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7. Closure properties

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

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All strategies require building DFAs from the regexes.

  1. 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.
  2. Use the indistinguishability algorithm to check if the two initial states of the two DFAs are indistinguishable.
  3. Check $ L(A_1) \subseteq L(A_2)$ and $ L(A_2) \subseteq L(A_1)$ . The condition $ L(A) \subseteq L(B)$ can be restated as $ 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.

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

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Build the DFA for that regex and then build the complement of that DFA. Transform it into a regex.

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)$ .

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Build the DFA $ A^R$ which accepts $ reverse(L(A))$ . Build the product construction between $ A$ and $ A^R$ .

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\} $.

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

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$ L/a = L((aaa \cup ba)^*(aa \cup b))$

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

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$ L/a = L(a^*)$

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

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Let $ A$ be a DFA which accepts $ L$ . We build a new DFA $ A/c$ which accepts $ L/c$ . This new DFA has the same number of states, initial state, same transitions and same alphabet. The set of final states is different though: $ F/c = \{q' \in K $ . In other words, the final states of $ A/c$ are those states which are $ c-$ predecessors of some final state of $ A$ . Note that some $ q'$ may be equal to $ q$ , and it may also be the case that no such predecessors exist.

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$.

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The easiest way is to rely on other closure properties: $ c/L = reverse(reverse(L)/c)$

Suffixes and prefixes

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.

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Suppose $ A$ is the accepting DFA of L. We build a new NFA from A by adding $ e-$ transitions from the initial state to all other states of A. Thus, we can nondeterministically skip any prefix $ x$ of any accepted word from $ L$ . This new DFA accepts $ \text{suffix(L)$

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 \} $.

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

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

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

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

Extras

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.