====== 9. Context-Free Languages ======

===== 9.1. Accepting and generating a CF language =====

Write a PDA as well as a CF grammar for the following languages. Specify - for each PDA, the way it accepts (by empty-stack or by final state). 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}\} $.

**9.1.2.** $ L = \{ A^{m} B^{m+n} C^{n} \ | \: n, m \geq 0 \} $

**9.1.3.** $ L = \{w \in \{a, b\}^* | \#_a(w) = \#_b(w) \} $

**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 \} $ \\

===== 9.2. Ambiguous grammars =====

Which of the following grammars are ambiguous? Justify each answer. Modify the grammar to remove ambiguity, wherever the case.

**9.2.1.**

$ S \leftarrow aA | A $ \\
$ A \leftarrow aA | B $ \\
$ B \leftarrow bB | \epsilon $

**9.2.2.**

$ S \leftarrow AS | \epsilon $ \\
$ A \leftarrow 0A1 | 01 | B $\\
$ B \leftarrow B1 | \epsilon $

**9.2.3.**

$ S \leftarrow ASB | BSA | \epsilon $\\
$ A \leftarrow aA | \epsilon $\\
$ B \leftarrow bB | \epsilon $

**9.2.4.** Write an ambiguous grammar for $ L(a^*) $.