Báo cáo khoa học: "Two Accounts of Scope Availability and Semantic Underspecification" pptx

We show that using a theory of scope availability based upon the function- argument structure of a sentence allows a deter- ministic, polynomial time test for the availabil- ity of a rea

T w o A c c o u n t s of S c o p e Availability and S e m a n t i c

U n d e r s p e c i f i c a t i o n

A l i s t a i r W i l l i s a n d S u r e s h M a n a n d h a r ,

D e p a r t m e n t of C o m p u t e r Science,

University of York, York Y010 5DD, UK

{agw, s u r e s h } @ c s , y o r k a c uk

A b s t r a c t

We propose a formal system for representing the

available readings of sentences displaying quan-

tifier scope ambiguity, in which partial scopes

may be expressed We show that using a theory

of scope availability based upon the function-

argument structure of a sentence allows a deter-

ministic, polynomial time test for the availabil-

ity of a reading, while solving the same problem

within theories based on the well-formedness

of sentences in the meaning language has been

shown to be NP-hard

1 I n t r o d u c t i o n

The phenomenon of quantifier scope ambigu-

ity has been discussed extensively within com-

putational and theoretical linguistics Given a

sentence displaying quantifier scope ambiguity,

such as Every man loves a woman, part of the

problem of representing the sentence's meaning

is to distinguish between the two possible mean-

ings:

Vx(ma (x) -+ 3y(woma (y) A lo e(x, y)))

where every man loves a (possibly) different

woman, or

where a single woman is loved by every man

One aspect of the problem is the generation of

all available readings in a suitable representa-

tion language Cooper (1983) described a sys-

tem of "storing" the quantifiers as A-expressions

during the parsing process and retrieving them

at the sentence level; different orders of quan-

tifier retrieval generate different readings of the

sentence However, Cooper's method generates

logical forms in which variables are not correctly

bound by their quantifiers, and so do not cor- respond to a correct sentence meaning This problem is rectified by nested storage (Keller, 1986) and the Hobbs and Shieber (1987) al- gorithm However, the linguistic assumptions underlying these approaches have recently been questioned Park (1995) has argued that the availability of readings is determined not by the well-formedness of sentences in the meaning lan- guage, but by the function-argument relation- ships within the sentence His theory proposes that only a subset of the well-formed sentences generated by nested storage are available to a speaker of English Although the theories have different generative power, it is difficult to find linguistic data that convincingly proves either theory correct

In the absence of persuasive linguistic data,

it is reasonable to ask whether other grounds exist for choosing to work with either of the two theories This paper considers the appli- cation of both theories to the problem of un- derspecified meaning representation, and the question of determining whether a set of con- straints represents an available reading of an ambiguous sentence or not We show that a constraint language based upon Park's linguis- tic theory (Willis and Manandhar, 1999) solves this problem in polynomial time, and contrast this with recent work based on dominance con- straints which shows that using the more per- missive theory of availability to solve the same problems leads to NP-hardness

2 U n d e r s p e c i f i c a t i o n

A recent area of interest has been with under- specified representations of an ambiguous sen- tence's meaning, for example, Quasi-Logical Form (QLF) (Alshawi and Crouch, 1992) and Underspecified Discourse Representation The-

ory (UDRT) (Reyle, 1995) We shall charac-

terise the desirable properties of an underspec-

ified meaning representation as:

1 the meaning of a sentence should be rep-

resented in a way that is not c o m m i t t e d to

any one of the possible (intended) meanings

of t h e sentence, and

2 it should be possible to incrementally intro-

duce partial information about t h e mean-

ing, if such information is available, a n d

w i t h o u t the need to undo work t h a t has

already been done

A principal aim of systems providing an un-

derspecified representation of quantifier scope is

the ability to represent partial scopings T h a t

is, it should be possible to state that some of

the quantifiers have some scope relative to each

other, while remaining u n c o m m i t t e d to the rel-

ative scope of the remaining quantifiers How-

ever, representations which simply allow partial

scopes to be stated without further analysis do

not adequately capture the behaviour of quanti-

tiers in a sentence Consider the sentence Every

representative of a company saw most samples,

represented in the style of QLF:

_:see(<+i every x _:rep.of(x,

<+j exists y co(y)>)>,

<+k most z sample(z)>)

A fully scoped logical form of this QLF is:

[+i,+k,+j] :see(<+i every x r e p o f ( x ,

<+j exists y co(y)>)>,

<+k most z sample(z)>)

where the list of quantifier labels indicates the rela-

tive scope of qnantifiers at that point in the sentence

Although this formula is well formed in the QLF

language, it does not correspond to a well formed

sentence of logic, seeming closer to the formula:

every (x, rep of (x, y), most (z, sample (z),

exists(y, co(y), see(x, z))))

where the variable y does not appear in the

scope of its quantifier A language such as

QLF will generally allow this scoping to be ex-

pressed, even t h o u g h it does not correspond to

a reading available to a speaker In QLF se-

mantics, a scoping which does not give rise to

any well formed readings is considered "uninter-

pretable"; ie there is no interpretation in which

an evaluation function maps the QLF onto a

t r u t h value

Our aim is to present a system in which there is a straightforward computational test of whether a well-formed reading of a sentence ex- ists in which a partial scoping is satisfied, with- out requiring recourse to the final logical form

T h e language CLLS (Egg et al., 1998) has re- cently been developed which correctly generates the well-formed readings by using dominance constraints over trees Readings of a sentence can be represented using a tree, where domi- nance represents outscoping, a n d quantifiers are represented using binary trees whose daughters correspond to the quantifiers' restriction and scope So for the current example, Every repre- sentative of a company saw most samples, the reading:

every(x, a(y, co(y), rep.o f ( x, y ) ),

m o s t ( z , sample(z), see(x, z) ) )

can be represented by the tree in figure 1, w h e r e the restrictions of a a n d most have been o m i t t e d for clarity Domination in t h e tree represents outscoping in the logical form

e v e r y / / ~

a • • m o s t

rep.o f • • see

Figure 1: Representing relative scope as a tree

Underspecification can be captured by defin- ing dominance constraints between nodes rep- resenting the quantifiers and relations in a sen- tence Readings of the sentence with a free variable are avoided by asserting that each re- lation containing a variable must be dominated

by that variable's quantifier, and an available reading of the sentence is represented by a tree

in which all the dominance constraints are sat- isfied So the ill-formed readings of the sen- tence can be avoided by stating that the relation

rep.of is d o m i n a t e d by the restriction of every

and the scope of a, while see is d o m i n a t e d by the scopes of b o t h a and most This is represented

in figure 2, where the dominance constraints are illustrated by d o t t e d lines

Further partial scope information can be introduced with additional dominance con- straints So the partial scope requirement that

2 9 4

• R o o t

e v e r y • ~ a • m o s t

i/%

r e p o f " "-~ see

Figure 2: Representing available scopes with

dominance constraints

most should outscope every would be captured

by a constraint stating t h a t t h e node represent-

ing most should d o m i n a t e the node representing

every in the constraints' solution

It is has been shown (Koller et al., 1998) that

determining the consistency of these constraints

is NP-hard In the rest of this paper, we show

that an alternative theory of scope availability

yields a constraint system that can be solved in

polynomial time

3 A l t e r n a t i v e A c c o u n t o f

A v a i l a b i l i t y

The NP-hardness result of the previous section

arises from the a s s u m p t i o n t h a t the availability

of scopings is determined by the well formedness

of the associated logical forms Park (1995) has

proposed an alternative theory of scope avail-

ability which states t h a t available scopes are

accounted for by relative scopes of arguments

around relations, whereby quantifiers may not

move across NP boundaries For example, con-

sider the sentence Every representative of a

company saw most samples, containing two rela-

tions, saw and of A r o u n d saw, every (represen-

tative of a company) can outscope most (sam-

ples), or vice versa, and around of, every (rep-

resentative) can outscope a (company), or vice

versa Park generalises this observation to the

claim that for any n-ary relation in a sentence,

there are n! possible orderings of quantified ar-

guments around that relation Other quanti-

tiers in the sentence should not "intercalate" be-

tween those which are single arguments to a re-

lation So in the example sentence there are four

possible scopes, because there are 2! = 2 scop-

ings around saw and 2! = 2 scopings around

of What is not possible is a reading where a

outscopes most which outscopes every; although

this can be represented by a well formed sen- tence of logic (with no u n b o u n d variables), it is not available to a speaker of English

By using this theory as t h e basis of under- specification, we can say:

• underspecification is to be captured by al- lowing different possible relative scope as- signments around the predicates, and

• partial scopes between arbitrary quanti- tiers in the sentence will be translated into the equivalent scoping of quantifiers a r o u n d their predicates

T h e chosen representation will be based u p o n a

sentence's quantifiers and relations (for exam-

ple, verbs a n d prepositions)

Quantifiers and the relations which determine their relative scope are represented by a set of elements under a strict partial order, where the ordering represents the relative scopes A strict order will be taken to be transitive, antisym- metric and irreflexive However, because the interaction between the predicates in the sen- tence has implications for possible scopings, it

is also necessary to consider the relationships between the ordered sets

Consider again the sentence Every m a n loves

a woman T h e quantifiers a n d relation in this sentence can be represented by a set of elements

{every, a, love} A strict partial order, ~-, is de- fined over the set which states t h a t the relation

love must be outscoped by b o t h quantifiers:

({every, a, love}, (every ~- love, a ~- love))

T h e partial order states that b o t h quantifiers outscope the verb, b u t says nothing about their scopes relative to each other This represents a completely underspecified meaning An unam- biguous reading of the sentence is represented when ~- defines a total order on the set So if

the relation e v e r y ~- a were added, t h e reading:

V x m a n ( x ) ~ 3 y w o m a n ( y ) A love(x, y)

e v e r y ~- a ~- love

would be represented Alternatively, adding

a ~- e v e r y to the underspecified form would rep- resent t h e reading:

3 y w o m a n ( y ) A V x m a n ( x ) -+ love(x, y)

a ~- e v e r y ~- love

T h e introduction of a further relation which

does not lead to a well formed sentence (such

as love ~- every) is shown by the irreflexivity of

~- being violated

While using a single set of elements correctly

accounts for the possible scopes of quantifiers in

the sentences discussed so far, relative clauses

and prepositional a t t a c h m e n t to NPs are more

complex Consider the sentence Every repre-

sentative of a company saw most samples T h e

presence of two binary relations, of a n d saw,

implies that there should be 2!.2! 4 readings

Continuing with the system developed so far,

these possibilities could be represented by a pair

of strictly partially ordered sets:

({every, most, s e e } , ( e v e r y N s e e , m o s t N s e e ) )

({every, a, o f } , (every ~ ' of, a ~ ' o f ) )

where the four possible ways of completing the

strict orders on the sets correspond to t h e four

available readings To represent relative scope

between arbitrary quantifiers in the sentence,

a further transitive relation, >, is defined Say

t h a t if (S, ~-) is a strictly partially ordered set in

the structure where x, y E S and x ~- y t h e n x >

y So for example, consider the pair of strictly

partially ordered sets:

({every, most, s e e } , ( e v e r y ~ m o s t ~ s e e ) )

({every, a, o f } , (a ~ ' every ~-' of))

which would represent the reading (in a format

similar to generalised quantifiers):

most(z, sample(z), see(x, z))))

T h e orders on the sets state t h a t every :> m o s t

see and a > every :> o f , a n d from the transi-

tivity of > it can be inferred (correctly) t h a t

a :> most Similarly, given the ambiguous sen-

tence and the partial scope requirement t h a t

a should outscope most, the required partial

scope can be obtained by adding the relations

a ~-~ every and every ~- most

T h e transitivity of > is not enough to cap-

ture all the available scope information Sup-

pose it were required t h a t m o s t should outscope

a There are two readings of t h e sentence which

satisfy this partial scope, those being:

m o s t ( z , sample(z),

every(x, a(y, co(y), rep.of (x, y)), see(x, z))) and

m o s t ( z , sample(z), a(y, co(y),

every(x, rep.oI (x, y), see(x, z))))

These readings are precisely those for which the object of see outscopes its subject; the partial scope is captured by the pair:

({every, most, see}, (most ~- every ~- see)) ({every, a, o f } , (every ~-' of, a ~-' of))

where there is no additional information a b o u t the relative scope of every and a However, t h e transitivity of -> alone does not capture t h e fact that m o s t .:> a follows from m o s t .:> every

We remedy this by defining a domination re- lation In the current case, say that every dom- inates a, which means t h a t a is nested within the Q N P whose head quantifier is every T h e n because quantifiers may not "intercalate" across

NP boundaries, anything t h a t outscopes every

also outscopes anything t h a t every dominates (here, a); if most outscopes one it must outscope both We capture this behaviour by p u t t i n g the sets into a tree structure, where each of the nodes is one of t h e strictly ordered sets repre- senting the scopes a r o u n d a relation For any node, N, each of the daughter nodes has (ex- actly) one element in c o m m o n with N , oth- erwise, any element appears only once in the structure So, consider again the sentence Ev- ery representative of a company saw most sam- ples T h e scope information of the underspeci- fled form is represented by the tree:

/

({every, a, o f } , ( e v e r y ~-' of, a ~ ' of))

Now, say that an element X dominates another element Y (denoted as X ~-~ Y) if X and Y are (distinct) elements in a set at some node, and X

is also in t h e parent node Also, ~-+ is transitive and irreflexive So in the example given:

every ~-+ a and every ~ o f ,

b u t every ~-+ every

We can now extend the definition of -> by saying that:

2 9 6

if (P,~-) is a node in the tree, a n d

x, y E P a n d x ~- y, t h e n x.>y a n d x.>z

where z is any t e r m t h a t y dominates

Also, > is transitive a n d irreflexive

This captures t h e scoping behaviour for nested

quantifiers So from the ambiguous representa-

tion of scopes:

({every, most, see}, (most every see))

I

where m o s t ~ every a n d every ~ a, it is pos-

sible to infer correctly t h a t m o s t .> a, whatever

t h e relation is between every a n d a

4 F o r m a l D e f i n i t i o n o f S c o p e

R e p r e s e n t a t i o n s

We now provide a formal description of t h e

structures described in section 3 T h e defini-

tion is divided into two parts First a scope

structure is defined, which is a tree s t r u c t u r e

whose nodes are sets under a strict order a n d

describes t h e correct possible scopings of quan-

tiffed a r g u m e n t s a r o u n d their relations Next, a

scope representation is defined, which is t h e pair

of a scope s t r u c t u r e and an outscoping relation,

•

>, which is defined over all t h e elements in t h e

structure

T h e analysis presented here differs from t h a t

of t h e previous section in t h a t t h e nodes in

the scope ~ structures are sets u n d e r a strict to-

tal order, r a t h e r t h a n u n d e r a p a r t i a l order

T h e s t r u c t u r e s therefore represent u n a m b i g u -

ous readings of t h e sentence Underspecifica-

tion will t h e n be c a p t u r e d in the constraint lan-

guage, r a t h e r t h a n in the u n d e r l y i n g structures,

as discussed in section 5

A scope s t r u c t u r e is a finite tree, where each

node of the tree is a finite, n o n - e m p t y set of el-

ements, P , taken from a set (9 = {a,/~,-),, }

u n d e r a strict total order For a n y node, each

d a u g h t e r node is also a strictly ordered set, such

t h a t each d a u g h t e r set di has exactly one el-

ement in c o m m o n with P , a different element

for each of the di An element can only a p p e a r

once in the tree, unless it is the c o m m o n node

between a m o t h e r a n d a daughter So:

is a correct scope s t r u c t u r e , because no element appears twice except c~ a n d 8, which a p p e a r in

m o t h e r / d a u g h t e r pairs (the ordering relations have been o m i t t e d for clarity)

A scope s t r u c t u r e is defined as a triple (P, ~- , :D), where P is a set of elements, ~- is a strict

t o t a l order over P a n d 7:) is t h e set of daughters

We say t h a t an element occurs in a scope struc-

t u r e if it is a m e m b e r of t h e set at a n y node in

t h e scope structure If (9 is a (countable) set

of elements, t h e n scope s t r u c t u r e s can be recur- sively defined as:

• If S = (Ps, >-s, {}), where P s is a finite,

n o n - e m p t y subset of (9 a n d >-s is a strict

t o t a l order on Ps, t h e n S is a scope struc- ture, where:

1 if x E Ps, t h e n x occurs in S,

• If R a n d S are scope structures such t h a t

R = (PR, ~ R , DR) a n d S = (Ps, ~-s, :DS), where no element occurs in b o t h R a n d

S, a n d there is some element a such t h a t

a E Pn, t h e n if T = (PT, N'T,~T), where

PT = {a} t2 Ps, T~T = {R} U :Ds a n d ~-T is

a strict t o t a l order on PT t h e n T is a scope structure, where:

1 If some element x occurs in either R

or S t h e n x occurs in T

2 If some element x occurs in R a n d x

a, t h e n a dominates x in T

3 If x a n d y occur in R a n d x d o m i n a t e s

y in R t h e n x d o m i n a t e s y in T

4 If x a n d y occur in S a n d x d o m i n a t e s

y in S t h e n x d o m i n a t e s y in T

If S is a scope structure, t h e n a node in S is defined as:

• If S is a scope s t r u c t u r e such t h a t S

(Ps, >-s, T~S), then:

- (Ps, >'-s) is a node in S

- if di E :Ds, t h e n a n y node in di is a node in S

Having defined scope structures, we now de- fine a scope representation, which is a pair

iS, ">s), where S is a scope s t r u c t u r e a n d ">s is

a relation between pairs of elements which oc- cur in S ">s represents outscoping between any

pair of elements in the structure, rather t h a n

j u s t between elements at a c o m m o n node

If S is a scope s t r u c t u r e such t h a t S =

( P s , ~ - s , 7 ) s ) , t h e n (S, > s ) is a scope represen-

tation, where ">s is the m i n i m u m relation such

that:

* If (P, ~-p) is a node in S and x, y E P and

x N-p y, t h e n x ">s Y

• If (P, ~-p) is a node in S and x, y E P and

x ~-p y, t h e n i f z is an element which occurs

in S a n d y d o m i n a t e s z in S then x ">s z

• ">s is transitive

If (S, ">s) is a well formed scope representation,

then ">s is a strict partial order over the set of

elements which occur in S

5 C o n s t r a i n t s f o r S c o p e

U n d e r s p e c i f i c a t i o n

We now consider a constraint language for rep-

resenting the available scopes in a sentence T h e

s t r u c t u r e of t h e sentence can b e defined in terms

of c o m m o n a r g u m e n t s to a relation (which is

represented b y m e m b e r s h i p of a c o m m o n set in

the scope s t r u c t u r e ) and the d o m i n a t i o n rela-

tion T h e constraint language is:

¢, ¢ ::= x o y C o m m o n set m e m b e r s h i p

x ¢ + y D o m i n a t i o n

x D y O u t s c o p i n g

~b A ¢ C o n j u n c t i o n

where x, y are m e m b e r s of a (countable) set of

constants, C O A l = { x , y, z, }

It is intended t h a t these constraints b e de-

fined over terms in an underspecified semantic

representation, such as Q L F or U D R T , with a

function m a p p i n g g r a m m a t i c a l o b j e c t s in the

representation onto m e m b e r s of C O N Repre-

senting the quantifiers and relations in the sen-

tence is sufficient for our current needs Con-

straints of the form x o y (where o is symmetric)

state either t h a t x a n d y represent c o m m o n ar-

guments to a relation, or t h a t x and y represent

a relation and a quantifier which quantifies over

it Constraints of the form x ~-4 y indicate t h a t

x is the head quantifier of a complex NP, in

which y, a n o t h e r g r a m m a t i c a l object (either a

quantifier or a relation), is nested

So for example, consider again the sentence

E v e r y representative of a c o m p a n y saw m o s t

samples, and assume t h a t t e r m s in the un-

derspecified representation representing the the

g r a m m a t i c a l o b j e c t s every, exists, most, rep.of and see m a p onto the elements e, a, m, o and s respectively, where {e, a, m , o, s } C C O N T h e n

the constraint representing the fully underspec- ified meaning is:

e o s A m o s A e o m A s o e A s o m A m o e

A

e o o A a o o A e o a A o o e A o o a A a o e

A

e c-~ a A e ~-+ o

A ei> s A e ~ o A m i > s A a D o

N o t e t h a t the s y m m e t r y of o is s t a t e d explic- itly in the constraint T h e (underspecified) con- straint is generated either from the g r a m m a r

or directly from t h e underspecified structure, so the inference rules for d e t e r m i n i n g the availabil- ity of a partial scope only generate constraints

of t h e form X t> Y These rules are discussed further in section 6 Underspecification is now

c a p t u r e d within the constraint language; note the parallels b e t w e e n the constraints of the form

X t> Y in this example and t h e partial orders used in section 3

T h e satisfiability of the constraints is given

in terms of the scope representations defined in section 4 A scope representation, (S, ">s), sat- isfies a constraint of the form X o Y if (P, >-p)

is a node in S such t h a t X ' , Y' E P s , X ' # Y ' ,

where some assignment function m a p s X and

Y onto X ' and Y' Similarly, constraints of the form X ~-+ Y are satisfied if X ' d o m i n a t e s Y'

in S, and constraints of the form X D Y are satisfied if X ' ">s Y' So the above constraint is satisfied by a set of scope s t r u c t u r e s of the form:

( { e v e r y , m o s t , see}, >-)

/ ( { e v e r y , a, o f } , ~-')

where the assignment function m a p s the con-

stants e , a , m , o and s onto t h e elements

e v e r y , a, m o s t , o f and see respectively, and

where e v e r y ~- see, m o s t ~- see, e v e r y ~-' o f

a n d a ~-' o f

We can now define the semantics for the con- straint language An assignment function, I[-~/,

m a p s constants of the constraint language onto

2 9 8

elements which occur in S a n d wffs of the con-

straint language onto one of t h e pair of values

{ t , f } I is a pair ((I),~4}, where (I) is a scope

representation, such t h a t (I) = (S, ">s}, and 4 is

a function m a p p i n g constants of t h e constraint

language onto t h e set of elements which occur

in S T h e denotation of t h e constraints is t h e n

given by:

constraint language

• I X o Y ] I = t if there is a n o d e in S, (P, N-p),

such t h a t IX~ I E P a n d [[y]]/ E P and

[[X]]I ~ [[y]]1, otherwise I X o y ] I = f

• I X ~ y ] I = t if IX~ I d o m i n a t e s ~y~I in

S, otherwise I X ~-+ y ~ I = f

• I X ~> Y ~ I = t if I Z ] I > s lynX, otherwise

otherwise [[¢ A ¢]]" f

Satisfiability A constraint set, A, is satisfiable

iff there is at least one I such t h a t I¢~ / = t

for all constraints ¢ where ¢ E A

T h e satisfiability of a constraint set represents

t h e existence of a reading of t h e sentence which

respects the partial scoping

6 A v a i l a b i l i t y o f Partial Scopes

We now t u r n to t h e question of d e t e r m i n i n g

w h e t h e r a partial scoping is available In sec-

tion 3 it was s t a t e d t h a t scope availability is

accounted for by the relative scope of quanti-

tiers a r o u n d their predicates It t u r n s out (al-

t h o u g h we do not prove it here) t h a t for any

partial scoping, there is a necessary and suffi-

cient set of scopings of quantifiers a r o u n d their

relations t h a t gives the partial scoping For ex-

ample, we showed t h a t for t h e sentence Every

representative of a company saw most samples,

t h e readings where most outscopes a are exactly

those where the subject of see outscopes its ob-

ject Therefore, from the constraint m o s t C> a,

it should be possible to infer m o s t E> every T h e

aim of t h e constraint solver is to d e t e r m i n e what

scopings of quantifiers a b o u t their relations are

required to obtain t h e required partial scoping,

a n d therefore to state w h e t h e r t h e partial scope

is available

A set of rules is defined on t h e constraints,

so t h a t additional scope information m a y be in- ferred T h e i n t r o d u c t i o n of f u r t h e r scope con- straints does not affect scope information al- ready present (monotonicity) T h e rules are given in figure 3, where F represents any con-

j u n c t i o n of literals a n d the associativity a n d

c o m m u t a t i v i t y of A are assumed T h e infer- ence rules S1, $2 and $3 o p e r a t e by recursively reducing t h e (arbitrary) outscoping constraint

X ~ > Z to X I > Y A Y E > Y ~ , where Y a n d Y~ represent a r g u m e n t s to a c o m m o n relation, a n d Y' either d o m i n a t e s or is equal to Z R e p e a t e d application of these constraints gives t h e set of scopes of quantifiers a r o u n d their relations for

t h e initial partial scoping T h e rules T r a n s

and D o r a t h e n generate t h e remaining possible scope constraints If a scope is unavailable, t h e n completing t h e transitive closure of D across t h e

s t r u c t u r e yields a constraint of the form X ~> X

We t h e n say that:

• A constraint set is in normal ]orm iff ap- plying t h e rules S1, $2, $3, Trans a n d D o m

does not yield any new constraints

If F is a constraint set in n o r m a l form then:

• F represents an available scoping iff it does

not contain a constraint of the form X ~> X

• F represents a complete scoping iff it rep-

resents an available scoping, and for every constraint of t h e form X o Y there is either

a constraint X D Y or a constraint Y D X

T h e condition for a scoping to be available fol- lows from t h e irreflexivity of -> T h e condition for a scoping to be complete states t h a t if two elements are a r g u m e n t s to a relation, or are a re- lation a n d one of its arguments, t h e n t h e y must have scope relative to each other This corre- sponds to considering sets u n d e r a total order,

r a t h e r t h a n u n d e r a partial order

C o m p l e x i t y I s s u e s Let F be a constraint representing an available scoping of a sentence,

and let X ~ > Y be a constraint representing a par-

tial scope between two t e r m s in t h a t sentence

T h e n the worst case of applying t h e inference rules to F A X ~> Y to s a t u r a t i o n t u r n s out to

be equivalent to completing the transitive clo- sure of i>, which is known to be soluble in b e t t e r

t h a n O ( n 3) time (Cormen et al., 1990), where

n is t h e n u m b e r of elements in t h e structure

S1 :

$ 2 :

$3 :

Trans:

Dora:

F A X o Y A X ~ X t AXtC> Y F X ~> Y AXtC> X

F A X o Y A Y ¢-4 Y ' A X t> Y t I- X i:> Y

F A X o Y A X , ~ X~ A Y , - + YI AXIC> y ' ~ - X ' D X AXC> Y

F A X t> Y A Y t > Z~- X c> Z

F A X o Y A X ~> Y A Y c-.+ Z t - X t> Z

where F is any conjunction of literals

Figure 3: Rules of inference

Application of rules $1, $2 and $3 to comple-

tion can be completed in linear time; if X i> Y

is a constraint between two arbitrary quanti-

tiers X and Y where X fi Y, t h e n exactly one

of the rules S1, $2 or $3 applies (lack of space

prevents us proving this here) If X o Y, t h e n

none of these three rules applies Application of

S1, $2 or $3 adds at most two new constraints,

of which at most one is a scope constraint XC>Y ~

where X fi Y~ At most n - 1 such constraints

are generated

Application of the rules S1, $2 a n d $3 re-

duces an arbitrary partial scope into relative

scopes of a r g u m e n t s around their relations If

a scoping is unavailable, this is represented by

the irreflexivity of C> being violated Testing for

this requires that the transitive closure of C> be

completed; this is known to be soluble in b e t t e r

t h a n cubic time We conclude t h a t testing for

the availability of a partial scope in this frame-

work can be achieved in better t h a n cubic time

in the worst case

7 C o n c l u s i o n a n d C o m m e n t s

A desirable property for an underspecified rep-

resentation of quantifier scope ambiguity is t h a t

there should be a computationally efficient test

for whether a partial scope is available or not

We have shown that accepting a theory of avail-

ability which states that scope availability is de-

termined by the function-argument structure of

a sentence allows the development of a test for

availability which is polynomial in t h e n u m b e r

of quantifiers and relations in a sentence, while

theories of availability based u p o n the logical

well-formedness of meaning representations has

been shown to be NP-hard

A c k n o w l e d g e m e n t s T h e authors would like

to t h a n k Alan Frisch, Mark Steedman and three

anonymous reviewers for useful comments T h e

first a u t h o r is funded by an E P S R C grant

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