Pushdown Automata deriving context free language

Question

I'm trying to understand how you derive a context free grammar (CFG) from a Pushdown Automata (PDA)?

I have the following PDA.

I believe the following context-free language can be derived..

{0^n1^n|n≥0}

My problem is I don't understand how you can build the context free grammar from a PDA and vice versa. How would I build a PDA from a CFG? I'd really appreciate if someone could walk me through this process.

Answer 1

To construct a PDA from CFG, we only need to use these two simple rules:

1. If the rule is of the form $S \rightarrow t$, where $t$ is a terminal, the transition is $t, t \rightarrow \epsilon$.
2. If the rule is of the form $S \rightarrow s$, where $s$ is any string (including variables and terminals), $|s| > 1$, the transition is $\epsilon, S \rightarrow s$.

The first rule is simple and straightforward, but the second rule requires a little bit of work. Let says, we have a CFG defined as follows: $$S \rightarrow aTXb$$ $$T \rightarrow XTS | \epsilon$$ $$X \rightarrow a | b$$ Initially, we have three states, where S stands for starting rule and \$ stands for stack symbol. Your book probably uses Z for stack symbol, so feel free to change it. It's just the matter of preference.

The restriction of a stack in PDA is that we can only push "one" symbol at a time. So to push a production rule onto a stack, we will break them into variables and terminals. Specifically, have a look at above picture, you can see that the first transition rule says push $S\$$ onto a stack, where $\$$ is stack symbol. This is not allowed in PDA, so we have to break the transition into two steps:

Note that this transition is a little special because it involves both stack variable $\$$ and the starting variable $S$. The next example of breaking long production rule should be more obvious. Firstly, all the production rule will go through the state after we pop $S$, i.e 2. We then apply the same procedure for any production rule that has length greater 1. So our first transition graph is as below:

In this graph, here are three transitions that need to be reconstructed. They are: $$\epsilon, \epsilon \rightarrow S\$$$ $$\epsilon, S \rightarrow XTS$$ $$\epsilon, S \rightarrow aTXb$$

Since we've already done $\epsilon, \epsilon \rightarrow S$, the next transition need to reconstructed is $\epsilon, S \rightarrow aTXb$. Note that the order of popping is from "right to left". So we pop $S \rightarrow T \rightarrow X$ respectively as below:

Reconstruct the last transition rule $\epsilon, S \rightarrow aTXb$, we have our final PDA as follows:

Reference

• Introduction to the Theory of Computation by Micheal Sipser
• Lecture notes from prof. Marvin K. Nakayama, New Jersey Institute of Technology