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'Greibach normal form'
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| Title of object: |
Greibach normal form |
| Canonical Name: |
GreibachNormalForm |
| Type: |
Definition |
| Created on: |
2009-05-11 22:42:35 |
| Modified on: |
2009-05-12 03:02:10 |
| Classification: |
msc:68Q42, msc:68Q45 |
Preamble:
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Content:
A formal grammar $G=(\Sigma,N,P,\sigma)$ is said to be in \emph{Greibach normal form} if every production has the following form:
$$A\to aW$$
where $A \in N$ (a non-terminal symbol), $a \in \Sigma$ (a terminal symbol), and $W\in N^*$ (a word over $N$).
A formal grammar in Greibach normal form is a context-free grammar. Moreover, any context-free language not containing the empty word $\lambda$ can be generated by a grammar in Greibach normal form. And if a context-free language $L$ contains $\lambda$, then $L$ can be generated by a grammar that is in Greibach normal form, with the addition of the production $\sigma \to \lambda$.
Let $L$ be a context-free language not containing $\lambda$, and let $G=(\Sigma,N,P,\sigma)$ be a grammar in Greibach normal form generating $L$. We construct a PDA $M$ from $G$ based on the following specifications:
\begin{enumerate}
\item $M$ has one state $p$,
\item the input alphabet of $M$ is $\Sigma$,
\item the stack alphabet of $M$ is $N$,
\item the initial stack symbol of $M$ is the start symbol $\sigma$ of $G$,
\item the start state of $M$ is the only state of $M$, namely $p$
\item there are no final states,
\item the transition function $T$ of $M$ takes $(p,a,A)$ to the singleton $\lbrace (p,W)\rbrace$, provided that $A\to aW$ is a production of $G$. Otherwise, $T(p,a,A)=\varnothing$.
\end{enumerate}
It can be shown that $L=L(M)$, the language accepted on empty stack, by $M$. If we further define $T(p,\lambda,\sigma):=\lbrace (p,\lambda)\rbrace$, then $M$ accepts $L\cup \lbrace\lambda\rbrace$. As a result, any context-free language is accepted by some PDA. |
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