With the standard logical forms produced by the syntactic and semantic translation components of current theoretical frameworks and implemented systems, it would seem that an inferencing
Trang 1J e r r y R Hobbs SRI International
M e n l o Park, California
i The P r o b l e m
C o n s i d e r the s e n t e n c e
In most democratic countries most politicians
can fool most of the people on almost every
issue most of the time
In the c u r r e n t l y standard ways of representing
q u a n t i f i c a t i o n in logical form, this sentence has
120 different readings, or q u a n t i f i e r scopings
Moreover, they are truly distinct, in the sense
that for any two readings, there is a model that
s a t i s f i e s one and not the other With the
standard logical forms produced by the syntactic
and semantic translation components of current
theoretical frameworks and implemented systems, it
would seem that an inferencing component must
process each of these 120 readings in turn in
order to produce a best reading Yet it is
obvious that people do not entertain all 120
possibilities, and people really do understand the
sentence The problem is not Just that
i n f e r e n c i n g is required for disamblguation It is
that people never do d l s a m b i g u a t e completely A
single q u a n t i f i e r scoping is never chosen (Van
L e h n [1978] and Bobrow and Webber [1980] have also
made this point.) In the currently standard
logical notations, it is not clear how this
vagueness can be represented 1
What is needed is a logical form for such
sentences that is neutral with respect to the
various scoplng possibilities It should be a
n o t a t i o n that can be used easily by an inferenclng
component That is, it should be easy to define
d e d u c t i v e operations on it, and the lo~ical forms
of typical sentences should not be unwieldy
Moreover, when the inferenclng component discovers
further information about dependencies among sets
of entities, it should entail only a minor
m o d i f i c a t i o n in the logical form, such as
c o n j o i n i n g a new proposition, rather than a major
restructuring Finally, since the notion of
"scope" is a powerful tool in semantic analysis,
there should be a fairly transparent relationship
between dependency information In the notation and
standard representations of scope
Three possible approaches are ruled out by
these criteria
i Representing the sentence as a
d i s j u n c t i o n of the various readings This is
impossibly unwieldy
I
Many people feel that most sentences exhibit too
few q u a n t i f i e r scope ambiguities for much effort
to be devoted to this problem, but a casual
inspection of several sentences from any text
should convince almost everyone otherwise
2 Using as the logical n o t a t i o n a triple
c o n s i s t i n g of an e x p r e s s i o n of the p r o p o s i t i o n a l content of the sentence, a store of q u a n t i f i e r structures (e.g., as in C o o p e r [1975], Woods [19781), and a set of c o n s t r a i n t s on how the quantifier structures could be unstored This would a d e q u a t e l y c a p t u r e the vagueness, but it is
d i f f i c u l t to imagine defining inference procedures that would work on such an object Indeed, Cooper did no inferenclng; Woods did little and chose a default reading h e u r i s t i c a l l y before doing so
3 Using a set-theoretlc notation like that
of (I) below, pushing all the universal quantifiers to the outside and the e x i s t e n t i a l quantifiers to the inside, and replacing the
e x i s t e n t i a l l y q u a n t i f i e d variables by Skolem functions of all the u n i v e r s a l l y q u a n t l f ~ e d variables Then when inferencing discovers a nondependency, one of the a r g u m e n t s is dropped from one of the Skolem functions One d i f f i c u l t y with this is that it yields r e p r e s e n t a t i o n s that are too general, being satisfied by models that correspond to none of the possible intended interpretations Moreover, in sentences in which one quantified noun phrase s y n t a c t i c a l l y embeds another (what Woods [1978] calls "functional nesting"), as in
Every representative of a c o m p a n y arrived
no r e p r e s e n t a t i o n that is neutral between the two
is immediately apparent W i t h wide scope, "a company" is existential, with narrow scope it is universal, and a shift in commitment from one to the other would involve s i g n i f i c a n t restructuring
of the logical form
The approach taken here uses the n o t i o n of the "typical element'" of a set, to produce a flat logical form of conjoined atomic predications A treatment has been worked out only for monotone increasing determiners; this is described in Section 2 In Section 3 some ideas about other determiners are discussed An inferenclng component, such as that explored in Hobbs [1976, 1980], capable of resolving coreference, doing coercions, and refining predicates, will be assumed (but not discussed) Thus, translating the q u a n t i f i e r scoping problem into one of those three processes will count as a solution for the purposes of this paper
This problem has received little a t t e n t i o n in linguistics and computational linguistics Those who have investigated the processes by which a rich knowledge base is used in interpreting texts have largely ignored quantifier ambiguities Those who have studied quantifiers have g e n e r a l l y noted that inferencing is required for
Trang 2n o t a t i o n that w o u l d a c c o m m o d a t e this inferencing
T h e r e are some exceptions B o b r o w and W e b b e r
[1980] discuss many of the issues involved, but it
is not e n t i r e l y clear what their proposals are
T h e w o r k of W e b b e r [1978] and M e l l l s h [1980] are
d i s c u s s e d below
2 M o n o t o n e I ~ c r e a s i n ~ D e t e r m i n e r s
2.1 A S e t - T h e o r e t i c N o t a t i o n
Let us represent the pattern of a simple
i n t r a n s i t i v e sentence w i t h a q u a n t i f i e r as "Q Ps
R" In '~ost m e n work," Q - "most", P = "man",
a n d R - "work" Q w i l l be referred to as a
d e t e r m i n e r A d e t e r m i n e r Q is m o n o t o n e i n c r e a s i n g
if and only if for any RI and R2 such that the
d e n o t a t i o n of R1 is a subset of the d e n o t a t i o n of
R2, "Q Ps RI" implies "Q Ps R2" (Barwlse and
C o o p e r [1981]) For example, letting RI - "work
hard" and R2 = "work", since "most m e n work hard"
i m p l i e s "most men work," the d e t e r m i n e r "most" is
m o n o t o n e increasing Intuitively, making the verb
p h r a s e more general doesn't change the truth
value Other m o n o t o n e i n c r e a s i n g determiners are
"every", "some", "many", "several", "'any" and "a
few" "No" and "few" are not
A n y noun phrase Q Ps w i t h a m o n o t o n e
i n c r e a s i n g d e t e r m i n e r Q involves two sets, an
i n t e n s i o n a l l y defined set denoted by the noun
phrase minus the determiner, the set of all Ps,
and a n o n c o n s t r u c t l v e l y specified set denoted by
the entire noun phrase The d e t e r m i n e r Q c a n be
v i e w e d as e x p r e s s i n g a relation between these two
sets Thus the sentence pattern Q Fs R can be
r e p r e s e n t e d as follows:
41) ( T s ) ( Q ( s , { x I P(x)}) & ( V Y ) ( ~ s -> R(y)))
T h a t is, there is a set s which bears the r e l a t i o n
Q to the set of all Ps, and R is true of every
e l e m e n t of s (Barwlse and C o o p e r call s a
"witness set".) "Most men work" w o u l d be
r e p r e s e n t e d
(~ s)(most(s,{x I man(x)})
& ( ~ y)(y~s -> w o r k ( y ) ) )
F o r c o l l e c t i v e predicates such as "meet" and
"agree", R would apply to the set rather than to
each of its elements
(3 s) 0(s,{x I F(x)}) ~ R(s)
S o m e t i m e s w i t h singular noun phrases and
d e t e r m i n e r s llke "a", "some" and "any" it will be
more convenient to treat the d e t e r m i n e r as a
r e l a t i o n between a set and one of its elements
( B Y) 0 ( y , { x I P(x)}) & R(y)
A c c o r d i n g to notation (i) there are two
aspects to quantification The first, which
concerns a relation between two sets, is discussed
in Section 2.2 The second aspect involves a
sets The approach taken here to this aspect of
q u a n t i f i c a t i o n is somewhat more radical, and depends on a v i e w of s e m a n t i c s that m i g h t be called "ontological promiscuity" This is
d e s c r i b e d briefly in S e c t i o n 2.3 T h e n in S e c t i o n 2.4 the s c o p e - n e u t r a l r e p r e s e n t a t i o n is presented
2.2 D e t e r m i n e r s as R e l a t i o n s b e t w e e n Sets
E x p r e s s i n g d e t e r m i n e r s as relations b e t w e e n sets allows us to express as axioms in a k n o w l e d g e base more refined p r o p e r t i e s of the d e t e r m i n e r s than can be c a p t u r e d by r e p r e s e n t i n g them in terms
of the standard quantlflers
First let us note that, w i t h the proper
d e f i n i t i o n s of "every" and "some",
( V sl,s2) e v e r y ( s l , s 2 ) <-> s l = s2 ( y x,s2) some(x, s2) <-> x~s2
f o r m u l a (I) reduces to the standard notation (This c a n be seen as e x p l a i n i n g why the
r e s t r i c t i o n is i m p l i c a t i v e in u n i v e r s a l
q u a n t i f i c a t i o n and c o n j u n c t i v e in e x i s t e n t i a l
q u a n t i f i c a t i o n )
A m e a n i n g postulate for "most" that is perhaps too m a t h e m a t i c a l is
( ~ s l , s 2 ) m o s t ( s l , s 2 ) -> Isll > i/2 Is21
Next, consider "any" Instead of trying to force an i n t e r p r e t a t i o n of "any" as a standard quantifier, let us take it to m e a n "a random element of"
(2) ( ~ x , s ) any(x,s) ~> x = random(s),
w h e r e "random" is a f u n c t i o n that returns a random element of a set This means that the
p r o t o t y p i c a l use of "any" is in sentences like
Pick any card
Let me surround this with caveats This can't be right, if for no other reason than that "any" is surely a more "primitive" n o t i o n in language than
"random" N e v e r t h e l e s s , m a t h e m a t i c s gives us firm intuitions about "random" and (2) may thus shed light o n some linguistic facts
M a n y of the linguistic facts about "any" can
be subsumed under two broad c h a r a c t e r i z a t i o n s :
i It requires a "modal" or " n o n d e f l n l t e " context For example, "John talks to any woman" must be interpreted d i s p o s i t l o n a l l y If we adopt (2), we can see this as deriving from the n a t u r e
of randomness It simply does not make sense to say of an actual entity that it is random
2 It n o r m a l l y acts as a u n i v e r s a l
q u a n t i f i e r outside the scope of the most immediate modal embedder T h i s is u s u a l l y the most natural
i n t e r p r e t a t i o n of "random"
Moreover, since "any" e x t r a c t s a single element, we can make sense out of cases in w h i c h
"any" fails to act llke "every"
58
Trang 3* I'Ii talk to e v e r y o n e but only to one person
J o h n w a n t s to m a r r y any Swedish woman
* John w a n t s to m a r r y every Swedish woman
(The second pair is due to Moore [1973].)
This a p p r o a c h does not, however, seem to
o f f e r an e s p e c i a l l y c o n v i n c i n g e x p l a n a t i o n as to
w h y "any" functions in q u e s t i o n s as an e x i s t e n t i a l
q u a n t i f i e r
2.3 O n t o l o g i c a l P r o m i s c u i t y
D a v i d s o n [1967] proposed a treatment of
a c t i o n sentences in w h i c h events are treated as
individuals This f a c i l i t a t e d the r e p r e s e n t a t i o n
of sentences with adverbials But v i r t u a l l y e v e r y
p r e d i c a t i o n that can be made in natural language
can be m o d i f i e d adverbially, be specified as to
time, function as a cause or effect of something
else, c o n s t i t u t e a belief, be nominalized, and be
r e f e r r e d to pronominally It is therefore
c o n v e n i e n t to extend Davidson's a p p r o a c h to all
p r e d i c a t i o n s , an a p p r o a c h that might be called
" o n t o l o g i c a l promiscuity" One abandons all
o n t o l o g i c a l scruples A similar approach is used
in many AI systems
We will use what might be called a
" n o m l n a l i z a t i o n " operator for predicates
C o r r e s p o n d i n g to every n-ary p r e d i c a t e p there
will be an n+l-ary predicate p" w h o s e first
a r g u m e n t can be thought of as a c o n d i t i o n of p's
being true of the subsequent arguments Thus, if
"see(J,B)" means that John sees Sill,
" s e e ' ( E , J , S ) " will mean that E is John's seeing of
Bill For the purposes of this paper, we can
c o n s i d e r that the primed and unprimed predicates
are related by the following a x i o m schema:
(3) ( ~ x,e) p'(e,x) -> p(x)
( V x ) ( ~ e ) p(x) -> p'(e,x)
It is beyond the scope of this paper to
e l a b o r a t e on the a p p r o a c h further, but it will be
assumed, and taken to extremes, in the r e m a i n d e r
of the paper Let me illustrate the extremes to
w h i c h it will be taken Frequently we want to
refer to the condition of two predicates p and q
h o l d i n g s i m u l t a n e o u s l y of x For this we will
refer to the entity e such that
a n d ' [ e , e l , e 2 ) & p*(el,x) & q'(e2,x)
Here el is the condition of p being true of x, e2
is the condition of q being true of X, and e the
c o n d i t i o n of the c o n j u n c t i o n being true
2.4 The S c o p e - N e u ¢ r a l R e p r e s e n t a t i o n
We will assume that a set has a typical
element and that the logical form for a plural
noun phrase will include reference to a set and
its ~z~ical element 2 The linguistic intuition
2 Woods [1978] mentions something llke this
approach, but rejects it because d i f f i c u l t i e s that
are worked out here would have to be worked out
pronouns and d e f i n i t e noun phrases as a n a p h o r s for plurals D e f i n i t e and indefinite g e n e r i c s can also be u n d e r s t o o d as r e f e r r i n g to the typical element of a set
In the spirit of o n t o l o g i c a l promiscuity, we simply a s s u m e that typical e l e m e n t s of s ~ ~re things that exist, and encode in meaning postulates the n e c e s s a r y r e l a t i o n s b e t w e e n a set's typical element and its real elements This move
a m o u n t s to r e i f y i n g the u n i v e r s a l l y q u a n t i f i e d variable The typical element of s will be
r e f e r r e d to as ~ ( s )
T h e r e are two very n e a r l y c o n t r a d i c t o r y properties that typical e l e m e n t s must have The first is the e q u i v a l e n t of u n i v e r s a l instantiation; real e l e m e n t s should inherit the properties of the typical element The second is that the typical element cannot itself be an element of the set, for that w o u l d lead to
c a r d i n a l l t y problems The two t o g e t h e r w o u l d imply the set has no elements 3
We could get around this p r o b l e m by p o s i t i n g
a special set of predicates that apply to typical elements and are s y s t e m a t i c a l l y related to the predicates that apply to real elements This idea should be rejected as being a d ho ~c, if aid did not come to us from an u n e x p e c t e d quarter the
n o t i o n of "grain size"
W h e n utterances predicate, it is n o r m a l l y at some degree of resolution, or "grain" At a fairly coarse grain, we might say that J o h n is at the post office "at(J,PO)" At a more refined grain, we have to say that he is at the stamp
w i n d o w "at(J,SW)'" We n o r m a l l y think of grain
in terms of distance, but more g e n e r a l l y we can move from entities at one grain to entities at a coarser grain by means of an a r b i t r a r y partition
F i n e - g r a i n e d entities in the same e q u i v a l e n c e class are i n d i s t i n g u i s h a b l e at the coarser grain
G i v e n a set S, c o n s i d e r the p a r t i t i o n that
c o l l a p s e s all e l e m e n t s of S into one element and leaves e v e r y t h i n g else unchanged We can view the typical element of S as the set of real elements seen at this c o a r s e r grain a grain at which, precisely, the elements of the set are
i n d i s t i n g u i s h a b l e Formally, we can define an
o p e r a t o r ~ w h i c h takes a set and a predicate as its arguments and produces what will be referred
to as an "indexed predicate":
T, if x=T(s) & (V yes) p(y),
<;'(s,p)(x) = F, if x=~(s) & ~ ( F y~s) p(y),
p(x) otherwise
We will frequently a b b r e v i a t e this "P5 " Note that predicate indexing gets us out of the above
3 An a l t e r n a t i v e a p p r o a c h w o u l d be to say that the typical element is in fact one of the real elements of the set, but that we will n e v e r know which one, and that furthermore, we will n e v e r
k n o w about the typical element any p r o p e r t y that
is not true of all the elements This a p p r o a c h runs into technical d i f f i c u l t i e s i n v o l v i n g the empty set
Trang 4" ~ ( s ) s"
true but tautologous
W e are now in a p o s i t i o n to state the
p r o p e r t i e s typical elements should have The
first implements u n i v e r s a l instantiation:
(4) ( U s , y ) p$(~(s)) & yes -> p(y)
T h a t is, the p r o p e r t i e s of the typical e l e m e n t at
the c o a r s e r g r a i n are also the p r o p e r t i e s of the
real e l e m e n t s at the finer grain, and the typical
e l e m e n t has those p r o p e r t i e s that all the real
e l e m e n t s have
Note that w h i l e we can infer a p r o p e r t y from
set m e m b e r s h i p , we cannot infer set m e m b e r s h i p
f r o m a property That is, the fact that p is
true of a typical element of a set s and p is true
of a n e n t i t y y, does not imply that y is an
e l e m e n t of s A f t e r all, we will want "three men"
to refer to a set, and to be able to i n f e r from
y's being in the set the fact that y is a man
But we do not want to infer from y's being a man
that y is in the set Nevertheless, we w i l l need
a n o t a t i o n for e x p r e s s i n g this s t r o n g e r r e l a t i o n
a m o n g a set, a typical element, and a d e f i n i n g
c o n d i t i o n In particular, we need it for
r e p r e s e n t i n g "every man", Let us d e v e l o p the
n o t a t i o n from the s t a n d a r d n o t a t i o n for
i n t e n s i o n a l l y defined sets,
by p e r f o r m i n g a fairly s t r a i g h t f o r w a r d , though
o n t o l o g i c a l l y promiscuous, syntactic t r a n s l a t i o n
o n it First, instead of v i e w i n g x as a
u n i v e r s a l l y q u a n t i f i e d variable, let us treat it
as the typical element of s Next, as a way of
g e t t i n g a handle on "p(x)", we will use the
nominalization o p e r a t o r to reify it, and r e f e r
to the c o n d i t i o n e of p (or p$) being true of the
t y p i c a l element x of s " p ~ (e,x)" E x p r e s s i o n
(6) can then be translated into the following flat
p r e d l c a t e - a r g u m e n t form:
( 7 ) set(s,x,e) & p ~ (e,x)
T h i s should be read as saying that s is a set
w h o s e typical element is x and w h i c h is defined by
c o n d i t i o n e, w h i c h is the c o n d i t i o n of p
( i n t e r p r e t e d at the level of the typical element)
b e i n g true of x The two critical properties of
the p r e d i c a t e "set" w h i c h make (7) e q u i v a l e n t to
(6) are the following:
(8) ~ s , x , e , y ) set(s,x,e) & p~ (e,x) & p(y) -> yes
(9) (~s,x,e) set(s,x,e) -> x " T ( s )
A x i o m schema (8) tells us that if an entity y has
the d e f i n i n g p r o p e r t y p of the set s, then y is an
e l e m e n t of s A x i o m (9), along with a x i o m schemas
(4) and (3), tells us that an element of a set has
the act's d e f i n i n g property
d i s t i n c t i o n b e t w e e n the d i s t r i b u t i v e and
c o l l e c t i v e r e a d i n g s of a sentence like
(I0) The m e n lifted the piano
For the c o l l e c t i v e reading the r e p r e s e n t a t i o n
w o u l d include "llft(m)" w h e r e m is the set of men For the d i s t r i b u t i v e reading, the r e p r e s e n t a t i o n
w o u l d have " l i f t ( ~ ( m ) ) " , w h e r e ~ ( m ) is the
t y p i c a l element of the set m To r e p r e s e n t the
a m b i g u i t y of (I0), we c o u l d use the d e v i c e
s u g g e s t e d in H o b b s [1982 I for p r e p o s i t i o n a l p h r a s e and other a m b i g u i t i e s , and wr~te "llft(x) & (x=m v
x- ~(m) )"
This a p p r o a c h i n v o l v e s a more t h o r o u g h use of typical e l e m e n t s than two p r e v i o u s a p p r o a c h e s
W e b b e r [1978] a d m i t t e d both set and p r o t o t y p e (my typical element) i n t e r p r e t a t i o n s of phrases like
"each man'" in o r d e r to have a n t e c e d e n t s for b o t h
"they" and "he", but she m a i n t a i n e d a d i s t i n c t i o n
b e t w e e n the two E s s e n t i a l l y , she treated "each man" as ambiguous, w h e r e a s the present a p p r o a c h
m a k e s both the typical element and the set
a v a i l a b l e for s u b s e q u e n t reference M e l l i s h [1980 1 uses =yplcal e l e m e n t s s t r i c t l y as an
i n t e r m e d i a t e r e p r e s e n t a t i o n that must be resolved into more s t a n d a r d n o t a t i o n by the end of processing He can do this because he is w o r k i n g
in a task d o m a i n physics p r o b l e m s in w h i c h sets are not just finite but small, and v a g u e n e s s
as to their c o m p o s i t i o n must be resolved W e b b e r did not a t t e m p t to use typical e l e m e n t s to d e r i v e
a s c o p e - n e u t r a l representation; M e l l i s h did so only in a limited way
Scope d e p e n d e n c i e s can n o w be r e p r e s e n t e d as relations among typical e l e m e n t s C o n s i d e r the
s e n t e n c e
(II) Most m e n love several women,
under the reading in which there is a d i f f e r e n t set of w o m e n for each man We can d e f i n e a
d e p e n d e n c y f u n c t i o n f w h i c h for each man returns the set of w o m e n w h o m that man loves
f(m) = {w [ woman(w) & love(m,w)}
The relevant parts of the initial logical form,
p r o d u c e d by a s y n t a c t i c and semantic t r a n s l a t i o n component, for s e n t e n c e (Ii) will be
(12) l o v e ( ~ ( m ) , ~ ( w ) ) & m o s t ( m , m l ) & m a n l ( ~ ( m l ) )
& several(w) & w o m a n l ( ~ ( w ) )
w h e r e ml is the set of all men, m the set of most
of them r e f e r r e d to by the noun phrase "most men", and w the set r e f e r r e d to by the noun phrase
"several women", and w h e r e "manl = ~ ' ( m l , m a n ) " and
"womanl = ~" (w,woman)' W h e n the i n f e r e n c l n g component d i s c o v e r s there is a different set w for each element of the set m, w can be viewed as refering to the typical element of this set of sets:
w - T ( { f < x > { x~m})
Trang 5d e f i n i t i o n of the d e p e n d e n c y function to the
typical element of m as follows:
f(~(m)) -Z({f(x) I x~m})
That is, f maps the typical element of a set into
the typical element of the set of images under f
of the e l e m e n t s of the set F r o m here on, we will
c o n s i d e r all d e p e n d e n c y functions so extended to
the typical elements of their domains
The i d e n t i t y "w - f ( ~ ( m ) ) " now
s i m u l t a n e o u s l y encodes the s c o p l n g i n f o r m a t i o n and
i n v o l v e s only existentially q u a n t i f i e d v a r i a b l e s
d e n o t i n g i n d i v i d u a l s in an ( a d m i t t e d l y
o n t o l o g l c a l l y promiscuous) domain E x p r e s s i o n s
llke (12) are thus the s c o p e - ~ e u t r a l
r e p r e s e n t a t i o n , and scoplng i n f o r m a t i o n is added
by c o n j o i n i n g such identities
Let us now c o n s i d e r several examples in w h i c h
p r o c e s s e s of i n t e r p r e t a t i o n result in the
a c q u i s i t i o n of scoplng information The first
will involve interpretation against a small model
The second will make use of world knowledge, while
the third illustrates the treatment of e m b e d d e d
q u a n t l f l e r s
First the simple, and classic, example
(13) E v e r y man loves some woman
The initial logical form for this sentence
includes the following:
lovel(r(ms),w) & m a n l ( ~ ( m s ) ) & woman(w)
w h e r e "lovel - @ ( m S , A x [ l o v e ( x , w ) ] ) ' " and "manl -
(ms,man)" Figure i i l l u s t r a t e s two small models
of this sentence M is the set of men {A,B}, W is
the set of w o m e n {X,Y}, and the arrows signify
love Let us assume that the process of
i n t e r p r e t i n g this sentence is Just the process of
i d e n t i f y i n g the e x i s t e n t i a l l y q u a n t i f i e d v a r i a b l e s
ms and w and possibly c o e r c i n g the predicates, in
a way that makes the sentence true 4
Figure I Two models of sentence (13)
In Figure l(a), "'love(A,X)" and "love(B,X)"
are both true, so we can use axiom schema (5) to
derive "lovel('~(M),X)" Thus, the
i d e n t i f i c a t i o n s "ms - M'" and "w = X'" result in the
s e n t e n c e being true
In Figure l(b), "love(A,X)" and "love(B,Y)"
are both true, but since these p r e d i c a t i o n s differ
4 B o b r o w and W e b b e r [1980] s i m i l a r l y show scoplng
i n f o r m a t i o n acquired by I n t e r p r e t a t l o n against a
small model
schema (5) First we d e f i n e a d e p e n d e n c y f u n c t i o n
f, m a p p i n g each man into a w o m a n he loves,
y i e l d i n g " l o v e ( A , f ( A ) ) " and "love(B,f(B))" We can now a p p l y axiom schema (5) to derive '" love2 ('~ (M), f ( ~ (M)) ) ", w h e r e "love2 =
~ ( M , A x [ l o v e ( x , f ( x ) ) ] ) " Thus, we can make the
s e n t e n c e true by i d e n t i f y i n g ms with M and w with f(~'(M)), and by c o e r c i n g "love" to "'love2" and
"woman" to " ~ (W,woman)" ,
In each case we see that the i d e n t i f i c a t i o n
of w is e q u i v a l e n t to solving the scope a m b i g u i t y problem
In our s u b s e q u e n t e x a m p l e s we will ignore the indexing on the predicates, until it must be
m e n t i o n e d in the case of e m b e d d e d quantifiers
Next c o n s i d e r an e x a m p l e in w h i c h w o r l d
k n o w l e d g e leads to d i s a m b l g u a t l o n :
T h r e e w o m e n had a baby
Before inferencing, the s c o p e - n e u t r a l
r e p r e s e n t a t i o n is
had(~Z~ws),b) & lwsI=3 & w o m a n ( ~ ( w s ) ) & baby(b)
Let us suppose the i n f e r e n c i n g component has axioms about the f u n c t i o n a l i t y of having a baby
s o m e t h i n g llke
( ~ x,y) had(x,y) -> x = m o t h e r - o f ( y )
and that we know about c a r d l n a l l t y the fact that for any function g and set s,
Ig(s)l ~ fsl
Then we know the following:
3 - lwsl = Imother-of(b) I ~ Ibl
This tells us that b cannot be an i n d i v i d u a l but must be the typical element of some set Let f be
a d e p e n d e n c y function such that
wEws & f(w) = x -> had(w,x)
that is, a function that maps each w o m a n into some baby she had Then we can identify b with
~ ( { f ( w ) I w ~ ws}), g i v i n g us the correct scope
Finally, let us return to i n t e r p r e t a t i o n with respect to small models to see how e m b e d d e d
q u a n t i f l e r s are represented C o n s i d e r
(14) Every r e p r e s e n t a t i v e of a company arrived
The initial logical f o r m i n c l u d e s
arrive(r) & set(rs,r,ea) & a n d ' ( e a , e r , e o )
& rep'(er,r) & o f ' ( e o , r , c ) & co(c)
That is, r arrives, w h e r e r is the typical element
of a set rs defined by the c o n j u n c t i o n ea of r's being a r e p r e s e n t a t i v e and r's being of c, where c
is a company We will c o n s i d e r the two models in
Trang 6{A,B,(C)}, K is the set of c o m p a n i e s {X,Y,(Z,W)},
there is an a r r o w from the r e p r e s e n t a t i v e s to the
c o m p a n i e s they represent, and the r e p r e s e n t a t i v e s
w h o arrived are circled
F i g u r e 2 Two m o d e l s of s e n t e n c e (14)
In Figure 2(a), "of(A,X)", "of(B,Y)" and "of(B,Z)"
a r e true Define a d e p e n d e n c y function f to m a p A
into X and B into Y T h e n "of(A,f(A))" and
" o f ( B , f ( B ) ) " are both true, so that
" o f ( ~ ( R ) , f ( ~ ( R ) ) ) " is also true Thus we have
the following identifications:
c = f ( Z ( R ) ) = ~ ( { X , Y } ) , r s = R , r -t(R)
In Figure 2(b) " o f ( B ~ " and "of(C,Y)'" are
both true, so "'of(~'(Rl),~)is also Thus we may
let c be Y and rs be RI, giving us the w i d e
r e a d i n g for "a company"
In the case w h e r e no one represents any
c o m p a n y and no one arrived, we can let c be
a n y t h i n g and rs be the empty set Since, by the
d e f i n i t i o n of o" , any predicate indexed by the
e m p t y set will be true of the typical element of
the empty set, " a r r l v e # ( ~ ( # ))" w i l l be true,
and the sentence will be satisfied
It is w o r t h pointing out that this approach
solves the problem of the classic "donkey
sentences" If in sentence (14) we had had the
v e r b phrase "hates it", then "it" would be
r e s o l v e d to c, and thus to w h a t e v e r c was resolved
to
So far the n o t a t i o n of typical elements and
d e p e n d e n c y functions has been introduced; it has
b e e n shown how scope i n f o r m a t i o n can be
r e p r e s e n t e d by these means; and an e x a m p l e of
i n f e r e n t i a l p r o c e s s i n g a c q u i r i n g that scope
i n f o r m a t i o n has been given N o w the precise
r e l a t i o n of this n o t a t i o n to standard notation
must be specified This can be done by means of
a n a l g o r i t h m that takes the inferential notation,
together with an i n d i c a t i o n of w h i c h p r o p o s i t i o n
is asserted by the sentence, and produces In the
c o n v e n t i o n a l form all of the readings consistent
w i t h the known d e p e n d e n c y information
First we must put the sentence into what w i l l
be called a "bracketed notation" We a s s o c i a t e
w i t h each variable v an indication of the
c o r r e s p o n d i n g quantifier; this is d e t e r m i n e d from
such pieces of the inferential logical form as
those involving the predicates "set" and "most";
"Quant(v)" The t r a n s l a t i o n of the r e m a i n d e r of the inferential logical form into b r a c k e t e d
n o t a t i o n is best shown by example For the sentence
A r e p r e s e n t a t i v e of every c o m p a n y saw a sample
the relevant parts of the i n f e r e n t i a l logical form are
see(r,s) & rep(r) & of(r,c) & co(c) & sample(s)
w h e r e "see(r,s) '° is asserted This is t r a n s l a t e d "
in a s t r a i g h t f o r w a r d way into (18) see(It I rep(r) & of(r,[c I co(c)l)],
Is I s a m p l e ( s ) ] ) This may be read "An r such that r is a
r e p r e s e n t a t i v e and r is of a c such that c is a company sees an s such that s is a sample
T h e n o n d e t e r m i n i s t i c a l g o r i t h m below
g e n e r a t e s all the scoplngs from the b r a c k e t e d notation The f u n c t i o n T O P B V S returns a llst of all the top-level b r a c k e t e d v a r i a b l e s in Form, that is, all the b r a c k e t e d variables except those
w i t h i n the brackets of some o t h e r variable - - in
n o n d e t e r m l n i s t i c a l l y g e n e r a t e s a separate process for each element in a list it is g i v e n as argument A f o u r - p a r t n o t a t i o n is used for
q u a n t i f i e r s (similar to that of Woods [1978])
" ( q u a n t i f i e r v a r l a b i e r e s t r i c t i o n body)"
G(Form) :
if [vlRl ~ B R A N C H ( T O P B V S ( F o r m ) ) then F o r m ~ (Quant(v) v B R A N C H ( { R , G ( R ) } ) Form~.~
if F o r m is w h o l e sentence then R e t u r n G(Form) else R e t u r n B R A N C H ( { F o r m , G ( F o r m ) } ) else Return F o r m
In this a l g o r i t h m the first BRANCH c o r r e s p o n d s to the choice in o r d e r i n g the t o p - l e v e l quantifiers The variable c h o s e n w i l l get the n a r r o w e s t scope The second BRANCH c o r r e s p o n d s to the d e c i s i o n of
w h e t h e r or not to give an embedded q u a n t i f i e r a wide reading The choice R c o r r e s p o n d s to a wide reading, G(R) to a n a r r o w reading The third BRANCH c o r r e s p o n d s to the d e c i s i o n of how w i d e a reading to give to an embedded quantifier
D e p e n d e n c y c o n s t r a i n t s can be built into this
a l g o r i t h m by r e s t r i c t i n g the elements of its argument that B R A N C H c a n choose If the variables
x and y are at the same level and y is d e p e n d e n t
on x, then the first BRANCH cannot choose x If y
is embedded under x and y is d e p e n d e n t on x, then the second BRANCH must c h o o s e G(R) In the third BRANCH, if any t o p - l e v e l b r a c k e t e d variable in
F o r m is d e p e n d e n t on any variable one level of recurslon up, then G(Form) must be chosen
A fuller e x p l a n a t i o n of this a l g o r i t h m and several further examples of the use of this notation are g i v e n in a longer v e r s i o n of this paper
62
Trang 7The approach of Section 2 will not work for
monotone decreasing determiners, such as "few" and
"no" Intuitively, the reason is that the
sentences they occur in make statements about
entities other than just those in the sets
referred to by the noun phrase Thus,
Few men work
is more a negative statement about all but a few
of the men than a positive statement about few of
them One possible representation would be
similar to (I), but wlth the implication reversed
(Bs)(q(s,{x I P(x)})
& ( ~ y)(P(y) & R(y) -> yes))
This is unappealing, however, among other things,
because the predicate P occurs twice, making the
relation between sentences and logical forms less
direct
Another approach would take advantage of the
above intuition about what monotone decreasing
determiners convey
(7 s)(Q(s,{x [ P(x)}) & ( ~ y ) ( y £ s - > - ~ R ( y ) ) )
That is, we convert the sentence into a negative
assertion about the complement of the noun phrase,
reducing this case tO the monotone increasing
case For example, "few men work" would be
represented as follows:
(~ s ) ( [ ~ w ( s , { x I man(x)})
& ( V y ) ( y ~ s - > ~ w o r k ( y ) ) ) 5
( T h i s f o r m u l a t i o n i s e q u i v a l e n t t o , b u t n o t
identical with, Barwlse and Cooper's [1981]
witness set condition for monotone decreasing
determiners.)
Some determiners are neither monotone
increasing nor monotone decreasing, but Barwlse
and Cooper conjecture that it is a linguistic
universal that all such determiners can be
expressed as conjunctions of monotone determiners
For example, "exactly three" means "at least three
and at most three" If this is true, then they
all yield to the approach presented here
Moreover, because of redundancy, only two new
conjuncts would be introduced by this method
Acknowledgments
I have profited considerably in this research
from discussions with Lauri Kartunnen, Bob Moore,
Fernando Pereira, Stan Rosenscheln, and Stu
Shleber, none of whom would necessarily agree with
what I have written, nor even view it with
sympathy This research was supported by the
Defense Advanced Research Projects Agency under
Contract No N00039-82-C-0571, by the National
Library of Medicine under Grant No IR01 LM03611-
5 " ~ w ' is pronounced "few bar"
01, and by the National Science Foundation under Grant No IST-8209346
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