====== Push down automata ======

**Exercise 1.** Write PDAs for the following languages:

**1.1.** $ L = \{\: w \in \{A,B\}^* \ | \:w \text{ is a palindrome}\} $

**1.2.** $ L = \{\: w \in \{A,B\}^* \ | \: \text{#}A(w) \neq \: \text{#}B(w) \} $

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

**1.4.** $ L = \{ A^{n}B^{n}C^{m}D^{m} | \: \ n,m \geq 0 \}  \cup  \{ A^nB^mC^mD^n | \: \ n,m \geq 0 \} $

**1.5.** $ L = \{ A^{i}B^{j}C^{k} | \: \text{ i=j or j=k} \} $

**Exercise 2.** Acceptance by empty stack.

Consider the following definition for an accepted word, by a PDA:
 $(q_{0}, w, Z_{0}) \vdash (q,e,e)$ where $ q $ is any state from K.

Prove that a language is CF if it can be accepted by a PDA via the empty-stack definition.

(**Hint**, you need to prove two parts)