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lfa:lab06-closure-properties [2021/11/08 18:10]
pdmatei 		lfa:lab06-closure-properties [2022/11/17 19:16] (current)
pdmatei
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 **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? **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?
 +
 +<​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>​
  
 **7.2.** Write the "​complement"​ regular expression for $(10 \cup 0)^*(1 \cup \epsilon)$. **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. **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\} $. 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\} $.
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 **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$. **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. **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 \}$. Let $ min(L) = \{ w \in L | \; \nexists x \in L, \; y \in \Sigma^* \setminus \{\epsilon\},​ \: \text{such that} \: xy = w \}$.