Introduction to Computational Linguistics

20. Greibach Normal Form 77
The reduction of the number of nonterminals has the same significance as in the
case of finite state automata: it speeds up recognition, and this can be significant
because not only the number of states is reduced but also the number of rules.
Another source of time efficiency is the number of rules that match a given
right hand side. If there are several rules, we need to add the left hand symbol for
each of them.
Definition 25 A CFG is invertible if for any pair of rules X → ⃗α and Y → ⃗α we
have X = Y.
There is a way to convert a given grammar into invertible form. The set of nonterminals
is ℘(N), the powerset of the set of nonterminals of the original grammar.
The rules are
(200) S → T 0 T 1 . . . T n−1
where S is the set of all X such that there are Y i ∈ T i (i < n) such that X →
Y 0 Y 1 . . . Y n−1 ∈ R. This grammar is clearly invertible: for any given sequence
T 0 T 1 · · · T n−1 of nonterminals the left hand side S is uniquely defined. What needs
to be shown is that it generates the same language (in fact, it generates the same
constituent structures, though with different labels for the constituents).
20 Greibach Normal Form
We have spoken earlier about different derivations defining the same constituent
structure. Basically, if in a given string we have several occurrences of nonterminals,
we can choose any of them and expand them first using a rule. This is
because the application of two rules that target the same string but different nonterminals
commute:
(201)
· · · X · · · Y · · ·
· · · ⃗α · · · Y · · · · · · X · · · ⃗γ · · ·
· · · ⃗α · · · ⃗γ · · ·
This can be exploited in many ways, for example by always choosing a particular
derivation. For example, we can agree to always expand the leftmost nonterminal,
or always the rightmost nonterminal.