Guy Perrier
LORIA,Universite Nancy2
BP 239
54506 Vanduvre-les-Nancy Cedex France
e-mail: [email protected]
Abstract
InteractionGrammars(IG)areanewlinguistic
for-malism which is based on descriptions of
under-speciedtreesintheframeworkofintuitionistic
lin-earlogic(ILL). Syntacticcomposition,which is
ex-pressedbydeductioninlinearlogic,iscontrolledby
asystemofpolarizedfeatures. Inthisway,parsing
amounts to generating models of tree descriptions
and it is implemented as a constraint satisfaction
problem.
Introduction
IGcanbepresentedasanattempttobringtogether
fundamental ideas,somecomingfromTree
Adjoin-ing Grammars (TAG) and others from Categorial
Grammars (CG), in order to overcome the specic
limitationsofeachoftheseformalisms. The
compu-tational and linguistic relevance of TAG lies in its
adjunction operation(Joshiet al.,1975;Krochand
Joshi,1985)butthesimplicityofitsmechanismhas
acounterpartintheinationofthelexiconsthatare
requiredfor expressingall grammaticalphenomena
of alanguage. Everytimethat awordisused in a
newsyntacticcontext,whichcandieronlybyword
orderforexample,anewelementarytree,which
en-codesthiscontext, mustbeaddedto thelexiconin
adirectmannerorbymeansofalexicalrule. Inthis
way,lexiconsquicklybecomecolossal,veryawkward
touseandveryhardtomaintain.
Recentworksaimtosolvethisproblemby
factoriz-inglinguisticinformationwiththenotionsof
under-specied trees and tree descriptions. These notions
wereintroducedforTAGbyVijay-Shankerwiththe
motivation of makingadjunction monotone
(Vijay-Shanker, 1992). Now, they are exploited fruitfully
in various directions: for structuringTAGlexicons
in hierarchical systemsof modules (Candito, 1999)
orforexpressingsemanticambiguity(Muskensand
Krahmer,1998;Eggetal.,1998)forinstance. Atthe
sametime, these notionsare a way of relaxing the
primitive adjunction operation in order to capture
linguisticphenomena: suchapossibilityisexploited
by(Rambowetal.,1995)withinD-TreeGrammars
and by (Kallmeyer, 1999) within Tree Description
Grammars.
Unfortunately, the counterpart of a more exible
frameworkisoftenover-generationandalossof
com-putationaleÆciencyin theabsence ofprinciplesfor
controlling syntactic composition. By looking at
CG, we can nd some answers to this
preoccupa-tion: a fundamental ideaof CG is that
grammati-calconstituentsareviewedasconsumableresources
(Retore,2000);theseresourcesaredividedinto
pos-itive and negative resourceswhich are
complemen-tary and search to neutralize themselves mutually.
The coreof CGis the Lambek Calculus (Lambek,
1958): bycombining resourcesensitivity and order
sensitivity, thislogic is agood candidate for
repre-senting the syntax of naturallanguagesbut at the
sametime,thiscombinationentailsarigiditywhich
limitsitsexpressive powergreatly. An appropriate
way of relaxing this rigidity constitutes an
impor-tantresearchareainCG(Moortgart,1996).
TheprincipleofIGistocombinethepowerfulnotion
ofunder-speciedtreedescriptionlinkedtotheTAG
philosophywithcontrolofsyntacticcompositionby
a system of polarities in accordance with the CG
philosophy. Moreprecisely,the basicobjectsof IG
aresyntactic descriptions which express
dependen-cies between syntactic constituents in the shape of
under-speciedtrees. Wordorderis referredtothe
same level as morphological information, which is
representedbyasystemoffeatureswhicharelinked
to the nodes of syntactic descriptions. Whereas a
feature is usually a pair (attribute, value), an IG
featureisatriplet(attribute,polarity,value) where
polarity can take one of the three values -1, 0 or
+1andbehaveslikeanelectrostatic charge: for
in-stance, a noun phrase which is waiting to receive
a syntactic function in a sentence, carries a
nega-tive feature of typefunct while a nite verbwhich
is searching for its subject, carries a positive
fea-turefunctwithvaluesubj. Attractionbetweenthese
dual features will make possible the fact that the
verbndsitssubjectand,simultaneously,thenoun
phrasends itsfunction in thesentence. (Muskens
andKrahmer,1998)alsorecognizedthenecessityof
descrip-position; the dierence with respect to IG lies in
thegranularityof thepolarizationwhich isner in
IG:in theirproposal,thepolarizedobjectsare
con-stituents, that is descriptionnodes, whereas in IG
oneconstituentcanincludeseveralfeatureswith
op-positepolarities.
Theframeworkwhichischosenforrepresenting
syn-tacticdescriptionsinthispaperisthatoflinearlogic
(Girard, 1987), more precisely a fragment of ILL
(Lincoln, 1992). The resource sensitivity of linear
logic allows one to express the fact that polarized
features behave as consumable resources in IG: a
positivefeaturehastonditsdualfeatureonceand
onlyonce. Ifwetrytouseclassicalorintuitionistic
logicformodellingIG,thecontractionand
weaken-ing rules, which are inherent in these logics, entail
a loss of resource-sensitivity: for instance, a verb
could take two subjects by appealing to the
con-tractionruleandsomenounphraseswouldnotneed
to ndtheirsyntacticrolein asentence by
appeal-ingto theweakening rule. Bydiscardingthesetwo
rules,linearlogicprovidesaframeworkthatexactly
correspondsto the\electrostatic"lawsthatcontrol
polarizedfeatures.
In this framework, parsingtakes the shape of
log-icaldeduction of closed syntactic descriptions from
any syntactic descriptions: adescription is said to
beclosed when it representsacompletely specied
syntactictreewhereallfeaturesareneutralized.
Iflinearlogicprovidesanelegantframeworkfor
rep-resenting IG, it gives no method for parsing
eÆ-cientlyandavoidingthecombinatoryexplosionthat
mayfollowfromtheexibilityoftheformalism. An
answerto thisproblem isgivenby theparadigmof
constraintsolving. Parsingaphrasecanberegarded
asgeneratingmodelsofthepartialdescriptionwhich
isprovidedbyalexiconforthewordsofthisphrase.
The process is monotone and can be expressed as
a constraint satisfaction problem. This
constraint-basedapproachwasinspiredbytheworkof(Duchier
and C., 1999; Duchier and Thater, 1999)on
domi-nance constraints. (Blache,1999)showsthe
advan-tagesofsuchanapproachbothfromalinguisticand
computationalviewpointwiththeformalismthathe
proposesandhecallsPropertyGrammars.
1 Syntactic descriptions as linear
logic formulas
IG are formally dened as an ILL theory. Basic
objects are syntactic descriptions which are
repre-sentedbylinearlogicformulasinthefollowingform:
Descr ::= DominjFeatjDescrDescr j
Descr&Descr
If a syntactic description concerns the dominance
relation between syntactic constituents, it has the
type Domin; if it concerns the features which are
ertiesofconstituents, ithasthe typeFeat. Finally,
adescriptioncanbe builtrecursivelyfrom two
de-scriptionsin twoways, which areexpressed by the
twolinearlogicconjunctions: themultiplicative
ten-sor()andtheadditivewith(&).
1.1 Multiplicativeand additive conjunction
ofresources in descriptions
AdescriptionD
1 D
2
requiresallresourcesofboth
descriptionsD
1 andD
2
whileadescriptionD
1 &D
2
requireseither the resourcesof D
1
ortheresources
ofD
2
but notboth. Thisuseofthetwolinearlogic
conjunctions is consistent with their left
introduc-tionrulesin thelinearsequentcalculus:
F
1 ;F
2 ; `G
F
1 F
2 ; `G
L F
1 ; `G
F
1 &F
2 ; `G
&
L1 F
2 ; `G
F
1 &F
2 ; `G
&
L2
In this way, it is possible to describe all syntactic
congurations of aword witha singlelexical entry
under the form ofasyntacticdescription: common
parts of thesecongurationsarefactorizedwhereas
specicpartsaredistributedintoalternationslinked
together with the connective with. For instance, a
possiblelexicalentryfortheniteverbvoitinFrench
has theshapeD
voit =D
1 (D
2
&D3)(D
4 &D5):
D
1
containsinformationrelatedtothesubjectwhich
iscommontoallusesoftheverbvoit;D
2
expresses
thecanonicalordersubject-verbinthesentencethat
isheadedbytheverbvoitwhereasD
3
expressesthe
reverseorderforwhichthesubjectmustberealized
under someconditions,suchasin thephraseMarie
que voit Jean; D
4
expresses that the verb has an
explicit object whereas D
5
corresponds to
circum-stances where thisobjectis notpresent,such asin
thesentenceJeanvoit.
1.2 Under-specication ofdominance
between constituents
AdescriptionoftypeDominhasthefollowingform:
Domin ::= Node>[LNodes]jNode>Nodej
Node>
Node
LNodes ::= jNodeLNodes
A predicate N > [N
1 ;:::;N
p
] states that the
constituent N is decomposed into the
sub-constituents N
1 ;:::;N
p
. The order between these
sub-constituents is only used for identifying each
one without any linguistic meaning; word order
is dealt with at the same level as morphological
informationbymeansoffeatures.
A predicate N
1 > N
2
expresses that N
2 is an
immediate sub-constituentofN
1
. Such apredicate
is used when only partial information on the
sub-constituentsofaphraseis available.
A predicate N
1 >
N
2
expresses that N
2
is
embed-ded in N
1
atanundetermineddepth. Forinstance,
if we continue with description D
1
related to the
(N
3 >[N
4 ;N
5 ])(N
4 > N
6
)which is interpreted
as follows: the verb phrase N
3
is constituted of
the verb N
4
and its object N
5 ; N
6
represents the
bare verb whereas N
4
represents the verb which
has been possibly modied by a clitic, a negation
oranadverb. Under-specicationofthedominance
relation N
4 >
N
6
leaves all these modications
open. Under-specication of dominance between
constituents goes beyond TAG adjunction in that
thenodeswhich areinadominancerelationdonot
necessarily have the same grammatical category
and thus linguistic phenomena like wh-extraction
canbeexpressedeasilyinthis way.
1.3 Polarized and under-specied features
Descriptionsof type Feat, relatedto features, have
thefollowingform:
Feat ::= Node:AttrPolVal j
Var2Dom jVar62Dom
Pol ::= j = j !
Val ::= ConstjVar
AfeatureNode:AttrPolValisatripletcomposed
ofanattributeAttr,apolarityPolandavalueVal
associated with a syntacticnode Node. Usually, a
feature is dened as a pair (attribute, value). In
IG, we add apolarity to this pair so that features
behavelikeelectrostatic charges: apositivefeature
Attr!ValseeksanegativefeatureAttr Valto
neutralize itand converselywhile aneutralfeature
Attr=Val onlyactsasalterthroughconstraints
on its value Val when it meets another feature of
typeAttr atthesamenode.
Inallcases,Valiseitheraconstantwhichisselected
from an innitecountable set Constof feature
val-ues ora variable which is selected from an innite
countable setVaroffeaturevariables;then,its
def-inition domain can be constrained bytwo kindsof
predicates: Val 2Dom andVal 62Dom; Dom isa
niteset ofelementstakenfrom Const.
Let us illustrate this presentation with a possible
lexicalentryforthepropernounJean:
DJean = (N>[])
(N:cat=np)(N :funct V1)
(N:ord!1)(N :phon= 0
Jean 0
)
(N:gen=m)(N:num=sg)
(N:pers=3)
Somefeatures areneutralbynature likeagreement
features: gen=m (gender=male), num=sg
(num-ber=singular), pers=3 (person=3). Others are
po-larizedbynature too: for instance,features oftype
funct which express syntacticfunctions. Inthe
ex-ample above, the feature of type funct is negative
becausethenounphraserepresentedbyNiswaiting
to receivea syntacticfunction (subject, object:::);
this function is not determined yet and thus it is
representedbyavariableV
1 .
Thephonologicalformofaconstituentisdetermined
by asystem of two features: phon which gives the
eective phonological form of the constituent and
ord which gives the order in which its immediate
sub-constituents must be concatened to build this
phonologicalform.Forinstance,wendtheformula
(N
1
:ord!V
2 )(V
2
2f12;21g)inthedescription
D
voit
toexpress thattheclause which hastheverb
voit as its head and is represented by node N
1 is
a concatenation subject-verb phrase (V
2
= 12) or
verbphrase-subject(V
2
=21). Whenanodehasno
children, two cases occur: the node has an empty
phonologicalformandthevalueofthefeatureordis
0orthenodeisalexicalanchorandthevalueofthe
featureordis1. Inthiscase,thefeaturephonisused
forretrievingtheeectivephonologicalform,which
canbeveriedin the descriptionD
Jean
.
Polariza-tionofphonologicalformsexpressesthatsome
con-stituents are capable of givingaphonological form
while others are waiting for one. As the previous
examplesshows, this polarityis not carried by the
feature phon but by the feature ord. The interest
of giving privilege to the feature ord with respect
to the feature phon, is twofold: we can determine
its value for a given node without being aware of
thephonological form of the children, the eective
phonologicalform will berebuilt stepbystep from
theleavesto theroot of the nal syntactic tree as
soonaspossible;anotherinterestisthat featuresof
typeordcanbedealt withlikeallotherfeatures; in
particular, we canapply to them the sametype of
constraints.
Finally,itis interestingto mentionthat value
shar-ing by dierent features is represented in an easy
waybyusing auniquevariableforthevaluesofthe
concernedfeatures.
2 Syntactic composition as
deduction in a linear theory
By choosing a logical framework for a formal
def-inition of IG, we nd a natural way of expressing
syntacticcompositionbymeansofdeductionin
lin-earlogicaccordingto theparadigm\parsingas
de-duction"of CG(forabroadsurveyofCGsee
(Re-tore,2000)). An interaction grammar islexicalized
in the sense that all linguistic resources are stored
inalexiconandtheseresourceswillbecombinedby
usinginferencerulesoftheILLdeductivesystemfor
buildingtheacceptablesentencesofthe
correspond-inglanguage. Sincesyntacticdescriptions useonly
afragmentofthis logicandifwechoosethe
frame-work of the sequent calculus, only seven ILL rules
areuseful:
F1;:::;Fn ` F1Fn id
` G
1; ` G 1
L F
1 ; F
2
; ` G
F
1 F
2
; ` G
L
F1; ` G
F
1 &F
2
; ` G
&
L1
F2; ` G
F
1 &F
2
; ` G &
L2
F[t=X]; ` G
8X F; ` G 8L
1
` F F;
2 ` G
sequentcalculus(Lincoln,1992),axiomidisdened
a bit dierently but this denition is equivalent to
theoriginaloneforthelogicalfragmentusedbyIG.
Rule 8
L
is a rstorder rulewhich is used here for
instantiating a node variable with aconcrete node
orafeaturevariablewithaconcretefeaturevalue.
Beside these general rules, we need proper axioms
toexpresspropertiesrelatedtodominancerelations,
feature polarities, feature values and phonological
forms. Concerning dominance relations, we have
thefollowingproperaxiomschemes:
N
Axiom scheme d
1
expresses that immediate
dom-inance is realized by a parent-children relation
whereas axiom schemes d
2
and d
3
express that
dominanceisrealizedby nitesequencesof
parent-children relations (L and L' represent sequencesof
nodevariables).
The behaviour of polarities is represented by the
followingproperaxiomschemes:
(N:AP V); (N:A=V) ` (N:AP V) p1
(N:A V); (N:A!V) ` (N:A=V) p2
Properties related to feature domains and values
areexpressedbythefollowingaxiomschemes:
`c2fcg[D
In both axiom schemes, D represents a concrete
nite set of feature valuestaken from Constand [
and n represent the usual operations of unionand
dierenceofsets.
Finally,three axiomschemes areused fordeducing
theeectivephonologicalformofaconstituentfrom
theorderofthephonologicalformsofitschildren:
N>[];(N:ord=0)`N>[](N:phon=`
respectively correspond to
emptycategoriesandlexicalanchors.
In scheme ph
3
, O is an abbreviation for
(N : ord = c()); is a permutation on [[1;p]]
which expresses an order for concatenating the
phonologicalforms v
1 ;:::;v
p
of the childrennodes
N
1 ;:::;N
p
of N andc()is abijectiveencodingof
this permutation with an integer. P
1
is an
abbre-viation for (N
1
an abbreviation for the product
(N : phon = v
A particular interaction grammar G is dened by
itsvocabularyVoc
G
andbyalexiconLex
G
;the
vo-cabularyVoc
G
includesthewordsusedforbuilding
G
thelexicon Lex
G
associatesasyntactic description
to each word of Voc
G
. Now, we have to combine
the resources provided by Lex
G
by means of the
inference rules and proper axioms of the linear
theory T which has just been dened to compose
well-formed andcomplete syntacticstructures ofG
undertheshapeofclosedsyntacticdescriptions. As
apreliminary,wehavetogiveaprecisedenitionof
aclosedsyntacticdescription:
A closedsyntacticdescription isap
articu-larsyntacticdescriptionintheshapeSF
where S andF, respectively, represent the
structuralandfeature parts of the
descrip-tionwith the following conditions:
1. Sisaproductofpredicatesintheform
(n > [n
represent concrete syntacticnodes,
and the structure dened by all these
parent-children relations isatree;
2. Fisaproductofpredicatesintheform
(n : attr = v), where n, attr and v
representconcreteatoms,andforeach
pair (n;attr)presentinF,thereis
ex-actly one feature(n:attr=v)in F.
3. Forevery syntacticnoden inS,there
isafeature(n:phon=v)inF.
Condition 1 guarantees that a closed syntactic
description represents a completely specied tree.
Condition2guarantees coherenceand neutralityof
thefeaturesystemwhichisattachedateach
syntac-ticnode. Condition 3 guarantees the phonological
well-formednessofthewholesyntacticstructure.
Now,letusexplain howGgeneratesclosed
syntac-ticdescriptionsfromnlexicalentriesD
w
1
;:::; D
w
n
corresponding to n words w
1
. For this, we need an additional description
D
root
to represent the root of the nal syntactic
treewhich hastheform:(N
0
representstheroot
ofthesyntactictreeand N
1
;:::; N
p
are thenodes
presentindescriptionsD
w1
;:::; D
wn . Then:
A closedsyntacticdescription D issaid to
begeneratedfromthewordsw
1 ;:::; w
n by
grammarGifthesequent8 ~
V representallnode
vari-ables andfeature variables that are free in
D
D describesatreewhichrepresentsthesyntaxofa
phrasegiven bythe feature phon of its root. If we
addthepredicate(N
0
wetransformthe generationof closedsyntactic
simple illustration of this mechanism. We assume
that a lexicon provides us with three descriptions
D
voit ,D
il andD
Jean
whichrespectivelycorrespond
to the nite verb voit, thepersonal pronounil and
the propernoun Jean. Asit wasdescribed in
sub-section 1.1, D
voit
has the shape D
1 (D
2 &D
3 )
(D
4 &D
5
)anditisschematized bythefollowing
di-agram:
ord = 21
ord = 1
&
&
funct -> subj
ord = 12
1
2
1
3
4
cat = np
funct -> obj
ord <-
ord = 12
2
3
4
5
D
3
D
D
4
D
5
cat = v
ord
<-cat = v
phon = ’voit’
ord <-
funct = subj
cat = np
1
2
3
4
6
ord -> 12 | 21
cat = s
cat = vp
ord = 1 | 12
ord -> 1
1
D
Toremain readable, the diagram includes only the
most signicant features of everynode. The
nota-tion ord ! 12j21 is an abbreviation for ord ! V
withV 2f12;21gandord meansthat thevalue
ofthefeatureordisundetermined.
Description D
il
has a structure that is similar to
D
voit :
ord = 21
11
8
cat = s
cat = vp
cat = v
cat = v
ord -> 12 | 21
cat = clit
phon = ’il’
cat = v
ord = 21
ord = 1
&
&
D
6
D
8
D
10
funct = subj
cat = np
typ = inter
7
8
9
10
11
12
13
7
ord
<-ord = 12
11
ord = 12
D
7
11
ord -> 0
funct <- subj
D
9
typ = decl
7
The rst additive component of description D
il ,
D
7 &D
8
represents achoice betweenthe absenceof
an explicit subject in the sentence beside the
per-andthepresenceof thissubjectsuch asin the
sen-tenceJean voit-il ?. The second alternativeentails
that thesentenceis interrogativeif we ignore
topi-calization,which explainsdescriptionD
8 .
Thesecond additive componentof description D
il ,
D
9 &D
10
,representsachoicebetweenthedeclarative
typeandtheinterrogativetypeofthesentencewhich
dependsontherelativeorderbetweentheverband
theclitic.
DescriptionD
Jean
isreducedtothefollowingsingle
node:
cat = np
ord -> 1
phon = ’Jean’
14
funct
<-From the description 8 ~
N 8 ~
V (D
root
D
voit
D
Jean
D
il
), it is possible to deduce three closed
syntacticdescriptionsD
a ,D
b andD
c
,which
respec-tivelyrepresentthesyntax ofthegrammatical
sen-tences:ilvoitJean,voit-ilJean? andJeanvoit-il?.
Inconcreteterms,thedeductionprocessthat leads
to these three solutions consists in plugging nodes
oftheinitialdescriptionswith theaim of
neutraliz-ingallpolarizedfeatureswhilerespectingdominance
and feature constrains. Let us detail the resulting
descriptionD
b
bymeansofthesyntactictreeit
spec-ies:
typ = decl
cat = vp
cat = s
cat = v
phon = ’voit’
phon = ’il’
cat = clit
cat = v
phon = ’ ’
cat = np
funct = subj
0 -1-7
2-8
3-9
4-10-11
12
6-13
5-14
phon = ’voit il’
phon = ’voit il Jean ’
phon = ’voit il Jean’
cat = np
phon = ’Jean’
funct = obj
The closed syntactic description that species the
treeaboverepresentsthesyntacticstructure ofthe
sentencevoit-il Jean ?. The numbersthat labelits
nodesarethetracesofthenodesofthedescriptions
thathavebeenpluggedin theparsingprocess.
3 A constraint-based
implementation
From the viewpoint of a computer scientist, a
lin-guisticmodelhastoshownotonlyexpressivepower
butalsocomputationaltractability. Intheprevious
section, we have shown that IG computations
re-ducetoILLproofs. Forthelogicalfragmentthatwe
non-L1 L2 L
takestheshapeofthreekindsofchoicepointsinthe
parsing process: selecting the pertinent branch for
every additive conjunction, identifying some node
variables and instantiating feature variables in an
appropriate manner. The NP-completeness of the
implicativefragmentofILL(Kanovich,1992)shows
that it is hopeless to nd a general parsing
algo-rithm forIG that works in polynomialtime in the
worstcases. Experiencehasshownthat,fortunately,
theseworstcasesrarelyoccurinparsingnatural
lan-guages. Nevertheless, the exibilityof IG entailsa
combinatory explosion of the parsingprocess ifwe
use a \generate and test" method and leads us to
choose amore appropriate method. The
specica-tionofourproblempromptsusin anaturalwayto
aconstraint-basedapproachasit wassuggestedby
some proposals for similar problems (Duchier and
C.,1999;DuchierandThater,1999).
Theproblemcanbeformulatedasfollows:
Given a syntacticdescription D
0
, nd all
closed syntactic descriptions D such that
8 ~
N 8 ~
V D
0
`D isprovableinthe theory T
( ~
N and ~
V respectively represent the node
variables N
1 ;:::;N
n
andthe features
vari-ables V
1 ;:::;V
m ofD
0 ).
A fundamental property of the deduction process
that leadstoasolutionismonotonicitysothat the
problem can be expressed as a constraint
satisfac-tion problem (CSP).A CSP is speciedfrom aset
ofvariables towhich constraintsare applied. Here,
weconsiderthreesetsofvariables,whichcorrespond
tothethreekindsofchoicepointsintheparsing
pro-cess:
1. the set fN
1 ;:::;N
n
g of syntactic node
vari-ables;
2. thesetfV
1 ;:::;V
m
goffeaturevariables;
3. the set fS
1 ;:::;S
p
g of selection variables;
ev-ery selection variable S
i
is an integer variable
which is associatedwith aconnective& ofD
0
andwhichisusedforindicatingtherankofthe
component of the correspondent additive
con-junctionthat isselectedinthededuction.
Selection andfeature variablesare consideredas
-nite domainvariables, which imply that allfeature
valuesareencodedasintegers(oneexceptionisthat
featuresoftypephonremainstrings).
Node variables are encoded indirectly via nite
set variables by using the method proposed in
(Duchier and C., 1999). Every node variable
N
i
is associated with ve nite set variables
eq(i); up(i); down(i); side(i) and alt(i) which are
usedforlocatingthenodeiwithrespecttothe
oth-ers in the system of dominance relations. Because
whichispresentinthedescriptionD
0
maybeabsent
from asolution. In this case,eq(i)=fig; alt(i)=
[[1;n]]nfig; up(i) = down(i) = side(i) = ;; in the
case that i is present in a solution, alt(i)
repre-sentsthenodesthatarenotselectedinthesolution
whereastheselected nodesare distributed into the
foursetseq(i); up(i); down(i)andside(i)according
totheirrelativeposition withrespecttoi.
Constraints onthevariables of theproblem are
di-videdintotwoparts:
generalconstraintsguaranteethatthesolutions
D areeectiveclosedsyntacticdescriptions;
specicconstraintsguaranteethatthesolutions
D aremodelsoftheinitialdescriptionD
0 .
3.1 Generalconstraints
Treenessconstraints Foreverynodei,the
parti-tionof[[1;n]]betweeneq(i); up(i); down(i); side(i)
andalt(i)guaranteesthatthesolutionisadirected
acyclicgraph(DAG).
For expressing that all dominance relations which
structureasolutionmustonlyberealizedby
parent-childrenrelations,wemustintroduceconstraintsin
whichvariablesof typeeq(i)andselectionvariables
appearforexpressingthateveryselectednode
vari-ablemust beidentied with anode variable which
istheparentin aselectedparent-childrenrelation.
Inorder to express that asolution is morethan a
DAG,thatisatree,wemustaddthefollowing
con-straint: for every selected parent-children relation,
the sets down(j) for the children j present in this
relationmust be disjoint. Such a conditioncan be
droppedifwewanttoextendtheformalismto take
into account resource sharing like coordination for
instance; in this case, syntactic structures are no
longertreesbut DAGs.
Neutrality constraints Feature neutrality of a
solutionisguaranteedbyconstraintswhichalso
ap-peal to variables of type eq(i) and selection
vari-ables: for each attributeAttr,weconsider twosets
of sets in the shape eq(i): the rst corresponds to
allselectedpredicatesinthe form(Ni:Attr V)
andthesecondtoallselectedpredicatesintheform
(Ni : Attr ! V). The elements of each of these
sets mustbedisjointsets and every elementof the
rstset must be identied withone elementof the
secondandconversely.
Other general constraints related to features and
phonologicalformsaretrivial.
3.2 Specic constraints
Such constraints are determined by D
0
.
Domi-nance constraints are easily implemented by
com-bining selection variables and variables of type
eq(i); up(i); down(i); side(i)(DuchierandThater,
and selection variables which are all nite domain
variables so that their implementation appeals to
classicaltoolsinthedomainofconstraint
program-ming.
3.3 A prototype parser forFrench
WehaveimplementedaprototypeparserforFrench.
It is written in the language Oz (Smolka, 1995)
which combines various aspects and modules,
in-cluding constraint programming. Though the
lin-guistic coverage of the lexicon is still limited, we
have learnt lessons from the rst experiments: in
particular,neutralityconstraintsplayacentralrole
forrestricting thesearch space, which conrms the
importance of polaritiesfor the computational
eÆ-ciency.
Conclusion
Starting from TAG and CG, we have presented a
linguisticformalismwhich aimsat bettercapturing
theexibilityof naturallanguagebyusing two
no-tionsas itsbasis: underspecication and polarities.
In some sense, they correspond to two important
properties of natural language: ambiguity and
re-sourcesensitivity.
Toregardparsingasaconstraintsatisfaction
prob-lem ts in with the exibility of the formalism in
terms of computational eÆciency but, at the same
time, it allows to go towards robustnessbeyond a
traditionalviewofparsinginwhichonly
grammati-calandcompletelyspeciedstructuresaretakeninto
account.
The success of IG does not essentially depend on
theformal propertiesthat are usuallyexhibited for
grammaticalformalisms: thecharacterizationofthe
classoflanguagesthataregeneratedbythese
gram-mars or the complexity of general parsing
algo-rithms. Formalpropertiesmatter but withrespect
toanessentialgoal:toextendthelinguisticcoverage
ofIGfromtoylexiconstomassivelexicaldatabases.
Forthis,IGhavesomeadvantagesbymakingit
eas-ily to factorize and modularize information: such
properties are decisive when one wants to extract
informationfrom alexical databaseeÆcientlyorto
updatedatawhilemaintainingthecoherenceofthe
wholebase.
ThesuccessofIGwillalsodependontheircapacity
tointegrateotherlinguisticlevelsthanthesyntactic
level,thesemanticlevelespecially.
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