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

Stabler - Lx 185/209 2003
(9) Example: Let Γ be the following sequence of definite clauses:
socrates_is_a_man.
socrates_is_dangerous.
socrates_is_mortal :- socrates_is_a_man.
socrates_is_a_dangerous_man :socrates_is_a_man,
socrates_is_dangerous.
Clearly, we can prove socrates_is_mortal and socrates_is_a_dangerous_man.
The proof can be depicted with a tree like this:
(10) Example: Acontextfreegrammar
where
socrates_is_a_dangerous_man
socrates_is_a_man socrates_is_dangerous
G =〈Σ,N,→, S〉,
1. Σ,N are finite, nonempty sets,
2. S is some symbol in N,
3. the binary relation (→) ⊆ N × (Σ ∪ N) ∗ is also finite (i.e. it has finitely many pairs),
For example,
ip → dp i1 i1 → i0 vp i0 → will
dp → d1 d1 → d0 np d0 → the
np → n1 n1 → n0
n1 → n0 cp
n0 → idea
vp → v1 v1 → v0 v0 → suffice
cp → c1 c1 → c0 ip c0 → that
Intuitively, if ip is to be read as “there is an ip,” and similarly for the other categories, then the rewrite
arrow cannot be interpreted as implies, since there are alternative derivations. That is, the rules (n1 →
n0) and (n1 → n0 cp) signify that a given constituent can be expanded either one way or the other.
In fact, we get an appropriate logical reading of the grammar if we treat the rewrite arrow as meaning
“if.” With that reading, we can also express the grammar as a prolog theory.
/*
* file: th2.pl
*/
ip :- dp, i1. i1 :- i0, vp. i0 :- will. will.
dp :- d1. d1 :- d0, np. d0 :- the. the.
np :- n1. n1 :- n0. n0 :- idea. idea.
n1 :- n0, cp.
vp :- v1. v1 :- v0. v0 :- suffice. suffice.
cp :- c1. c1 :- c0, ip. c0 :- that. that.
In this theory, the proposition idea can be read as saying that this word is in the language, and ip :dp,
i1 says that ip is in the language if dp and i1 are. The proposition ip follows from this theory.
After loading this set of axioms, we can prove ?- ip. Finding a proof corresponds exactly to finding a
derivation from the grammar.
9