A Formalism for Complex Word Meanings
Manfred Pinkal and Michael Kohlhase
Universitat des Saarlandes, Germany
{kohlhase@ags|pinkal@col i}.u ni-sb.d e
Abstract
In this paper we revisit
Puste-jovsky's proposal to treat
ontologi-cally complex word meaning by
so-calleddottedpairs. Weusea
higher-order feature logicbased on Ohori's
record -calculus to model the
se-mantics of words like book and
li-brary, in particular their behavior
inthe context of quanticationand
cardinalitystatements.
1 Introduction
The treatment of lexical ambiguity is one of
themainproblemsinlexicalsemanticsandin
themodelingofnaturallanguage
understand-ing. Pustejovsky's framework of the
\Gen-erative Lexicon" made a contribution to the
discussion by employing the concept of type
coercion, thus replacing the enumeration of
readingsbythesystematiccontext-dependent
generation of suitable interpretations, in the
case of systematic polysemies (Pustejovsky,
1991; Pustejovsky, 1995). Also, Pustejovsky
pointed to a frequent and important
phe-nomenon in lexical semantics, which at rst
sight looks as another case of polysemy, but
issignicantlydierent innature.
(1) Thebookis blue/ontheshelf.
(2) Maryburnedthebook.
(3) Thebookis amusing.
(4) Maryunderstandsthebook.
(5) Thebookis beautiful.
(6) Marylikesthebook.
(7) Maryread thebook.
Examples (1)-(4) suggest an inherent
ambi-guityof thecommon nounbook: blue,on the
shelf, and burn subcategorize for a physical
object,whileamusing andunderstand require
an informationalobjectasargument. (5)and
(6) are in fact ambiguous: The statements
may refer either to the shape or the content
of thebook. However, athoroughanalysis of
thesituationshowsthatthereisathird
read-ing where the beauty of the book as well as
Mary's positive attitude are due to the
har-mony between physical shape and
informa-tional content. The actionof reading, nally,
is not carried outon a physicalobject alone,
nor on a pure informational object as
argu-ment, but requires an object which is
essen-tially a combination of the two. This
indi-cates asemanticrelationwhichisconjunctive
or additive in character, rather than a
dis-junctionbetweenreadingsasintheambiguity
case. Inadditiontothemorephilosophical
ar-gument, the assumption of a basically
dier-entsemanticrelationissupportedby
observa-tionsfromsemanticcomposition. Ifthe
physi-cal/informationaldistinctioninthesemantics
of book were just an ambiguity, (8) and (9)
wouldnotbeconsistentlyinterpretable,since
the sortal requirements of the nounmodier
(amusing and on the shelf, resp.) are
incom-patible with the selection restrictions of the
verbs burn and understand, respectively.
(8) Mary burnedan amusingbook.
(9) Maryunderstandsthebookontheshelf.
represents them as \dotted pairs" made up
form two (or more) ontologically simple
ob-jects, and being semantically categorized as
\dotted types", e.g.,PIinthe caseof book.
He convincingly argues that complex types
are omnipresent in the lexicon, the
phys-ical/informational object distinction being
just a special case of a wide range of dotted
types, including container/content (bottle),
aperture/panel (door) building/institution
(library).
The part of the Generative Lexicon
con-cept which was not concerned with
onto-logically complex objects, i.e., type
coer-cion and co-composition mechanisms using
so-called qualia information, has triggered a
line of intensive and fruitfulresearch in
lexi-calsemantics, whichledto progressin
repre-sentation formalisms and tools for the
com-putational lexicon (see e.g. (Copestake and
Briscoe,1995;Dolling,1995; Busaand
Bouil-lon, forthcoming; Egg, 1999)). In contrast,
aproblemwithPustejovsky'sproposalabout
the complex objects is that the dotted-pair
notation has been formally and semantically
notclear enoughto form a startingpoint for
meaning representationand processing.
In thispaper, we present a formallysound
semantic reconstruction of complex objects,
using a higher-order feature logic based on
Ohori's record -calculus (1995) which has
beenoriginally developed forfunctional- and
object-oriented programming. We do not
claim that our reconstruction provides a full
theory of theof thepeculiarkindof
ontolog-ical objects, but it appears to be usefulas a
basisforrepresenting lexicalentries forthese
objects and modeling the composition
pro-cess in which they are involved. We will not
onlyshow thatthebasicexamplesabove can
be treated, but also that our treatment
pro-videsa straightforwardsolutionto some
puz-zles concerning the behavior of dotted pairs
in quanticational, cardinality and identity
statements.
(10) Mary burnedevery bookinthelibrary.
(11) Mary understood every bookinthe
(13) Allnew books areon theshelf.
(14) Thebookonyourbook-shelfis theone
Isawinthelibrary.
In (10), the quantication is about physical
objects,whereasin(11),itconcernsthebooks
quainformationalunit. (12)isambiguous
be-tween a number-of-copies and a
number-of-titlesreading. Therespectivereadingsin(10)
and (11) appearto be triggeredby thesortal
requirements of the verbal predicate, as the
ambiguity in (12) is due to the lack of a
se-lection restriction. However, (13) { uttered
towards a customer in a book store { has a
natural reading where the quantication
re-lates to the information level and the
pred-icate is about physical objects. Finally, (14)
hasareadingwherearelationofnon-physical
identityisascribed toobjectswhich areboth
referred to byphysicalproperties.
2 The Record--Calculus F
Inordertoreducethecomplexityofthe
calcu-lus,wewillrstintroduceafeature-calculus
F and thenextendit to F
. F,is an
exten-sionofthesimplytyped-calculusbyfeature
structures (which we will call records). See
Figure 1 for the syntactical categories of the
raw terms.
We assume the base types e (for
individu-als) and t (for truth values), and a set L =
f`
1 ;`
2
;::: goffeatures. Thesetofwell-typed
T ::= ejtjT !T 0
jff` 1
:T 1
;:::;` n
:T n
gg
(Types: ;;:::)
M ::= Xjcj(MN)jX
T
:MjM:`
jff` 1
=M
1 ;:::;`
n
=M
n gg
(FormulaeA;B;:::)
::= ;j;[c:T] (Signature)
::= ;j ;[X:T] (Environment)
Figure 1: Syntax
terms isdenedbytheinferencerulesin
Fig-ure2forthetypingjudgment `
A:. The
sumptions in the signature and the
(lo-cal)typeassumptions (thecontext)forthe
variables. As usual, we say that a term A
is of type (and often simply write A
to
indicate this), i `
A: is derivable by
these rules. We will call a type a record
[c:]2
`
c:
[X:]2
`
X:
`
A: ! `
C:
`
AC:
;[X:]`
A: ` X
:A: !
`
A:ff:::;`:;:::gg
` A:`: ` A 1 : 1 ::: ` A n : n ` ff` 1 =A 1 ;:::;` n =A n gg
Figure2: Well-typed termsinF
type(with features `
i
), i it is of the form
ff` 1 : 1 ;:::;` n : n
gg . Similarly,we callan F
-term A a record, i it has a record type.
Note that recordselectionoperator \." can
only be applied to records. In a slight abuse
ofnotation,wewillalsouseitonrecordtypes
and haveA
:`::`.
It is well-known that type inference with
these rules isdecidable(as a consequence we
will sometimes refrain from explicitly
mark-ing types in our examples), that well-typed
terms have uniquetypes, and thatthe
calcu-lus admits subject reduction,i.e that theset
ofwell-typedtermsisclosedunderwell-typed
substitutions.
The calculus F is equipped with an
(op-erational) equality theory,given by the rules
inFigure3 (extendedtocongruencerelations
on F-termsin the usualway). The rst two
arejustthewell-knownequalityrulesfrom
-calculus (we assume alphabetic renaming
record dereferencing operation \:". Here we
know that these rules form a canonical (i.e.
terminatingandconuent),andtype-safe
(re-duction does not change the type)reduction
system, and that we therefore have unique
-normal forms. The semanticsof F
is a
straightforward extention of that of the
sim-ply typed -calculus: records are interpreted
as partial functions from features to objects,
and dereferencing isonlyapplication ofthese
functions. With this semantics it is easy to
show that the evaluation mapping is
well-typed (I
' (A
)2D
) and that theequalities
in Figure3 aresound (i.e. if A=
B, then
I
'
(A)=I
' (B)). (X :A)B ! [B=X]A
X 2= free(A)
(X:AX) !
A
ff:::;`=A;::: gg :`!
A
`
A:ff`
1 : 1 ;:::;` n : n gg ff` 1 =A:` 1 ;:::;` n =A:` n gg!
A
Figure 3: OperationalEqualityforF.
Up to now, we have a calculus for
so-called closed records that exactly
pre-scribe the features of a record. The
se-mantics given above also licenses a slightly
dierent interpretation: a record type =
ff` 1 : n ;:::;` n : n
gg is descriptive, i.e. an
F-term of type would only be required to
have at least the features `
1 ;:::`
n
, but may
actually have more. This makes it
neces-sary to introduce a subtyping relation ,
since a record ff`=A
gg will now have the
types ff`:gg and ffgg . Of course we have
ff`:gg ffgg , since the latter is less
restric-tive. The higher-order feature logic F
we
ure 4. The rst rule species that record
kn
ff`
1 :
1 ;:::;`
n :
n
ggff`
1 :
1 ;:::;`
n :
k gg
`
A:
`
A:
2BT
0
0
0
! !
0
Figure4: The open recordcalculusF
types that prescribe more features are more
specic, and thus describe a smaller set of
objects. The second ruleisa standard
weak-ening rulefor thesubtype relation. We need
the reexivityrule forbase types in order to
keepthelastrule,whichinducesthesubtype
relationonfunctiontypesfromthatofits
do-main and range types simple. It states that
function spaces can be enlarged by
enlarg-ing the range type orby making the domain
smaller(intuitively,every functioncan be
re-strictedtoa smallerdomain). We saythat
iscovariant(preservingthedirection)inthe
rangeandcontravariantinthedomaintype
(invertingthedirection).
ForF
,we have thesame meta-logical
re-sults as for F
(the type-preservations,
sub-ject reduction, normal forms, soundness,...)
except for the unique type property, which
cannotholdbyconstruction. Insteadwehave
theprincipal type property,i.e. every F
-termhasa uniqueminimaltype.
To fortify our intuition about F
, let us
take a look at the following example: It
should be possible to apply a function F
of type ff`
1
:gg ! to a record with
features `
1 ;`
2
, since F only expects `
1 .
The type derivation in Figure 5 shows that
Fff`
1 =A
1
1 ;`
2 =A
2
2
ggis indeedwell-typed.
In the rst block, we use the rulesfrom
Fig-ure4 (in particular contravariance) to
estab-lishasubtyperelationthatisusedinthe
sec-ond blocktoweaken thetype of F,sothat it
gumentrecordthathasonefeaturemorethan
the feature`
1
requiredbyF'stype.
12
ff`1:1;`2:2ggff`1:1gg
ff`1:1gg!ff`1:1;`2:2gg!
F:ff`1:1gg!
F:ff`1:1;`2:2gg!
`
A
i
:
i
`ff`1=A 1
1 ;`2=A
2
2
gg :ff`1:1;`2:2gg
`
Fff`
1 =A
1
1 ;`
2 =A
2
2 gg :
Figure 5: A F
examplederivation
3 Modeling ontologicallycomplex
objects
We start with the standard Montagovian
analysis (Montague, 1974),onlythatwe base
it on F
instead of the simply typed
-calculus.
For our example, it will be suÆcient to
take the set L of features as a superset of
fP;I;Hg (where the rst stand for physical,
and informationalfacetsofanobject). Inour
fragment we use the extension F
to
struc-ture type e into subsets given by types of
the form ff`
1
:e;:::;`
n
:egg . Note that
throw-ingawayallfeatureinformationandmapping
each such type to a type E in our examples
will yielda standard Montagoviantreatment
of NL expressions, where E takes the role
that e has in standard Montague grammar.
Linguistic examples are the proper name
Mary, whichtranslatestomary 0
:ffH:egg,shelf
which translates to shelf 0
:ffP:egg ! t, and
the common noun book which translates to
book 0
:ffP:e;I:egg!t.
Apredicatelikeblue requiresaphysical
ob-jectasargument. Tobeprecise,theargument
need not be an object of type ffP:egg, like a
shelf or a table. blue can be perfectly
an informationalobject, an institution,oran
aperture. At rst glance, this seems to be a
signicantdierencefromkindpredicateslike
shelf andbook. However,itisOKtointerpret
thetypeassignmentforkindpredicatesalong
with property denoting expressions: In both
cases,theoccurrenceofafeature`meansthat
` occurs inthe type of the argument object.
Thus, ff`:egg ! t is a sortal characterization
fora predicateAwith thefollowingimpact:
1. A has a value for feature `, possibly
among otherfeatures,
2. the semantics of A is projective, i.e.,
theapplicabilityconditionsof A and
ac-cordingly the truth value of the
result-ing predication is only dependent of the
valueof `.
Note that 1. is exactly thebehavior that we
have built the extension F
forand that we
have discussedwiththe exampleinFigure 5.
We willnowcome to 2.
Although type eneveroccursas argument
type directlyinthetranslationofNL
expres-sions,representationlanguageconstantswith
type-e arguments are usefulin the denition
of the semantics of lexical entries. E.g., the
semantics of book can be dened using the
basic constant book
of type e ! e ! t,
asx:(book
(x:P;x:I)),wherebook
expresses
the book-specic relation holding between
physicaland informationalobjects 1
.
ThefragmentinFigure6provides
represen-tationsforsomeofthelexicalitemsoccurring
in theexamples of Section 1, in terms of the
basicexpressions
mary
:e; shelf
;blue
;amusing
:e!t
on
;book
;burn
;understand
:e!e!t;
read
:e!e!e!t
Observe that the representations nicely
re-ect the distinction between linguistic arity
of the lexical items, which is given by the
-prex(e.g.,two-placeinthecaseofread),and
1
Pustejovskyconjecturesthattherelationholding
amongdierent ontologicallevelsis morethanjusta
setof pairs. Werestrict ourselvesto theextensional
Word Meaning/Type
Mary ffH =mary
gg :ffH:egg
shelf x:(shel f
(x:P)):ffP:egg!t
book x:book
(x:P;x:I)
ffP:e;I:egg!t
amusing x:amusing
(x:I)
ffI:egg!t
on xy :on
(x:P;y:P)
ffP:egg!ffP:egg!t
burn xy :burn
(x:H;y:P)
ffP:egg!ffP:egg!t
underst. xy :understand
(x:H;x:I)
ffH:egg!ffI:egg!t
read xy :read
(x:H;y:P;y:I)
ffH:egg!ffP:e;I:egg!t
Figure6: A tinyfragment of English
the \ontological arity" of the underlying
ba-sicrelations(e.g.,the3-place-relationholding
between a person, the physical object which
is visually scanned, and the content which is
acquired by that action). In particular, all
of themeaningsare projective,i.e. they only
pickoutthe featuresfrom thecomplex
argu-ments and make them available to the basic
predicate. Therefore, we can reconstruct the
meaning term R = xy:read
(x:H;y: P;y: I)
of read ifwe only know therelevant features
(wecallthemselectionrestrictions)ofthe
ar-guments, and writeRasread
[fHgf P;Ig].
Theinterpretationofsentence(2)viabasic
predicates is shown in (15) to (17). For
sim-plicity,thedenite nounphraseis translated
by an existential quantier here. (15) shows
theresultof thedirectone-to-one-translation
of lexical items into representation language
constants. In (16), these constants are
re-placed by -terms taken from the fragment.
(17)isobtainedby-reductionand-equality
from (16): in particular, ffH =mary
:Hgg is
replaced bythe-equivalent mary
.
(15) 9v:book 0
(v)^burn 0
(ffH =mary
gg ;v)
(16) 9v:(x:book
(x:P;x:I))(v)^
(xy:burn
(x:H;x:P) )(ffH =mary
gg;v)
(17) 9v:book
(v:P;v:I)^burn
(mary
(4)and(7),respectively,demonstratehowthe
predicates understand and read pick out
ob-jects of appropriate ontological levels. (20)
and(21)areinterpretationsof(8)and(9)
re-spectively,wherenestedfunctorscomingwith
dierent sortal constraints apply to one
ar-gument. The representations show that the
functors select there appropriate ontological
levellocally,therebyavoidingglobal
inconsis-tency.
(18) 9v(book
(v:P;v:I))^
(understand
(mary
;v:I))
(19) 9v(book
(v:P;v:I))^
(read
(mary
;v:P;v:I))
(20) 9v(book
(v:P;v:I))^amusing
(v:I)^
(burn
(mary
;v:P))
(21) 9v(book
(v:P;v:I))^9ushelf
(v:P)^
on
(v:P;u:P)^
(understand
(mary
;v:I))
Thelexicalitemsbeautiful andlike in(5)and
(6), resp.,arepolysemousbecauseofthelack
of strict sortal requirements. They can be
representedasrelationalexpressions
contain-ing a parameter for the selection restrictions
which has to be instantiated to a set of
fea-turesbycontext. like,e.g., can betranslated
to like[S] 0
, with like[fPg] 0
, like[fIg] 0
, and
like[fP;Ig] 0
as(someofthe)possiblereadings.
Ofcourse thispresupposes theavailabilityof
a setof basicpredicateslike
i
of dierent
on-tological arities.
4 Quantiers and Cardinalities
We now turn to the behavior of
non-existential quantiers and cardinality
oper-ators in combination with complex objects.
Thechoiceoftheappropriateontologicallevel
for an application of these operators may
be guided by the sortal requirements of the
predicates used (asin (10)-(12)), but as(13)
demonstrates it is not determined by the
lexical semantics. We represent quantiers
and cardinalityoperators assecond-order
re-lations, according to the theory of
gener-rameterized by a context variable S L for
selection restrictions in the same manner as
the predicates like and beautiful. The value
of S may depend on the general context as
wellasonsemanticpropertiesoflexicalitems
inthe utterance.
We dene the semantics of a
parameter-ized quantier Qj
S
by applying its
respec-tive basic, non-parameterizedvariants to the
S-projectionsof theirargument predicatesP
andQtofeaturesinS,whichwewriteasPj
S
and Qj
S
,respectively. Formally Pj
f`1;:::;`ng is
x
1 :::x
n
:9u:P(u)^x
1 =u:`
1
^:::^x
n =u:`
n
A rst proposal is given in (22). (23)
gives the representation of sentence (13) in
the\bookstorereading"(omittingthe
seman-tics of new and representing on the shelf as
anatomicone-placepredicate,forsimplicity),
(24) the reduction of (23) to ordinary
quan-ticationontheS-projections,whichis
equiv-alent totherst-orderformula(25), whichin
turn can be spelled out as (26) using basic
predicates.
(22) Qj
S
(P;Q),Q(Pj
S ;Qj
S )
(23) everyj
fIg (book
0
;on shelf 0
)
(24) every
book 0
j
fIg
;on shelf 0
j
fIg
(25) 8x:9u:(x=u:I^book 0
(u))
=)9v:x=v:I^onshelf 0
(v)
(26) 8x:9u:(x=u:I^book
(u:P;u:I))
=)9v:x=v:I^onshelf
(v:P)
As one can easily see, the instantiation of S
to fIg triggers the wanted 89 reading (\for
all books(asinformationalobjects)there isa
physical object on the shelf"), where the
in-stantiation to fPg would have given the 88
reading, since on shelf 0
is projective for P
only,and asa consequencewehave
onshelf 0
j
fPg
=x:9u:on shelf 0
(u)^x=u:P
=x:9u:on shelf
(u:P)^x=u:P
,x:9u:on shelf
(x)^x=u:P
,x:on shelf
forward.
The proposed interpretation may be too
permissive. Takeasituation,wherenew
pub-lications are alternatively available as book
and on CD-ROM. Then (22)-(26) may come
out to be true even if no book at all is on
the shelf (only one CD-ROM containing all
new titles). We therefore slightlymodifythe
general scheme (22) by (27), where the
re-striction of the quantier is repeated in the
nuclearscope.
(27) Qj
S
(P;Q),
Q( Pj
S
;(x:P(x)^B(x))j
S )
For ordinary quantication, this does not
cause anychange, because of the
monotonic-ity of NL quantiers. In our case of
level-specic quantication, itguarantees that the
secondargumentcoversonlyprojections
orig-inating from the right type of complex
ob-jects. We give the revised rst-order
repre-sentation corresponding to (26)in(28).
(28) 8x:9u:(x=u:I^book
(u:P;u:I))
=)9v:x=
v:I^book
(v:P;v:I)^on shelf
(v:P)
5 Conclusion
Our higher-order feature logic F
provides
a framework for the simple and
straightfor-ward modeling of ontologically complex
ob-jects, including the puzzles of quantication
and cardinality statements. In this
frame-work,anumberofinterestingempirical
ques-tionscan befurtherpursued:
Theontologyforcomplexobjectscanbe
in-vestigated. Sofar,weconstrainedourselvesto
thesimplest case of \dotted pairs",and may
even have taken over a wrong classication
fromtheliterature,talkingaboutthedualism
ofphysicalandinformationalobjects,wherea
type/token distinctionmighthave beenmore
adequate. Therealityaboutbooks(aswellas
bottlesandlibraries)mightbemorecomplex,
however,includingboththeP/Idistinctionas
wellashierarchicaltype/token structures.
The linguistic selection restrictions are
we may have to take distinguish exocentric
and endocentriccases of dotted pairs,aswell
as projective and non-projectiveverbal
pred-icates.
Anotherfruitfulquestionmightbewhether
theframeworkcouldbeusedtoreconsiderthe
mechanismoftypecoercioningeneral: Itmay
bethatatleastsomecasesofreinterpretation
may be better described by adding an
onto-logicallevel, andthuscreating acomplex
ob-ject, rather thanby switchingfrom one level
to another.
Wewouldliketo concludewithavery
gen-eral remark: The data type of feature
struc-tures asemployed inour formalismhas been
widely used in grammar formalisms, among
otherthingstoincorporatesemantic
informa-tion. In this paper, a logical framework for
semantics is proposed, which itself has
fea-ture structures asa part of themeaning
rep-resentation. Itmaybeworthwhiletoconsider
whether this property can be used to tell a
new story about treating syntax and
seman-ticsin auniform framework.
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