Consider the context-free grammar G below
$$\matrix{ S & \to & {aSb|X} \cr X & \to & {aX|Xb|a|b,} \cr } $$
where S and X are non-terminals, and a and b are terminal symbols. The starting non-terminal is S.
Which one of the following statements is CORRECT?
Consider the pushdown automation (PDA) P below, which runs on the input alphabet {a, b}, has stack alphabet {$$\bot$$, A}, and has three states {s, p, q}, with s being the start state. A transition from state u to state v, labelled c/X/$$\gamma$$, where c is an input symbol or $$\in $$, X is a stack symbol, and $$\gamma$$ is a string of stack symbols, represents the fact that in state u, the PDA can read c from the input, with X on the top of its stack, pop X from the stack, push in the string $$\gamma$$ on the stack, and go to state v. In the initial configuration, the stack has only the symbol $$\bot$$ in it. The PDA accepts by empty stack.
Which one of the following options correctly describes the language accepted by P?
Consider the following languages:
$$\eqalign{ & {L_1} = \{ ww|w \in \{ a,b\} *\} \cr & {L_2} = \{ {a^n}{b^n}{c^m}|m,\,n \ge 0\} \cr & {L_3} = \{ {a^m}{b^n}{c^n}|m,\,n \ge 0\} \cr} $$
Which of the following statements is/are FALSE?