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I am looking for a context-sensitive grammar $G$ for the language $$L = \left\{w \in \left\{0,1\right\}^* | \; |w|_0 \geq |w|_1 \right\}$$ where $|w|_0$ means the amount of the terminal $0$ in the word $w$.

I know how to create a PDA for this by "remembering" the amout of $1$'s, but I do not know how to create a Grammar (respectively the production rules).

Could you please help me finding the answer? Is there a general way of "converting" a PDA definition to grammar rules?

Thanks in advance!

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1 Answer 1

up vote 1 down vote accepted

Since you can recognize the language with a PDA, it must be context-free, so we can do better than a context-sensitive grammar. Here’s a context-free grammar that should work; $S$ is the initial symbol, $\lambda$ is the empty string, $E$ is intended to generate words with an equal number of $0$’s and $1$’s, and $U$ is intended to generate words with more $0$’s than $1$’s.

$$\begin{align*} &S\to U\\ &U\to E0U\mid E0E\\ &E\to 0E1E\mid 1E0E\mid\lambda \end{align*}$$

(I don’t really need $U$: I could use $S$ instead.)

The basic idea is that there has to be at least one extra $0$; the second production generates it, possibly preceded by a string with an equal number of $0$’s and $1$’s; what follows it may but need not have excess $0$'s.

There is an algorithm for converting a non-deterministic PDA to a CFG, but it’s fairly complicated. (The opposite direction is easier.) One version is outlined in these notes starting at page 9. These notes do it a little differently and in pretty full detail.

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Thank you, great! –  muffel Oct 10 '11 at 14:28

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