====== 9. Context-Free Grammars ======

===== 9.1. Generating a CF language =====

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}\} $.

<hidden Solution 9.1.1>

$ S \leftarrow aSa\ |\ bSb \ |\ a \ |\ b \ |\ \epsilon $ \\

</hidden>

**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 =====

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 $

<hidden Solution 9.2.1>

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

</hidden>

**9.2.2.**

$ S \leftarrow AS\ |\ \epsilon $ \\
$ 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.**

$ S \leftarrow ASB\ |\ BSA\ |\ \epsilon $\\
$ 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 =====

**Definition:** A regular grammar is a CF grammar where the production rules follow one of these 2 patterns: 
  - all of them are of the form:
       * $ X \leftarrow aY $
       * $ X \leftarrow Y $
       * $ X \leftarrow a $
       * $ X \leftarrow \epsilon $
  - OR all of them are of the form:
       *  $ X \leftarrow Ya $
       * $ X \leftarrow Y $
       * $ X \leftarrow a $
       * $ X \leftarrow \epsilon $

**9.3.1.** (warmup) Find a DFA or an NFA that accepts the language generated by the following regular grammar:

$ X \leftarrow 0X\ |\ 1Y $

$ 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
  * prove that any regular grammar generates a regular language: **start** from a regular grammar G and **construct** an NFA $ N $ such that $ L(N) = L(G) $
  * prove that any regular language is generated by a regular grammar: **start** from a DFA $ A $ and **construct** a regular grammar such that $ L(G) = L(A) $
  * write the proofs for both definitions (the one with $ X \leftarrow aY $ and the one with $ X \leftarrow Ya $). 
  * **Hint**: for the first definition, think top-down; for the second one, think bottom-up;
  * **Hint 2**: use induction in the proofs