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lfa:2024:lab11 [2025/01/08 22:45]
cata_chiru 		lfa:2024:lab11 [2025/01/09 00:14] (current)
cata_chiru [11.2. Closed under CFLs]
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 **11.2.1.** Concat is a closure property. **11.2.1.** Concat is a closure property.
 +
 +<hidden Solution 11.2.1.>
 +Let \( A \) and \( B \) be context-free languages over an alphabet \( \Sigma \), and let \( G_A = (V_A, \Sigma, R_A, S_A) \) and \( G_B = (V_B, \Sigma, R_B, S_B) \) be context-free grammars that generate \( A \) and \( B \) respectively. By renaming the variables if necessary, we assume that \( V_A \) is disjoint from \( V_B \), and that neither variable set contains the variable \( S \). We now construct a new CFG by writing:
 +
 +\[
 +G = (V_A \cup V_B \cup \{S\}, \Sigma, R_A \cup R_B \cup \{S \to S_AS_B\}, S)
 +\]
 +
 +</​hidden>​
 +
 +
  
 **11.2.2.** Union is a closure property. **11.2.2.** Union is a closure property.
 +
 +<hidden Solution 11.2.2.>
 +Let \( A \) and \( B \) be context-free languages over an alphabet \( \Sigma \), and let \( G_A = (V_A, \Sigma, R_A, S_A) \) and \( G_B = (V_B, \Sigma, R_B, S_B) \) be context-free grammars that generate \( A \) and \( B \) respectively. By renaming the variables if necessary, we assume that \( V_A \) is disjoint from \( V_B \), and that neither variable set contains the variable \( S \). We now construct a new CFG by writing:
 +
 +\[
 +G = (V_A \cup V_B \cup \{S\}, \Sigma, R_A \cup R_B \cup \{S \to S_A | S_B\}, S)
 +\]
 +
 +</​hidden>​
  
 **11.2.3.** Kleene Star is a closure property. **11.2.3.** Kleene Star is a closure property.
 +
 +<hidden Solution 11.2.3.>
 +Let \( A \) be a context-free languages over an alphabet \( \Sigma \), and let \( G_A = (V_A, \Sigma, R_A, S_A) \) be the context-free grammar that generate \( A \). We now construct a new CFG by writing:
 +
 +\[
 +G = (V_A \cup \{S\}, \Sigma, R_A \cup \{S \to SS | S_A | \epsilon\}, S)
 +\]
 +
 +</​hidden>​
 +
  
 **11.2.4.** Reverse is a closure property. **11.2.4.** Reverse is a closure property.
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 **11.2.5.** Intersection with a regular language is a closure property. **11.2.5.** Intersection with a regular language is a closure property.
 +
 +<hidden Solution 11.2.5.>
 +Let \( L_1 \) be a context-free language and \( P = (K_1, \Sigma, \Gamma, \Delta_1, q^{1}_{0}, F_1) \)  ​
 +be its respective PDA.  ​
 +
 +Let \( L_2 \) be a regular language and \( A = (K_2, \Sigma, \delta, q_{20}, F_2) \) be its respective DFA.  ​
 +
 +We build the following PDA \( (K, \Sigma, \Gamma, \Delta, q_0, F) \) where:  ​
 +
 +- \( K = K_1 \times K_2 \)  ​
 +
 +- \( q_0 = (q^{1}_{0}, q^{2}_{0}) \)  ​
 +
 +Transition rules for \( \Delta \):  ​
 +
 +1. \( ((q_1, q_2), c, \alpha, (q_1', q_2'), \beta) \in \Delta \)  ​
 +   iff \( (q_1, c, \alpha, q_1', \beta) \in \Delta_1 \) and \( \delta(q_2, c) = q_2' \).  ​
 +
 +2. \( ((q_1, q_2), \epsilon, \alpha, (q_1', q_2), \beta) \in \Delta \)  ​
 +   iff \( (q_1, \epsilon, \alpha, q_1', \beta) \in \Delta_1 \).  ​
 +
 +- \( F = F_1 \times F_2 \)  ​
 +
 +The PDA accepts \( L(P) \cap L(A) \).
 +</​hidden>​
  
 **11.2.6.** Difference with a regular language is a closure property. **11.2.6.** Difference with a regular language is a closure property.
 +<hidden Solution 11.2.6.>
 +Let \( L_1 \) be a context-free language and \( L_2 \) a regular language.
 +
 +$ \overline{L_2} $ is regular, as complement is a closure property for regular languages.
 +
 +From 11.2.5, we know that the intersection is closed between CFLs and Regular Languages, and write $ L_1 \setminus L_2 = L_1 \cap \overline{L_2} $.
 +
 +</​hidden>​