Question

Give context-free grammars to generate the following languages. Each CFG should have at most two variables.

$\\ (a) \ L_1 = \{ w \ | \ w \in \{0, 1\}^* \text{ and } w \text{ contains a number of 1's that is divisible by 3}\} \\ \-\ \-\ \-\ \-\ \-\ \Sigma_1 = \{ 0,1 \} \\ \\ (b) \ L_2 = \{ w \ | \ \text{the bottom row of w is the reverse of the top row of w}\} \\ \-\ \-\ \-\ \-\ \-\ \Sigma_2 = \left\{ \left[\begin{array}{c} 0 \\ 0 \end{array} \right ], \left[\begin{array}{c} 0 \\ 1 \end{array} \right ], \left[\begin{array}{c} 1 \\ 0 \end{array} \right ], \left[\begin{array}{c} 1 \\ 1 \end{array} \right ], \right\} \\ \\ (c) \ L_3 = \{ w \ | \ w \text{ is a balanced string of parentheses and brackets}\} \\ \-\ \-\ \-\ \-\ \-\ \Sigma_3 = \{ (,),[,] \} \\$