Introduction to Computational Linguistics

18. Context Free Grammars 68
A context free grammar can be seen as a statement about sentential structure,
or as a device to generate sentences. We begin with the second viewpoint. We
write ⃗α ⇒ ⃗γ and say that ⃗γ is derived from ⃗α in one step if there is a rule X → ⃗η,
a single occurrence of X in ⃗α such that ⃗γ is obtained by replacing that occurrence
of X in ⃗α by ⃗η:
(176) ⃗α = ⃗σ ⌢ X ⌢ ⃗τ, ⃗γ = ⃗σ ⌢ ⃗η ⌢ ⃗τ
For example, if the rule is < number >→< number >< digit > then replacing
the occurrence of < number > in the string 1 < digit >< number > 0 will give
1 < digit >< number >< digit > 0. Now write ⃗α ⇒ n+1 ⃗γ if there is a ⃗ρ such that
⃗α ⇒ n ⃗ρ and ⃗ρ ⇒ ⃗γ. We say that ⃗γ is derivable from ⃗α in n + 1 steps. Using the
above grammar we have:
(177) < number >⇒ 4 < digit >< digit >< digit >< number >
Finally, ⃗α ⇒ ∗ ⃗γ (⃗γ is derivable from ⃗α) if there is an n such that ⃗α ⇒ n ⃗γ.
Now, if X ⇒ ∗ ⃗γ we say that ⃗γ is a string of category X. The language generated
by G consists of all terminal strings that have category S ∈ Σ. Equivalently,
(178) L(G) := {⃗x ∈ A ∗ : there is S ∈ Σ : S ⇒ ∗ ⃗x}
You may verify that the grammar above generates all strings of digits from the
symbol < number >. If you take this as the start symbol, the language will consist
in all strings of digits. If, however, you take the symbol < digit > as the start
symbol then the language is just the set of all strings of length one. This is because
even though the symbol < number > is present in the nonterminals and the rules,
there is no way to generate it by applying the rules in succession from < digit >. If
on the other hand, Σ contains both symbols then you get just the set of all strings,
since a digit also is a number. (A fourth possibility is Σ = ∅, in which case the
language is empty.)
There is a trick to reduce the set of start symbols to just one. Introduce a new
symbol S ∗ together with the rules
(179) S ∗ → S
for every S ∈ Σ. This is a different grammar but it derives the same strings.
Notice that actual linguistics is different. In natural language having a set of start