Introduction to Computational Linguistics

19. Parsing and Recognition 72
The trees used in linguistic analysis are often ordered. The ordering is here
implicitly represented in the string. Let C = 〈⃗γ 1 , ⃗γ 2 〉 be an occurrences of ⃗σ and
D = 〈⃗η 1 , ⃗η 2 〉 be an occurrence of τ. We write C ⊏ D and say that C precedes D
if ⃗γ 1 ⃗σ is a prefix of ⃗η 1 (the prefix need not be proper). If one underlines C and D
this definition amounts to saying that the line of C ends before the line of D starts.
(190) abccddx
Here C = 〈a, cddx〉 and D = 〈abcc, x〉.
19 Parsing and Recognition
Given a grammar G, and a string ⃗x, we ask the following questions:
• (Recognition:) Is ⃗x ∈ L(G)?
• (Parsing:) What derivation(s) does ⃗x have?
Obviously, as the derivations give information about the meaning associated with
an expression, the problem of recognition is generally not of interest. Still, sometimes
it is useful to first solve the recognition task, and then the parsing task. For if
the string is not in the language it is unnecessary to look for derivations. The parsing
problem for context free languages is actually not the one we are interested
in: what we really want is only to know which constituent structures are associated
with a given string. This vastly reduces the problem, but still the remaining
problem may be very complex. Let us see how.
Now, in general a given string can have any number of derivations, even infinitely
many. Consider by way of example the grammar
(191) A → A | a
It can be shown that if the grammar has no unary rules and nor rules of the form
X → ε then a given string ⃗x has an exponential number of derivations. We shall
show that it is possible to eliminate these rules (this reduction is not semantically
innocent!). Given a rule X → ε and a rule that contains X on the right, say