7. Closure properties

7.1.1. Identify different strategies to verify that two regular expressions generate the same language. State them as algorithmic procedures.

7.1.2. Write a regex for the complement of $ L((10 \cup 0)^*(1 \cup \epsilon))$ .

7.1.3. Suppose $ A$ is a DFA. Write another DFA which accepts only those words $ w$ such that both $ w \in L(A)$ and $ reverse(w) \in L(A)$ .

Let $ L \subseteq \Sigma^* $ be a language and $ c \in \Sigma $ a symbol. The quotient of $ L $ and $ c $ is the language defined as $ L/c = \{ w \in \Sigma^* | wc \in L\} $.

7.2.1. Let $ L = L((aaa \cup ba)^*(ab)^* )$. What is the language $ L/a $ ?

7.2.2. Let $ L = L(a^*)$. What is the language $ L/a $ ?

7.2.3. Prove that if $ L $ is a regular language, then $ L/c$ is a regular language $ \forall c \in \Sigma$.

7.2.4. Let $ L \subseteq \Sigma^* $ be a language and $ c \in \Sigma $ a symbol. Then $ c / L = \{ w \in \Sigma^* | cw \in L \} $. Prove that if $ L $ is a regular language, then $ a / L $ is a regular language,$ \forall a \in \Sigma$.

7.3.1. Prove that $ \text{suffix}(L) = \{ w \in \Sigma^* | \exists x \in \Sigma^*, \: \text{such that} \: xw \in L \} $ is a closure property.

Let $ min(L) = \{ w \in L | \; \nexists x \in L, \; y \in \Sigma^* \setminus \{\epsilon\}, \: \text{such that} \: xy = w \}$.

Example: If $ L = \{ aab, bab, aa \} $, then $ min(L) = \{ bab, aa \} $.

7.4.1. Describe $ min(L)$ without mathematical notation.

7.4.2. What is the language $ min(L(a^*)) $ ?

7.4.3. What is the language $ min(L(a^*b)) $ ?

7.4.4. Prove that $ min $ has the closure property with respect to the regular languages.

7.4.5. Let $ L$ be a regular language and $ R = \{xy \mid yx \in L\}$ . Both $ x$ and $ y$ can be empty strings. Show that $ R$ is regular.

7.5.1. (Solved during lecture) Define the reversal of a language $ L $ as $ rev(L) = \{ w \in \Sigma^* | rev(w) \in L \}$, where $ rev(c_1c_2 \dots c_n) = c_nc_{n - 1} \dots c_1$ , with $ c_i \in \Sigma, 1 \leq i \leq n $. Show that reversal is a closure property.

7.5.2. Invent a language transformation or an operation between languages. See if you can show that it is a closure property.