G) regular?ïs undecidable. Here is a possible approach. Let N be some language that is known to be context-free but not regular (for example, {a^nb^n | n ≥ 0}). Now consider the language L = (N#Σ^*) ∪ (Σ^*#L(G)), where # is some symbol that is not in L(G) or N. Prove that L is always context-free but is regular if and only if L(G) = Σ^*. This, by itself, does not complete the question, so you have to complete all the remaining steps as well as proving the claim. Also, you should think about why I put part (1) together with this question? You are free to ignore this hint, but if you do so, I will mark you just as rigorously as the people who used the hint.$

Added by Mark R.

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