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--- lfa:2024:lab10 [2024/12/16 18:21]
cata_chiru
+++ lfa:2024:lab10 [2024/12/16 19:34] (current)
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@@ Line 5: / Line 5: @@
 Write CF grammar for the following languages. For each grammar, make sure it is not ambiguous. (Start with any CF grammar that accepts L. Then write another non-ambiguous grammar for the same language).
-**9.1.1.** $ L = \{\: w \in \{A,B\}^* \ | \:w \text{ is a palindrome}\} $.
+**9.1.1.** $ L = \{\: w \in \{a,b\}^* \ | \:w \text{ is a palindrome}\} $.
 <hidden Solution 9.1.1>
-$ S \leftarrow ASA\ |\ BSB \ |\ A \ |\ B \ |\epsilon $ \\
+$ S \leftarrow aSa\ |\ bSb \ |\ a \ |\ b \ |\ \epsilon $ \\
 </hidden>
-**9.1.2.** $ L = \{ A^{m} B^{m+n} C^{n} \ | \: n, m \geq 0 \} $
+**9.1.2.** $ L = \{ a^{m} b^{m+n} c^{n} \ | \: n, m \geq 0 \} $
+<hidden Solution 9.1.2>
+$ S \leftarrow AB $ \\
+$ A \leftarrow aAb\ |\ \ \epsilon $ \\
+$ B \leftarrow bBc\ |\ \ \epsilon $ \\
+</hidden>
 **9.1.3.** $ L = \{w \in \{a, b\}^* | \#_a(w) = \#_b(w) \} $
+<hidden Solution 9.1.3>
+$ S \leftarrow aBS\ |\ bAS\ |\ \epsilon $ \\
+$ A \leftarrow a\ |\ bAA $ \\
+$ B \leftarrow b\ |\ aBB $ \\
+</hidden>
 **9.1.4.** $ L = \{w \in \{a, b\}^* | \#_a(w) \neq \#_b(w) \} $
 **9.1.5.** $ L = \{a^ib^jc^k | i = j \lor j = k \} $ \\
+<hidden Solution 9.1.5>
+$ S \leftarrow C\ |\ A $ \\
+$ C \leftarrow Cc\ |\ B $ \\
+$ B \leftarrow aBb\ |\ \ \epsilon $ \\
+$ A \leftarrow aA\ |\ D $ \\
+$ D \leftarrow bDc\ |\ \ \epsilon $ \\
+</hidden>
 ===== 9.2. Ambiguity =====
@@ Line 30: / Line 59: @@
 $ A \leftarrow aA\ |\ B $ \\
 $ B \leftarrow bB\ |\ \epsilon $
+<hidden Solution 9.2.1>
+$ S \leftarrow A $ \\
+$ A \leftarrow aA\ |\ B $ \\
+$ B \leftarrow bB\ |\ \epsilon $
+</hidden>
 **9.2.2.**
@@ Line 36: / Line 73: @@
 $ A \leftarrow 0A1\ |\ 01\ |\ B $\\
 $ B \leftarrow B1\ |\ \epsilon $
+<hidden Solution 9.2.2>
+$ S \leftarrow AS\ |\ A $ \\
+$ A \leftarrow 0A1\ |\ B $\\
+$ B \leftarrow B1\ |\ \epsilon $
+</hidden>
 **9.2.3.**
@@ Line 42: / Line 88: @@
 $ A \leftarrow aA\ |\ \epsilon $\\
 $ B \leftarrow bB\ |\ \epsilon $
+<hidden Solution 9.2.3>
+$ S \leftarrow aS\ |\ bS\ |\ \epsilon $ \\
+</hidden>
 **9.2.4.** Write an ambiguous grammar for $ L(a^*) $.
+<hidden Solution 9.2.4>
+$ S \leftarrow aS \ |\ A\ |\ \epsilon $\\
+$ A \leftarrow aA\ |\ \epsilon $\\
+</hidden>
 ===== 9.3 Regular Grammars =====
@@ Line 64: / Line 124: @@
 $ Y \leftarrow 1Y\ |\ 0\ |\ \epsilon $
+<hidden Solution 9.3.1>
+Y produces $ 1^* (0 | \epsilon) $
+X produces $ 0^* 1Y \rightarrow 0^* 1^+ (0 | \epsilon) $
+</hidden>
 **9.4.1.** Prove that the languages accepted by regular grammars are exactly the regular languages