Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
10.5.1 Representing the derivations
Consider the context free grammar G1 =〈Σ,N,→〉, where
Σ ={p, q, r, ¬, ∨, ∧},
N ={S}, and
→ has the following 6 pairs in it:
S → p S → q S → r
S →¬S S → S ∨ S S → S ∧ S
This grammar is ambiguous since we have two derivation trees for ¬p ∧ q:
S
¬ S
S
p
∧ S
q
Here we see that the yield ¬p ∧ q does not determine the derivation.
One way to eliminate the ambiguity is with parentheses. Another way is to use Polish notation. Consider
the context free grammar G2 =〈Σ,N,→〉, where
Σ ={p, q, r, ¬, ∨, ∧},
N ={S}, and
→ has the following 6 pairs in it:
S
¬ S
p
S
∧ S
S → p S → q S → r
S →¬S S →∨SS S→∧SS
With this grammar, we have just one derivation tree for ∧¬pq, and just one for ¬∧pq:
Consider the minimalist grammar G2 =〈Σ,N,Lex,F〉,where
Σ ={p, q, r, ¬, ∨, ∧},
N ={S}, and
Lex has the following 6 lexical items built from Σ and N:
p :: S q :: S r :: S
¬ :: =S S ∨ :: =S =S S ∧ :: =S =S S
This grammar has ambiguous expressions, since we have the following two different two derivations of ¬p ∧q:
∧q : =S S
∧ :: =S =S S q :: S
¬p ∧ q : S
¬p : S
¬ :: =S S p :: S
q
¬p ∧ q : S
¬ :: =S S p ∧ q : S
∧q : =S S
∧ :: =S=SS q :: S
p :: S
These correspond to trees that we might depict with X-bar structure in the following way:
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