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

Stabler - Lx 185/209 2003
(11) Predicate prolog
language: atoms a ={a− z}{a − zA − Z0 − 9_} ∗ | ′ {a − zA − Z0 − 9_ @#$% ∗ ()} ∗′
variables v ={A − Z_}{a − zA − Z0 − 9_} ∗
terms T ::= v | a | a(S)| 0 − 9 +
sequence of terms S ::= T | T,S (n.b. one or more terms in the sequence)
predications p ::= a | a(S)
conjunctions C ::= ɛ. | p, C
goals G ::= ?-C
definite clauses D ::= p:-C
(Notation) We write f n for an atom f that forms a term with a sequence of n terms as arguments. A term f 0 is
an individual constant.
We write pn for an atom p that forms a predication with a sequence of n termsasarguments. A0-place
predication p0 is a proposition.
inference: G, Γ ⊢ G [axiom] for any set of definite clauses Γ and any goal G
G, Γ ⊢ (?-p, C)
G, Γ ⊢ (?-q1,...,qn,C)θ
if (q:-q1,...,qn) ∈ Γ ,mgu(p,q)= θ
N.B. For maximal generality, we avoid confusing any variables in the goal with variables in an axiom
with the following policy: every time a definite clause is selected from Γ , rename all of its variables
with variables never used before in the proof. (Standard implemented prolog assigns these new
variables numerical names, like _1295.)
semantics: amodelM=〈E,2,[[·]]〉, where
[[ fn ]] : En → E. Whenn = 0, [[ f0 ]] ∈ E.
[[ pn ]] : En → 2. When n = 0, [[ p0 ]] ∈ 2.
[[ v]] : [V → E] → E, such that for s : V → E, [[ v]] ( s ) = s(v).
[[ fn (t1,...tn)]] : [V → E] → E, wherefors : V → E, [[ fn (t1,...,tn)]](s) = [[ fn ]] ( [[ t1]](s), . . . , [[tn]](s))
[[ pn (t1,...,tn)]] : [V → E] → 2, where for s : V → E, [[ pn (t1,...,tn)]](s) = [[ pn ]] ( [[ t1]](s), . . . , [[tn]](s))
[[ A, B]] : [V → E] → 2, where for s : V → E, [[ A, B]] ( s ) = min{[[ A]](s), [[B]](s))}.
[[ ɛ ]] : [V → E] → 2, where for s : V → E, [[ ɛ ]] ( s ) = 1
⎧
⎨0
if ∃s ∈ [V → E], [[A]] ( s ) = 1and[[ B]] ( s ) = 0
[[ B:-A]] =
⎩1
otherwise
[[ ?-A]] = 1 − min{[[ A]] ( s ) | s : V → E}
metatheory: G, Γ ⊢ A iff G, Γ ⊨ A, for any goals G, A and any definite clause theory Γ .
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