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lfa:lab06-closure-properties [2021/11/08 18:11]
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?
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 +<​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 $(10 \cup 0)^*(1 \cup \epsilon)$.