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--- lfa:2024:lab11 [2025/01/08 22:56]
cata_chiru
+++ lfa:2024:lab11 [2025/01/09 00:14] (current)
cata_chiru [11.2. Closed under CFLs]
@@ Line 164: / Line 164: @@
 **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 \( \Delta \):
+. \( ((q_1, q_2), c, \alpha, (q_1', q_2'), \beta) \in \Delta \)
+   iff \( (q_1, c, \alpha, q_1', \beta) \in \Delta_1 \) and \( \delta(q_2, c) = q_2' \).
+. \( ((q_1, q_2), \epsilon, \alpha, (q_1', q_2), \beta) \in \Delta \)
+   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.
+<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>