I gotta make a CFG and PDA for the grammar that has perfectly nested parentheses and brackets.

$\qquad\begin{align} S &\to [S] \\ S &\to (S) \\ S &\to SS \\ S &\to \varepsilon \end{align}$

Not sure if this is correct, or how to make the PDA from it?

  • $\begingroup$ Try using the standard construction from the proof that CFG and NPDA are equally powerful! Does "perfectly nested" exclude $([)]$ here? $\endgroup$
    – Raphael
    Nov 19, 2012 at 18:11

2 Answers 2


The language you study is a classic, the one-sided Dyck language (on two pairs of brackets). You can directly make a PDA by considering the following property of nested strings: every symbol closing bracket you read should match the last unmatched opening bracket. Keep the unmatched $[$ and $($ on the stack and you are ready to go.


A: your CFG looks good.

B: There is a very well-known method of converting CFGs to PDAs.

Check https://www.youtube.com/watch?v=MJ9xNavURY8

Or the Wikipedia article on Pushdown automata

also this question has some interesting details.

But mainly - google "Converting CFG ro PDA".


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