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lfa:lab06-closure-properties [2020/11/05 15:42]
pcristian created 		lfa:lab06-closure-properties [2022/11/17 19:16] (current)
pdmatei
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-====== Closure properties ======+====== ​7. Closure properties ======
  
-1. Show that $(10 \cup 0)^*(1 \cup \epsilon)$ ​ and $(1 \cup \epsilon)(00^*1)^*0^*$ are equivalent regular expressions.+**7.1.** Show that $(10 \cup 0)^*(1 \cup \epsilon)$ ​ and $(1 \cup \epsilon)(00^*1)^*0^*$ are equivalent regular expressions. ​Are there several strategies?
  
-2. Write the "​complement"​ regular expression for $(10 \cup 0)^*(1 \cup \epsilon)$.+<​hidden>​ All strategies require building DFAs from the regexes. 
 +  - You can use an existing minimisation algorithm to find the minimal DFAs. Then, label each state from each DFA from 0 to |K|. Fix an ordering of the symbols of the alphabet. Sort the transitions by symbol. Make a textual representation of each DFA which includes the number of states and the sorted transition function. ​ If the textual representations of the two DFAs are identical then they accept the same language. 
 +  - Use the indistinguishability algorithm to check if the two initial states of the two DFAs are indistinguishable. 
 +  - Check $math[L(A_1) \subseteq L(A_2)] and $math[L(A_2) \subseteq L(A_1)]. The condition $math[L(A) ​\subseteq L(B)] can be restated as $math[L(A) \cap complement(L(B)) = \emptyset]. Build the complement DFA for B, and use the product construction with A. Check if the resulting DFA has any final state accessible from the initial one. 
 +</​hidden>​
  
-3. Define the reversal of a language $L$ as $ rev(L) = \{ w \in \sigma^* | rev(w) \in L \}$, where $rev(c_1c_2..c_n) = c_nc_(n - 1)..c_1$, $c_i \in \sigma, 1 \leq i \leq n$. Show that reversal is a closure property.+**7.2.** Write the "​complement"​ regular expression for $(10 \cup 0)^*(1 \cup \epsilon)$. 
 + 
 +**7.3.** (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. 
 + 
 +===== 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.
  
-4. Let $L \subset \sigma^*$ be a language and $c \in \sigma$ a symbol. The quotient of $L$ and $c$ is the language defined as $L/c = { w \in \sigma^* | wc \in L}$