Báo cáo khoa học: "CONNECTION RELATIONS AND QUANTIFIER SCOPE" pot

Let us take the following sentence from Kempson/Cormack 1981 as an example: Two examiners marked six scripts.. The quantifier two examiners can have wide scope over the quantifier six sc

C O N N E C T I O N R E L A T I O N S A N D Q U A N T I F I E R S C O P E

Long-in Latecki

University of Hamburg Department of Computer Science Bodenstedtstr~ 16, 2000 Hamburg 50, Germany e-mail: latecki@rz.informatik.uni-hamburg.dbp.de

A B S T R A C T

A f o r m a l i s m will be presented in this

paper which makes it possible to realise the

idea of assigning only one scope-ambiguous

representation to a sentence that is ambiguous

with regard to quantifier scope The scope

d e t e r m i n a t i o n r e s u l t s in e x t e n d i n g this

representation with additional context and

world k n o w l e d g e conditions If there is no

scope determining information, the formalism

can work further with this scope-ambiguous

representation Thus scope information does

not have to be completely determined

0 I N T R O D U C T I O N

Many natural language sentences have

more than one possible reading with regard to

quantifier scope T h e most w i d e l y used

methods for scope determination generate all

possible readings of a sentence with regard to

quantifier scope by applying all quantifiers

w h i c h o c c u r in t h e s e n t e n c e in all

combinatorically possible sequences These

methods do not make use of the inner structure

a n d m e a n i n g of a quantifier At best,

q u a n t i f i e r s are c o n s t r a i n e d by external

conditions in order to eliminate some scope

relations The best k n o w n m e t h o d s are:

determination of scope in LF in GB (May 1985),

Cooper Storage (Cooper 1983, Keller 1988) and

t h e a l g o r i t h m of H o b b s a n d S h i e b e r (Hobbs/Shieber 1987) These m e t h o d s assign, for instance, six possible readings to a sentence with three quantifiers Using these methods, a sentence m u s t be disambiguated in order to receive a semantic representation This means that a scope-ambiguous sentence necessarily has several semantic representations, since the formalisms for the representation do not allow for scope-ambiguity

It is hard to imagine that h u m a n beings disambiguate scope-ambiguous sentences in the same way The generation of all possible combinations of sequences of quantifiers and the assignment of these sequences to various readings seems to be cognitively inadequate The problem becomes even more complicated

w h e n natural l a n g u a g e quantifiers can be

i n t e r p r e t e d d i s t r i b u t i v e l y as well as collectively, which can also lead to further readings Let us take the following sentence from Kempson/Cormack (1981) as an example: Two examiners marked six scripts The two quantifying n o u n phrases can in this case be interpreted either distributively

or collectively The quantifier two examiners can have wide scope over the quantifier six scripts, or vice versa, which all in all can lead

to various readings Kempson a n d Cormack assign four possible readings to this sentence,

D a v i e s (1989) e v e n eight (A d e t a i l e d

discussion will follow.) No one, however, will

m a k e the claim that people will first assign

all possible representations with regard to the

scope of the quantifiers and their distribution,

a n d w i l l t h e n e l i m i n a t e c e r t a i n

interpretations according to the context; but

this is t o d a y ' s s t a n d a r d p r o c e d u r e in

linguistics In m a n y cases, it is also almost

impossible to d e t e r m i n e a preferred reading

The difficulties that people have w h e n t h e y

are forced to disambiguate such sentences (to

explicate all possible readings) point to the

fact t h a t p e o p l e o n l y a s s i g n an u n d e r -

determined scope-ambiguous representation in

the first place

Such a r e p r e s e n t a t i o n of t h e e x a m p l e

sentence w o u l d only contain the information

that w e are dealing w i t h a marking-relation

b e t w e e n examiners a n d scripts, a n d that we

are always d e a l i n g w i t h two examiners a n d

six scripts This representation does not contain

a n y information about scope On the basis of

this representation one m a y in a given context

d e r i v e a r e p r e s e n t a t i o n w i t h a d e t e r m i n e d

scope But it m a y also be the case that this

information is sufficient in order to understand

the sentence if no scope-defining information is

given in the context, since in m a n y cases h u m a n

beings do not disambiguate such sentences at

all T h e y u s e u n d e r d e t e r m i n e d , scopeless

interpretations, because their knowledge often

need not be so precise If a disambiguation is

carried out, then this process is d o n e in a very

natural w a y on the basis of context and world

knowledge This points to the assumption that

scope d e t e r m i n a t i o n b y h u m a n b e i n g s is

p e r f o r m e d on a semantic level and is d e d u c e d

on the basis of acquired knowledge

I will present a formalism which works in

a similar way This formalism will also show

that it is not necessary to w o r k with m a n y

sequences of quantifiers in order to determine

the various readings of a sentence with regard

to quantifier scope

W i t h i n this f o r m a l i s m it is possible to

r e p r e s e n t a n a m b i g u o u s s e n t e n c e w i t h an

ambiguous representation which need not be disambiguated, but can be disambiguated at a later stage The r e a d i n g s can e i t h e r be specified m o r e clearly b y giving additional conditions, or they can be d e d u c e d from the basic a m b i g u o u s reading b y inference Here, the i n n e r s t r u c t u r e a n d t h e m e a n i n g of quantifiers play an important role The process

of disambiguation can only be performed w h e n

a d d i t i o n a l i n f o r m a t i o n t h a t restricts the

n u m b e r of possible readings is available As an

e x a m p l e of such i n f o r m a t i o n , I will treat anaphoric relations

I n t u i t i v e l y s p e a k i n g , t h e d i f f e r e n c e

b e t w e e n a s s i g n i n g a n u n d e r t e r m i n e d representation to an ambiguous sentence and assigning a disjunction of all possible readings

to this sentence corresponds to the difference between the following statements*:

"Peter owns between 150 and 200 books."

and

"Peter owns 150 or 151 or 152 or or 200 books."

It g o e s w i t h o u t s a y i n g t h a t b o t h

s t a t e m e n t s a r e e q u i v a l e n t , since w e can understand "150 or 151 or or 200" as a precise specification of " b e t w e e n 150 a n d 200" Nevertheless, there are procedural differences

in processing the two pieces of information; and there are cognitive differences for h u m a n beings, since w e w o u l d never explicitly utter the second sentence If we could represent

"between 150 and 200" directly by a simple formula and not b y giving a disjunction of 51 elements, t h e n we m a y certainly gain great procedural and representational advantages The deduction of readings in semantics does not of c o u r s e e x c l u d e a c o n s i d e r a t i o n of syntactic restrictions T h e y can be i m p o r t e d into the semantics, for e x a m p l e by passing syntactic information with special indices, as

* The comparison stems from Christopher Habel

described in Latecki (1991) Nevertheless, in

this paper I will abstain from taking syntactic

restrictions into consideration

1 S C O P E - A M B I G U O U S

R E P R E S E N T A T I O N AND SCOPE

D E T E R M I N A T I O N

The aims of the representation presented

in this paper are as follows:

1 A s s i g n i n g an a m b i g u o u s semantic

representation to an ambiguous sentence (with

regard to quantifier scope and distributivity),

from which further readings can later be

inferred

2 The connections between the subject and

objects of a sentence are explicitly represented

by relations The quantifiers (noun phrases)

constitute restrictions on the domains of these

relations

3 Natural language sentences have more

than one reading with regard to quantifier

scope (and distributivity), but these readings

are not independent of one another The target

representation makes the logical dependencies

of the readings easily discernible

4 The construction of complex discourse

referents for anaphoric processes requires the

construction of complex sums of existing

d i s c o u r s e r e f e r e n t s In c o n v e n t i o n a l

approaches, this can lead to a combinatorical

explosion (cf Eschenbach et al 1989 and 1990)

In the representation which is presented here,

the discourse referents are immediately

available as d o m a i n s of the relations

Therefore, we need not construe any complex

discourse referents Sometimes we have to

specify a discourse referent in more detail,

which in turn can lead to a reduction in the

number of possible readings

I now present the formalism

The representational language used here is

second-order predicate logic However, I will

mainly use set-theoretical notation (which

can be seen as an abbreviation of the

corresponding notation of second-order logic) I choose this notation because it points to the semantic content of the formulas and is thus more intuitive

Let R ~ XxY be a relation, that means, a sub-set of the product of the two sets X and Y The domains of R will be called Dom R and Range R, with

Dom R={x~ X: 3y~ Y R(x,y)} and Range R={y~ Y: 3x~ X R(x,y)}

I make the explicit assumption here that all relations are not empty (This assumption only serves in this paper to make the examples simpler.)

In the formalism, a verb is represented by a relation whose d o m a i n is d e f i n e d b y the

arguments of verbs Determiners constitute restrictions on the domains of the relation These restrictions correspond to the role of determiners in Barwise's and Cooper's theory

of generalized quantifiers (Barwise and Cooper 1981) This means for the following sentence:

(1.1) Every boy saw a movie

that there is a relation of seeing between boys and movies

In the formal notation of second-order logic

we can describe this piece of information as follows:

(1.1.a) 3X2 (Vxy (X2(x,y) ~ Saw(x,y) &

Boy(x) & M0vie(y) )) X2 is a second-order variable over the domain of the binary predicates; and Saw,

Boy, and Movie are second-order c o n s t a n t s which represent a general relation of seeing, the set of all boys, and the set of all movies, respectively We will abbreviate the above formula by the following set-theoretical formula:

(1.1.b) 3saw (saw ~ Boy x Movie)

In this formula, w e view s a w as a sorted

variable of the sort o f t h e b i n a r y seeing-

relations T h e variable s a w corresponds to the

variable X2 in (1.1.a)

(1.1.b) describes an i n c o m p l e t e semantic

r e p r e s e n t a t i o n of s e n t e n c e (1.1) Part of the

certain k n o w l e d g e that d o e s not d e t e r m i n e

scope in the case of sentence (1.1) is also the

information that all b o y s are involved in the

relation, w h i c h is easily describable as:

D o m s a w = B o y W e obtain this i n f o r m a t i o n

f r o m the d e n o t a t i o n of the d e t e r m i n e r every

In this w a y w e h a v e a r r i v e d at the scope-

ambiguous representation of (1.1):

(1.1.c) 3 s a w (saw ~ Boy x Movie &

D o m saw=Boy)

It m a y be that the information p r e s e n t e d

in (1.1.c) is sufficient for the interpretation of

s e n t e n c e (1.1) A p r e c i s e d e t e r m i n a t i o n of

quantifier scope n e e d not be important at all,

since it m a y be irrelevant w h e t h e r each b o y

saw a different m o v i e (which corresponds to

the w i d e scope of the universal quantifier) or

w h e t h e r all boys saw the s a m e m o v i e (which

c o r r e s p o n d s to t h e w i d e s c o p e of t h e

existential quantifier)

Classic p r o c e d u r e s will in this case

i m m e d i a t e l y g e n e r a t e t w o r e a d i n g s w i t h

definite scope relations, w h o s e notations in

predicate logic are given below

(1.2.a) Vx(boy(x) ~ 3y(movie(y) & saw(x,y)))

(1.2.b) 3y(movie(y) & Vx(boy(x) ~ saw(x,y)))

We can also obtain these representations in

our formalism by simply adding n e w conditions

to (1.1.c), w h i c h force the disambigiuation of

(1.1.c) w i t h r e g a r d to q u a n t i f i e r scope To

obtain r e a d i n g (1.2.b), w e m u s t come to k n o w

that t h e r e is o n l y o n e movie, w h i c h can be

formaly writen b y I Range saw I =1, w h e r e I I

d e n o t e s the cardinality function To obtain reading (1.2.a) from (1.1.c), w e do not need any

n e w information, since the two formulas are equivalent This situation is d u e to the fact that (1.2.b) implies (1.2.a), w h i c h m e a n s that (1.2.b) is a special case of (1.2.a) This relation can be easly seen b y c o m p a r i n g the resulting formulas, which correspond to readings (1.2.a) and (1.2.b):

(1.3.a) 3saw (saw c Boy x Movie &

D o m saw=Boy) (1.3.b) 3saw (saw ~ Boy x Movie &

D o m saw=Boy & I Range saw I =1)

So, w e have (1.3.b) => (1.3.a)

As I have stated above, h o w e v e r , it is not

v e r y useful to d i s a m b i g u a t e r e p r e s e n t a t i o n (1.1.c) i m m e d i a t e l y It m a k e s m o r e sense to leave r e p r e s e n t a t i o n (1.1.c) u n c h a n g e d for further processing, since it m a y be that in the

d e v e l o p m e n t a n e w c o n d i t i o n m a y a p p e a r which determines the scope For instance, we can obtain the additional condition in (1.3.b),

w h e n sentence (1.1) is followed by a sentence containing a p r o n o u n refering to a movie, as in sentence (1.4)

(1.4) It w a s "Gone with the Wind"

Since it refers to a movie, the image of the saw-relation (a subset of the set of movies) can contain only one element Thus, the resolution

of the r e f e r e n c e results in an extension of representation (1.1.c) by the condition

I Range saw I = 1 Therefore, we get in this case only one reading (1.3.b) as a representation of sentence (1.1), which corresponds to wide scope

of the existential quantifier T h u s in the

c o n t e x t of (1.4) w e h a v e d i s a m b i g u a t e d sentence (1.1) with r e g a r d to quantifier scope

w i t h o u t h a v i n g first g e n e r a t e d all possible readings (in o u r case these w e r e (1.2.a) and (1.2.b))

Let us n o w a s s u m e that sentence (1.5)

follows (1.1)

(1.5) All of t h e m w e r e m a d e b y Walt Disney

Studios

Syntactic theories alone are of no help

here for finding the correct discourse referent

for them in sentence (1.1), since there is no

n u m b e r agreement b e t w e e n them and a movie

The plural n o u n them, however, refers to all

movies the boys have seen This causes great

problems for standard anaphora theories and

plural theories, since there is no explicit object

of reference to w h i c h them could refer (cf

Eschenbach et al 1990; Link 1986) Thus, the

usual procedure would be to construe a complex

reference object as the s u m of all movies the

boys have seen With m y representation, we

do not n e e d such p r o c e d u r e s because the

d i s c o u r s e r e f e r e n t s a r e a l w a y s available,

n a m e l y as d o m a i n s of the relations In the

context of (1.1) and (1.5), the p r o n o u n them

(just as it in (1.4)) refers to the image of the

relation s a w , w h i c h additionally serves the

p u r p o s e of d e t e r m i n i n g the quantifier scope

Here, just as in the p r e c e d i n g cases, the

representation (1.1.c) has to be seen as the

" s t a r t i n g r e p r e s e n t a t i o n " of (1.1) T h e

i n f o r m a t i o n that them is a plural n o u n is

represented by the condition I Range saw I > 1,

w h i c h in t u r n l e a d s to t h e f o l l o w i n g

representation:

(1.6) 3saw (saw ~ BOy x Movie &

Dom saw=Boy & I Range saw I >1)

The representation (1.6) is not a m b i g u o u s

with regard to quantifier scope The universal

quantifier has w i d e scope o v e r the w h o l e

sentence, d u e to the condition I Range saw I > 1

The r e a d i n g p r e s e n t e d in (1.6) is a f u r t h e r

specification of (1.3.a), w h i c h at the s a m e

time e x c l u d e s r e a d i n g (1.3.b) T h u s (1.6)

c o n t a i n s m o r e i n f o r m a t i o n t h a t f o r m u l a

(1.2.a), which is equivalent to (1.3.a)

A classical scope d e t e r m i n i n g s y s t e m can only choose o n e of the r e a d i n g s (1.2.a) a n d (1.2.b) H o w e v e r , if it chooses (1.2.a), it will not win a n y n e w information, since (1.2.b) is a special case of (1.2.a) So, q u a n t i f i e r scope can

not be completely determined b y such a system

In o r d e r to indicate further a d v a n t a g e s of this r e p r e s e n t a t i o n formalism, let us take a look at the following sentence (cf Link 1986): (1.7) Every boy saw a different movie

Its representation is generated in the same

w a y as that of (1.1), the only difference being that the w o r d different c a r r i e s a d d i t i o n a l information about the relation s a w different

r e q u i r e s t h a t t h e r e l a t i o n be injective Therefore, the formula (1.1.c) is e x t e n d e d b y the condition ' s a w is 1-1' The f o r m u l a (1.8) thus represents the o n l y r e a d i n g of sentence (1.7), in w h i c h s c o p e is c o m p l e t e l y determined; the universal quantifier has wide scope

(1.8) 3saw (saw ~ Boy x Movie &

D o m saw=Boy & saw is 1-1)

2 S C O P E - A M B I G U O U S

R E P R E S E N T A T I O N F O R

S E N T E N C E S W I T H N U M E R I C

Q U A N T I F I E R S

So far, I have not stated exactly h o w the representation of sentence (1.1) was generated

In o r d e r to d o so, let us take a n e x a m p l e sentence with numeric quantifiers:

(2.1) Two examiners m a r k e d six scripts

It is certainly not a n e w observation that this sentence has m a n y interpretations w i t h

r e g a r d to quantifier scope a n d distributivity,

w h i c h can be s u m m a r i z e d to a f e w m a i n readings H o w e v e r , their exact n u m b e r is controversial While K e m p s o n a n d C o r m a c k

(1981) assign four readings to this sentence (see

also Lakoff 1972), Davies (1989) assigns eight

readings to it I quote here the readings from

(Kempson/Cormack 1981):

Uniformising:

Replace "(Vx~ Xn)(3Y)" by "(3Y)(Vx~ Xn)"

10 There were two examiners, and each of

t h e m marked six scripts (subject n o u n phrase

with wide scope) This interpretation could be

true in a situation with two examiners and 12

scripts

20 There were six scripts, and each of these

was m a r k e d by two examiners (object n o u n

phrase with wide scope) This interpretation

c o u l d be t r u e in a situation w i t h twelve

examiners and six scripts

30 The incomplete g r o u p interpretation:

Two examiners as a group marked a group of six

scripts between them

40 The complete g r o u p interpretation: Two

examiners each m a r k e d the same set of six

scripts

K e m p s o n a n d C o r m a c k represent these

readings with the help of quantifiers over sets

in the following way:

10 (3X2)(Vx~ X2)(3S6)(Vs~ S6)Mxs

20 (3S6)(Vs~ S6)(3X2)(Vx~ X2)Mxs

30 (3X2)(3S6)(Vx~ X2)(Vs~ S6)Mxs

40 (3X2)(3S6)(Vx~ X2)(3s~ S6)Mxs &

(Vs~ $6)(3x~ X2)Mxs

Here, X 2 is a s o r t e d variable w h i c h

denotes a two-element set of examiners, and S 6

is a sorted variable that denotes a six-element

set of scripts

K e m p s o n a n d C o r m a c k d e r i v e these

r e a d i n g s f r o m an initial f o r m u l a in the

conventional w a y by changing the order and

d i s t r i b u t i v i t y of quantifiers This fact is

discernible from their derivational rules and

the following quotation:

Generalising:

Replace "(3x~ Xn)" by "(Vx~ Xn)"

"What we are proposing, then, as an

a l t e r n a t i v e to t h e c o n v e n t i o n a l ambiguity account is that all sentences

of a form corresponding to (42) [here: 2.1] have a single logical form, which

is t h e n subject to the p r o c e d u r e of generalising and uniformising to yield

t h e v a r i o u s i n t e r p r e t a t i o n s of the sentence in use." ( K e m p s o n / C o r m a c k (1981), p 273)

Only in reading 40 the relation between

e x a m i n e r s a n d s c r i p t s is c o m p l e t e l y characterized For the other f o r m u l a s there are several possible a s s i g n m e n t s b e t w e e n examiners a n d scripts w h i c h m a k e these formulas valid

At this point I want to make an important observation, namely that these four readings are not totally i n d e p e n d e n t of one another I

am, h o w e v e r , n o t c o n c e r n e d w i t h logical implications between these readings alone, but rather with the fact that there is a piece of information which is contained in all of these readings a n d w h i c h does not necessitate a

d e t e r m i n a t e d quantifier scope This is the information which - cognitively speaking - can

be extracted from the sentence by a listener without determining the quantifier scope The difficulties which people have w h e n they are forced to disambiguate a sentence containing numeric quantifiers such as (2.1) without a specific context point to the fact that only such

a scopeless representation is assigned to the sentence in the first place On the basis of this representation one can then, within a given context, d e r i v e a r e p r e s e n t a t i o n w i t h a definite scope We can describe the scopeless piece of information of sentence (2.1), which all readings have in common, as follows We

k n o w that we are d e a l i n g w i t h a marking-

relation b e t w e e n examiners and scripts, and

that we are a l w a y s d e a l i n g w i t h t w o

examiners or with six scripts In the formalism

d e s c r i b e d in this p a p e r this piece of

information is represented as:

(2.2) 3mark ( mark c Examiner x Script &

(IDommarkl=2 v IRangemarkl 6))

It may be that this piece of information is

sufficient in order to understand sentence (2.1)

If there is no scope-determining information in

the given context, people can understand the

sentence just as well If, for example, we hear

the following utterance,

(2.3) In preparation for our workshop, two

examiners corrected six scripts

it m a y be without any relevance what the

relation b e t w e e n examiners and scripts is

exactly like The only important thing may be

that the examiners corrected the scripts and

that we have an idea about the number of

examiners and the number of scripts

Therefore, we have assigned an under-

determined scope-ambiguous representation

(2.2) to sentence (2.1), which constitutes the

maximum scopeless content of information of

this sentence The lower line of (2.2) represents

a scope-neutral part of the information which

is contained in the meaning of the quantifiers

two examiners and six scripts This fact

indicates that the meaning of a quantifier has

to be structured internally, since a quantifier

contains scope-neutral as well as scope-

determining information Distributivity is an

example of scope-determining information

Then what happens in a context which

contains scope-determining information? This

context just provides restrictions on the

domains of the relation These restrictions in

turn contribute to scope determination We

may, for instance, get to know in a given

context that there were twelve scripts in all,

which excludes the condition I Range mark I =6

in the disjunction of (2.2) We then know for certain that there were two examiners and that each of them marked six different scripts

Consequently, the quantifier two examiners

acquires wide scope, and we are dealing with a distributive reading Thus, in this context we have completely disambiguated sentence (2.1) with regard to quantifier scope; and that

s i m p l y on the basis of the scopeless, incomplete representation (2.2) On the other

h a n d , s t a n d a r d p r o c e d u r e s (the m o s t important were listed at the beginning) first have to generate all representations of this sentence by considering all combinatorically possible scopes together with distributive and collective readings

3 C O N C L U D I N G R E M A R K S

A c o g n i t i v e l y a d e q u a t e m e t h o d for dealing with sentences that are ambiguous with regard to quantifier scope has been described in this paper An underdetermined scope-ambiguous representation is assigned to

a scope-ambiguous sentence and then extended

by additional conditions from context and world knowledge, which further specify the meaning of the sentence Scope determination

in this procedure can be seen as a mere by- product The quantifier scope is completely determined w h e n the representation which was generated in this w a y corresponds to an interpretation with a fixed scope Of course, this only works if there is scope-determining information; if not, one continues to work with the scope-ambiguous representation

I use the l a n g u a g e of s e c o n d - o r d e r predicate logic here, but not the whole second- order logic, since I need deduction rules for scope derivation, but not deduction rules for second-order predicate logic (which cannot be completely stated) One could even use the formalism for scope determination alone and then translate the obtained readings into a

f i r s t - o r d e r f o r m a l i s m H o w e v e r , t h e

f o r m a l i s m l e n d s itself v e r y e a s i l y to

representation and processing of the derived

semantic knowledge as well

A C K N O W L E D G M E N T S

I would like to thank Christopher Habel,

Manfred Pinkal and Geoff Simmons

B I B L I O G R A P H Y

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