Edit this page Backlinks This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== 7. Closure properties ====== **7.1.** Identify **different strategies** to verify that two regular expressions generate the same language. State them as algorithmic procedures. **7.2.** Write a regex for the complement of $math[L((10 \cup 0)^*(1 \cup \epsilon))]. ===== Quotients ===== 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\} $. **7.4.1.** Let $ L = L((aaa \cup ba)^*(ab)^* )$. What is the language $ L/a $ ? **7.4.2.** Let $ L = L(a^*)$. What is the language $ L/a $ ? **7.4.3.** Prove that if $ L $ is a regular language, then $ L/c$ is a regular language $ \forall c \in \Sigma$. **7.5.** 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$. ===== Suffixes and prefixes ===== **7.6.** Prove that $ \text{suffix}(L) = \{ w \in \Sigma^* | \exists x \in \Sigma^*, \: \text{such that} \: xw \in L \} $ is a closure property. ===== Other transformations ===== 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 \} $. **7.7.1.** What is the language $ min(L(a^*)) $ ? **7.7.2.** What is the language $ min(L(a^*b)) $ ? **7.7.3.** Prove that $ min $ has the closure property with respect to the regular languages. ===== Extras ===== **7.8.1.** (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.