Introduction to Computational Linguistics

19. Parsing and Recognition 76
Definition 24 A CFG is in standard form if all the rules have the form X →
Y 1 Y 2 · · · Y n , with X, Y i ∈ N, or X → ⃗x. If in addition n = 2 for all rules of the first
form, the grammar is said to be in Chomsky Normal Form.
There is an easy way to convert a grammar into standard form. Just introduce a
new nonterminal Y a for each letter a ∈ A together with the rules Y a → a. Next
replace each terminal a that cooccurs with a nonterminal on the right hand side of
a rule by Y a . The new grammar generates more constituents, since letters that are
introduced together with nonterminals do not form a constituent of their own in
the old grammar. Such letter occurrences are called syncategorematic. Typical
examples of syncategorematic occurrences of letters are brackets that are inserted
in the formation of a term. Consider the following expansion of the grammar (??).
(197)
< term >→ < number >| (+)
| (*)
Here, operator symbols as well as brackets are added syncategorematically. The
procedure of elimination will yield the following grammar.
(198)
< term >→ < number >| Y ( < term > Y + < term > Y )
Y ( →(
Y ) →)
Y + →+
Y ∗ →*
| Y ( < term > Y + < term > Y )
However, often the conversion to standard form can be avoided however. It is
mainly interesting for theoretic purposes.
Now, it may happen that a grammar uses more nonterminals than necessary.
For example, the above grammar distinguishes Y + from Y ∗ , but this is not necessary.
Instead the following grammar will just as well.
(199)
< term >→ < number >| Y ( < term > Y o < term > Y )
Y ( →(
Y ) →)
Y o →+ | *