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:

${\displaystyle A\to aA_{1}A_{2}\cdots A_{n}}$

where ${\displaystyle A}$ is a nonterminal symbol, ${\displaystyle a}$ is a terminal symbol, ${\displaystyle A_{1}A_{2}\ldots A_{n}}$ is a (possibly empty) sequence of nonterminal symbols not including the start symbol and ${\displaystyle S}$ is the start symbol.

Observe that the grammar does not have left recursions.

Every context-free grammar can be transformed into an equivalent grammar in Greibach normal form.[1] 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.[2] 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 n, any top-down parser will halt at depth n.