====== Grammars  ======

=== Writing Grammars === 

**1.** Write grammars for the following languages:

**1.1.** $ L = \{a^{m}b^{m + n}c^{n} | n, m \geq 0 \} $ \\
**1.2.** $ L = \{a^ib^jc^k | i = j \lor j = k \} $ \\
**1.3.** $ L = \{w \in \{a, b\}^* | \#_a(w) \neq \#_b(w) \} $

=== Ambiguity ===

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

**2.1.**

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

**2.2.**

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

**2.3.**

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

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

=== Regular Grammars ===

**4.** Is the language described by the following grammar regular? If so, write a regular expression for it.

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

**5.** Write a regular expression for the language described by:

$ S \leftarrow aX $\\
$ X \leftarrow bY | S $\\
$ Y \leftarrow aX | bS | \epsilon $

**6.** Write a regular grammar for $ L((0 \cup 1^*)^*01^*) $.

=== Chomsky Normal Form === 

**7.** Remove "$ \epsilon \text{-rules} $" from the following grammar:

$ A \leftarrow \epsilon | B $\\
$ B \leftarrow b $\\
$ B \leftarrow ABC | BAC $\\
$ C \leftarrow AC | c$

**8.** Remove the "unit rules" from the previous grammar, after "$ \epsilon \text{-rules} $" have been removed.

**9.** Apply the CNF conversion rules to the solution for **1.1.** Does the accepted language stay the same?