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 variables. 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:

$A \to a A_1 A_2 \cdots A_n$

or

$S \to \varepsilon$

where $A$ is a nonterminal symbol, $a$ is a terminal symbol, $A_1 A_2 \ldots A_n$ is a (possibly empty) sequence of nonterminal symbols not including the start symbol, $S$ is the start symbol, and ε is the empty word.^[1]

Observe that the grammar does not have left recursions.

Every context-free grammar can be transformed into an equivalent grammar in Greibach normal form.^[2] 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(n⁴) in the general case and O(n³) if no derivation of the original grammar consists of a single nonterminal symbol, where n is the size of the original grammar.^[3] This conversion can be used to prove that every context-free language can be accepted by a real-time 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 n, any top-down parser will halt at depth n.

See also

• Backus–Naur form
• Chomsky normal form
• Kuroda normal form

References

1. Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages and Computation. Reading, Massachusetts: Addison-Wesley Publishing. p. 95. ISBN 0-201-02988-X.
2. Greibach, Sheila (January 1965). "A New Normal-Form Theorem for Context-Free Phrase Structure Grammars". Journal of the ACM 12 (1). doi:10.1145/321250.321254.
3. Blum, Norbert; Koch, Robert (1999). "Greibach Normal Form Transformation Revisited". Information and Computation 150 (1): 112–118. doi:10.1006/inco.1998.2772. CiteSeerX: 10.1.1.47.460.