Introduction to Computational Linguistics

20. Greibach Normal Form 78
Recall that a derivation defines a set of constituent occurrences, which in turn
constitute the nodes of the tree. Notice that each occurrence of a nonterminal
is replaced by some right hand side of a rule during a derivation that leads to a
terminal string. After it has been replaced, it is gone and can no longer figure in a
derivation. Given a tree, a linearization is an ordering of the nodes which results
from a valid derivation in the following way. We write x ⊳ y if the constituent of
x is expanded before the constituent of y is. One can characterize exactly what it
takes for such an order to be a linearization. First, it is linear. Second if x > y then
also x ⊳ y. It follows that the root is the first node in the linearization.
Linearizations are closely connected with search strategies in a tree. We shall
present examples. The first is a particular case of the so–called depth–first search
and the linearization shall be called leftmost linearization. It is as follows. x ⊳ y
iff x > y or x ⊏ y. (Recall that x ⊏ y iff x precedes y. Trees are always considered
ordered.) For every tree there is exactly one leftmost linearization. We shall
denote the fact that there is a leftmost derivation of ⃗α from X by X ⊢ l G
⃗α. We can
generalize the situation as follows. Let ◭ be a linear ordering uniformly defined
on the leaves of local subtrees. That is to say, if B and C are isomorphic local trees
(that is, if they correspond to the same rule ρ) then ◭ orders the leaves B linearly
in the same way as ⊳ orders the leaves of C (modulo the unique (!) isomorphism).
In the case of the leftmost linearization the ordering is the one given by ⊏. Now
a minute’s reflection reveals that every linearization of the local subtrees of a tree
induces a linearization of the entire tree but not conversely (there are orderings
which do not proceed in this way, as we shall see shortly). X ⊢G ◭ ⃗α denotes
the fact that there is a derivation of ⃗α from X determined by ◭. Now call π a
priorization for G = 〈S, N, A, R〉 if π defines a linearization on the local tree H ρ ,
for every ρ ∈ R. Since the root is always the first element in a linearization, we
only need to order the daughters of the root node, that is, the leaves. Let this
ordering be ◭. We write X ⊢ π G ⃗α if X ⊢◭ G
⃗α for the linearization ◭ defined by π.
Proposition 26 Let π be a priorization. Then X ⊢ π G ⃗x iff X ⊢ G ⃗x.
A different strategy is the breadth–first search. This search goes through the tree
in increasing depth. Let S n be the set of all nodes x with d(x) = n. For each n,
S n shall be ordered linearly by ⊏. The breadth–first search is a linearization ∆,
which is defined as follows. (a) If d(x) = d(y) then x ∆ y iff x ⊏ y, and (b) if
d(x) < d(y) then x ∆ y. The difference between these search strategies, depth–first
and breadth–first, can be made very clear with tree domains.