Differences
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lfa:2024:lab11 [2025/01/09 00:09] cata_chiru |
lfa:2024:lab11 [2025/01/09 00:14] (current) cata_chiru [11.2. Closed under CFLs] |
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| Let \( L_1 \) be a context-free language and \( P = (K_1, \Sigma, \Gamma, \Delta_1, q^{1}_{0}, F_1) \) | 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. | 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: | We build the following PDA \( (K, \Sigma, \Gamma, \Delta, q_0, F) \) where: | ||
| + | |||
| - \( K = K_1 \times K_2 \) | - \( K = K_1 \times K_2 \) | ||
| + | |||
| - \( q_0 = (q^{1}_{0}, q^{2}_{0}) \) | - \( q_0 = (q^{1}_{0}, q^{2}_{0}) \) | ||
| - | |||
| - | Let \( L_2 \) be a regular language and \( A = (K_2, \Sigma, \delta, q_{20}, F_2) \) be its respective DFA. | ||
| Transition rules for \( \Delta \): | Transition rules for \( \Delta \): | ||
| 1. \( ((q_1, q_2), c, \alpha, (q_1', q_2'), \beta) \in \Delta \) | 1. \( ((q_1, q_2), c, \alpha, (q_1', q_2'), \beta) \in \Delta \) | ||
| - | if \( (q_1, c, \alpha, q_1', \beta) \in \Delta_1 \) and | + | iff \( (q_1, c, \alpha, q_1', \beta) \in \Delta_1 \) and \( \delta(q_2, c) = q_2' \). |
| - | \( \delta(q_2, c) = q_2' \). | + | |
| 2. \( ((q_1, q_2), \epsilon, \alpha, (q_1', q_2), \beta) \in \Delta \) | 2. \( ((q_1, q_2), \epsilon, \alpha, (q_1', q_2), \beta) \in \Delta \) | ||
| - | if \( (q_1, \epsilon, \alpha, q_1', \beta) \in \Delta_1 \). | + | iff \( (q_1, \epsilon, \alpha, q_1', \beta) \in \Delta_1 \). |
| - | Accepting states: | ||
| - \( F = F_1 \times F_2 \) | - \( F = F_1 \times F_2 \) | ||
| Line 192: | Line 192: | ||
| **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.> | <hidden Solution 11.2.6.> | ||
| - | Let \( A \) be a context-free language and \( B \) a regular language. $ \overline{B} $ 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 $ A \setminus B = A \cap \overline{B} $. | + | 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> | </hidden> | ||