Greibach Normal Form

In formal language theory, a context-free grammar is in Greibach normal form (GNF) if the right-hand sides of all production rules start with a terminal symbol, optionally followed by some non-terminals. A non-strict form allows one exception to this format restriction for allowing the empty word (epsilon, ε) to be a member of the described language. The normal form was established by Sheila Greibach and it bears her name.

More precisely, a context-free grammar is in Greibach normal form, if all production rules are of the form:
:$A \to a A_1 A_2 \cdots A_n$

where $A$ is a nonterminal symbol, $a$ is a terminal symbol, and
$A_1 A_2 \ldots A_n$ is a (possibly empty) sequence of nonterminal symbols.

Observe that the grammar does not have left recursions.

Every context-free grammar can be transformed into an equivalent grammar in Greibach normal form. Various constructions exist. Some do not permit the second form of rule and cannot transform context-free grammars that can generate the empty word. For one such construction the size of the constructed grammar is O(⁴) in the general case and O(³) if no derivation of the original grammar consists of a single nonterminal symbol, where is the size of the original grammar. This conversion can be used to prove that every context-free language can be accepted by a real-time (non-deterministic) pushdown automaton, i.e., the automaton reads a letter from its input every step.

Given a grammar in GNF and a derivable string in the grammar with length , any top-down parser will halt at depth .

See also

* Backus–Naur form
* Chomsky normal form
* Kuroda normal form

References

*
* {{cite book, author=György E. Révész, title=Introduction to Formal Languages, url=https://books.google.com/books?id=3s7CAgAAQBAJ&q=%22Greibach+normal+form%22, date=17 March 2015, publisher=Courier Corporation, isbn=978-0-486-16937-8

Formal languages