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Relaxing Underspecified Semantic Representations for Reinterpretation
Alexander Koller, Joachim Niehren, Kristina Striegnitz
To cite this version:
Alexander Koller, Joachim Niehren, Kristina Striegnitz. Relaxing Underspecified Semantic Repre-
sentations for Reinterpretation. 6th Meeting on Mathematics of Language, 1999, Orlando, Florida,
United States. pp.74–87. �inria-00536811�
Reinterpretation
AlexanderKoller (kolleroli.uni-sb.de)
Dept.ofComputational Linguistis,UniversityoftheSaarland
JoahimNiehren (niehrenps.uni-sb.de)
ProgrammingSystemsLab,UniversityoftheSaarland
KristinaStriegnitz (krisoli.uni-sb.de)
Dept.ofComputational Linguistis,UniversityoftheSaarland
Abstrat. Type andsortonitsinsemantisare usuallyresolvedby aproess
ofreinterpretation,whihintroduesanoperator intothesemanti representation.
We elaborateonthefoundations ofareent approahtoreinterpretationwithina
frameworkforsemantiunderspeiation.Inthisapproah,relaxedunderspeied
semanti representations are inferred from the syntati struture, leaving spae
for subsequent addition of reinterpretation operators. Unfortunately, a strutural
danger of overgeneration is inherent to the relaxation of underspeied semanti
representations.Weidentifytheproblemanddistinguishstruturalpropertiesthat
avoidit.Wefurthermoredeveloptehniquesforprovingthesepropertiesandapply
themtoprovethesafety ofrelaxationinaprototypialsyntax/semantisinterfae.
Indoingso,wepresentsomenovelpropertiesoftreedesriptions intheonstraint
languageoverlambdastrutures(CLLS).
Keywords:onstraints,naturallanguagesemantis,reinterpretation,treedesrip-
tions,underspeiation
1. Introdution
In natural language semantis, sort or type onits as well as as-
sumptions of the ontext a sentene is uttered in may ause meaning
shiftsofwordsorexpressions(Bierwish,1983;Dolling,1994;Nunberg,
1995; Pustejovsky,1995). An exampleisthe followingsentene.
(1) Peter begana book.
The probleminthis sentene is thatPeter an onlybegin an ativity.
Soinunderstandingit,wemustllinwhat,exatly,Peterbeginstodo
with thebook{ for example, readingit, writing it, et. Thismeaning
shift leads to readings suh as (2) or(3), in whih more material has
beenadded,asindiatedbytheboxes.Thisproessofaddingmaterial
is alledreinterpretation.
(2) 9x:book(x)^begin(peter; read(peter,x )):
(3) 9x:book(x)^begin(peter; write(peter,x) )
Pustejovsky (1995) assumed that the reinterpretation operator is
determinedbypurelylexialinformation.Sinethenithasbeennotied
thattheinformationdeterminingthereinterpretationproessomesa-
tuallyfromavarietyofsoure,inludingontextandworldknowledge.
This impliesthatthe naldeisionona spei reinterpretationoper-
ator an onlybemade after semanti onstrution.Inan extension to
Pustejovsky's approah,Lasarides andCopestake (1998) usedefaults
whih an be overridden after semanti onstrution to aount for
thisfat. Egg (2000) notiedthat the atualinsertion of a reinterpre-
tation operator an be modeled in an elegant, monotoni way within
a framework forsemantiunderspeiation(vanDeemter andPeters,
1996;Reyle,1993;Bos,1996;Pinkal,1996;Eggetal.,1998).Insteadof
derivingandthendestrutivelyhangingafullyspeisemantirepre-
sentationforasentene,hehangeshissemantionstrutionproessto
produea relaxed desriptionof thesentene meaning,whih ontains
a \gap" at the reinterpretation site. A reinterpretation operator an
then belled into the gap non-destrutively.Suh a desriptionould
lookroughlyasin(4).
(4) 9x:book(x)^begin(peter; ::: x :::)
It is possible to derive suh relaxed desriptions ompositionally,
and reinterpretation indeed beomes a monotoni proess of making
desriptions more spei. However, relaxation is in general a proess
thatbearsa dangerof overgeneration.Roughlyspeaking,itispossible
thatmaterialalreadypresentinthedesriptionouldslipintothegap,
giving therelaxed desriptionunintended readings.
This artile is a formal investigation of the relaxation operation
in the framework of the Constraint Language for Lambda Strutures
(CLLS) (Egg et al., 1998), where underspeied semanti repre-
sentations are expressed by tree desriptions subsuming dominane
onstraints. We dene what it means for a relaxation to be safe (the
overgenerationproblemisavoided)andopen(arbitrarymaterialanbe
lled into the gap). Then we develop general tehniquesfor reasoning
aboutstruturalpropertiesoftreedesriptionsinCLLS{mostnotably,
hains of fragments { and use these tehniques to prove that every
relaxed desription produed by the syntax/semantis interfae of a
ertain toy grammaris open and safe. We believe that thisresult an
be lifted to more serious grammars while using the presented proof
tehniques.
Planof theartile. Setion2introduestreedesriptionsinCLLS.In
tationandmaketheovergeneration problemonrete.InSetion4,we
formalize thenotionsof relaxation,safety, andopenness. InSetion 5,
we dene fragments and hains of fragments and prove some useful
properties of these objets. Setion 6 presents the prototypial toy
grammar for deriving relaxed underspeiedrepresentations. Finally,
we prove that all relaxations that an be derived bythis interfae are
safe andopen inSetion7.
2. Semanti Underspeiation in CLLS
This setion introdues the Constraint Language for Lambda Stru-
tures (CLLS) as a formalism for semanti underspeiation. Briey,
the idea of underspeiation is that beause the number of read-
ings of an ambiguous sentene may grows hyper-exponentially with
the number of ambiguities, it an make senseto derive only a single,
\underspeied"desriptionof themeaning fromthesyntax andthen
to work with this desription instead of the readings for as long as
possible.Readings areenumerated onlybyneed.
CLLS (Egg et al., 2000; Egg et al., 1998; Koller et al., 1998) an
beused fortheunderspeieddesriptionof-terms.Tehnially,itis
a languageoftree desriptionsbased ondominaneonstraints, whih
are used in various appliations throughout omputational linguistis
(Marus et al.,1983; Vijay-Shanker, 1992; Rambow etal., 1995; Gar-
dent and Webber, 1998). Dominane onstraintsappear, to a varying
degree of expliity, in many frameworks of sope underspeiation
(Reyle, 1993; Bos, 1996; Muskens, 1995), and their omputationalas-
petsare ratherwellunderstood(Bakofenetal.,1995;Vijay-Shanker
etal.,1995;Kolleretal.,1998;DuhierandNiehren,2000;Kolleretal.,
2000). CLLS is interpreted over -strutures, onservative extensions
of tree strutures. Below, we will rst dene -strutures and then
two dierent ways of desribing them: by (onjuntive) logiformulas
and by onstraint graphs. Then we give a quik overview about the
appliationtosope.Finally,weintroduetherst-orderlanguageover
CLLS, whih will turn out later to be useful for talking about CLLS
desriptions.
2.1. Lambda Strutures
A -struture represents a -term uniquely,up to renamingof bound
variables. It is an ordinary rst-order tree struture extended with
a partial -binding funtion. -strutures an be drawn as tree-like
graphs,withdashedarrowsrepresenting-binding,asinFigure1.The
lam u
0
u
1
ar u
2
var u
3
represents
! x:ar(x)
Figure 1. A-strutureandthe-termitrepresents.
tree strutures we onsiderarebasedon onstrutor trees;that is,the
label of a node determines the number of its hildren. Construtor
treesarebasiallygroundtermsoveragivensignature;forinstane,the
onstrutortreeunderlyingthe-strutureinFigure1islam(ar var).
We assume a signature =flam;;var ;ar ;driver;:::g of funtion
symbols written as f;g. Every funtion symbol f 2 has an arity
ar(f)0.Welet a;b range over onstants{ funtionsymbols of arity
0.Fordesribing-terms,weassumethefollowingfuntionsymbolsto
belong to : aunarysymbollam for-abstration,thebinary symbol
for appliation, and the onstant var for ourrenes of -bound
variables.
We dene an (unlabeled) tree in the standard way as a direted
graph (V;E) where V is a niteset of nodes u, v and E V V a
nitesetof edges e. Theindegree ofeah nodeinV isat most1;eah
tree has exatly one root, i.e. a node with indegree 0. We all a node
with outdegree0 aleaf ofthetree.
A (nite) onstrutor tree over is a triple(V;E;L) where (V;E)
is a tree, L :V ! a node labeling and L :E !N an edge labeling,
suh thatany node u 2V hasexatly one outgoingedge withlabelk
foreah1k ar(L(u)),andno otheroutgoingedges.The symbolL
is overloaded to serve bothas a node and an edge labeling; there will
beno danger ofonfusion.
Denition 2.1. A (partial) -struture is a quadruple (V,E,L,)
onsisting of a onstrutor tree (V;E;L) over and a partial fun-
tion : L 1
(var ) L 1
(lam) mapping ourrenes of lambda bound
variables to their lambda binder.
We will freely identify the -struture (V;E;L;) with a rst-
order struture with domain V and the following relations: binding
(u)=v,dominane
,and,foreahfuntionsymbolf 2,labeling.
Dominaneand labelingaredenedforu;v;u
1
;:::;u
n
2V by:
u
v i there isa pathfrom u to v in(V;E);
lam X
Y
var Z
represents
!
X:lam(Y)^Y
Z ^
Z:var ^(Z)=X ^
X6=Z
Figure 2. Aonstraintgraphrepresentingaoverlap-freeonstraint.
For illustration, we briey return to the -struture for x:ar (x) in
Figure 1.
Inthis-struture,therelationsu
1 :(u
2
;u
3 ),u
0
u
2
,and(u
3 )=
u
0
hold,among others.
Note that not every var node ina partial -strutureis neessarily
mapped to anything by the binding funtion. This deviates from the
standarddenitionof (total)-strutures,whereallvarnodesmustbe
bound,butisneessaryinorderto maintainsomeintermediate results
in Setion4. However, the underspeieddesriptions we willuse will
generallyenfore thatallrelevant varnodeshavebinders.Notefurther
thattreesand-struturesanbedenedbyspeifyingthesetofnodes
asa\treedomain"{asetofwordsoverthenaturalnumbers.Thetwo
denitions areequivalent, butthe graph denitionis more onvenient
forthepurposesof thispaper.
2.2. Desriptions in CLLS
CLLS is a language for desribing -strutures. For the purposes of
the present artile, we onne ourselves to the sublanguage of CLLS
providingdominane, labeling, and -bindingonstraints, but note in
passing that the fulllanguage also ontains parallelism and anaphora
onstraints; fordetails, see(Egget al.,2000).
Figure 2 shows a desription in CLLS, both as a graph and as a
logialformula.A-strutureMsatisesthisdesriptioniX denotes
a node inMthatis labeled by lamand hasasingle hild,denoted by
Y;thishilddominates(isananestorof)anothernode,denotedbyZ,
whihislabeledbyvarand -boundbythenodereferredto byX.For
instane, the -struture in Figure 1 satises all of these onstraints,
given thatX denotesits root,Y its innernode, and Z its right leaf.
WeassumeaninnitesetVarsof(node)variableswhihwewill,gen-
erally,denoteX;Y;Z ;U;V;W.ThesyntaxofthefragmentofCLLSwe
onsiderisdenedinFigure3.Itprovides onjuntionsof atomion-
straints for labeling (X: f(X
1
;:::;X
n
)), dominane(X
Y), lambda
binding ((X)= Y), and inequality(X6=Y). We freelyabbreviate on-
' ::= X:f(X
1
;:::;X
n ) (f
jn 2)
j X
Y
j (X)=Y
j X6=Y
j '^' 0
:
Figure 3. SyntaxofafragmentofCLLS.
straints, suh as X
Y ^ Y
X by X=Y and X
Y ^ X 6= Y by
X +
Y.
CLLS is interpreted over the lass of -strutures, whih provide
relations for the interpretations of all relation symbols, in the usual
Tarskian way (see Setion 2.5 for more details). Note that the same
notationisusedforrelationsymbolsinonstraintsandtheorrespond-
ing relations in a -struture. They an generally be distinguishedby
ontext, as relations are always applied to nodes u whereas relation
symbols areappliedto variablesX.
2.3. Constraint Graphs
For underspeied desriptions, we will only use overlap-free on-
straintswhihontainsuÆientlymanyinequalitiesforexpressingthat
anytwo labeledvariablesdenote distintnodes.
Denition 2.2. A onstraint ' is alled overlap-free i for any two
distint labeling onstraints X: f(X
1
;:::;X
n
) and Y:g(Y
1
;:::;Y
m )
ourring in ' it holds that X6=Y is in' as well (even if f =g).
For easier readability,we willusually draw overlap-free onstraints
as onstraint graphs: Figure 2 is an example of an overlap-free on-
straint and its graph. The nodes of a onstraint graph stand for
variables of the onstraint, and the various types of edges stand for
labeling, dominane, and binding onstraints. Furthermore, the on-
straintgraphrepresentsaninequalityonstraintforeahpairoflabeled
nodes;soitreallystandsforanoverlap-freeonstraint.Despiteasuper-
ial(andintended)similarity,itisimportanttodistinguishonstraint
graphsfrom -strutures.
Constraint graphs ontain \rigid fragments", whih are trees with
solid edges whoseleavesmayormaynot be labeled.Beause therep-
resented onstraint is overlap-free, the variables orresponding to the
innernodesoftwofragmentsmustneverbemappedto thesame node;
every maaboss lam
a meeting lam
attend var var
Figure 4. Semantirepresentationofasopeambiguity.
that is,fragmentsmay notoverlap. However, an unlabeled leaf of one
fragmentmaystillbemappedtothesamenodeastherootofanother.
Fragmentswillbedened formallyinSetion 5.
2.4. Sope Underspeifiation
We now illustrate briey how CLLS an be used to model a sope
ambiguity,asinthe followingexample.
(5) Every maa boss attendsa meeting.
In one readingofthesentene,allmaa bossesattendthesame meet-
ing, while in the other one, there is not neessarily a single meeting
whihallofthemaabossesattend.Underthisseondreadingitwould
e.g. bepossiblethatevery maabosshashis ownmeeting.
Itsunderspeiedsemanti representationisgiven inFigure4.The
onstraint graphontains three rigidfragments: Thefragmentsat the
toprepresent thetwo quantiers\every maaboss" and\a meeting",
and theone at the bottom the verb semantis. Sope ambiguities are
haraterizedbyinvolvingtwoormorequantierswhoserelativesope
isnotxed.InFigure4,thisisaountedforbyimposingtheonstraint
thatbothofthequantierfragmentsmustdominate thethirdone. As
thereanbenoupwardbranhingintrees,thisenforesthatoneofthe
quantier fragments has to be above the other in eah tree desribed
bythegraph,buttheexat ordering isleft open.
However,thisonstraint hasnotonlythetwoobvioussolutions;the
satisfying -strutures ould be muh larger, as long as they ontain
the material required in the onstraint. In other words, there might
be nodes in a solution of a onstraint ' whih are not referred to by
anyvariable of '.Note that CLLSdiersfrom mostother underspe-
iation formalisms inthisrespet(Alshawiand Crouh,1992; Reyle,
1993; Bos, 1996), whih is an essential prerequisite for underspeied
reinterpretationasonsideredinthispaper.
2.5. First-Order Formulas
CLLS onstraints are only onjuntions of atomi onstraints. How-
ever, it will be useful later to use arbitrary rst-order formulas
over these atomi onstraints. Their funtion isto failitate reasoning
about underspeied semanti representations; they are not used as
underspeied semanti representations themselves, in order to save
unneessary omputational omplexityand also beause ourbasi no-
tions of safety and hains (e.g. Lemma 4.7 and thus Proposition 4.8)
depend on the fat that CLLS does not support quantiation over
nodes.
The set of (free) node variables of a rst-order formula over
CLLS onstraints is denoted by Var(). A variable assignment into
a -struture (V;E;L;) is a partial funtion : Vars V. We
write Dom() for the domain of . A solution of a formula on-
sists of a -struture M and a variable assignment into M with
Var() Dom () under whih evaluates to true. We also saythat
(M;) satises and write M; j= if (M;) is a solution of .
We write j= 0
and say that entails 0
if every solution of
interpretingall variablesin Var(
0
) isa solutionof 0
.
Inidentally,we willonlyuse the propositional onnetives of rst-
order logi,disjuntionand negation, butnotquantiation.Negation
allows to express disjointness between two nodes in a tree struture,
meaning that neither of them dominates the other, by the following
formulaX?Y:
X?Y =
def
:X
Y ^:Y
X
3. Underspeied Reinterpretation
We already saw an example of reinterpretation in the introdution,
namely (1). Expressed in -terms of higher-order logis, whih are
the basis of the semanti representation formalism introdued in the
previoushapter,thesemantisof(1)shouldlooklike(6)forinstane.
(6) a(book)(x:begin(peter;read(peter;x) ))
However, traditional semanti onstrution would rather derive some-
thing like (7).
(7) a(book)(x:begin(peter;x))
So,whatreinterpretationhastodohere,intuitively,istollinsemanti
material that was not present on the surfae (read(peter;) in this
a book
lam
begin
var
peter
Figure 5. Relaxedsemantirepresentationof(1)
example) at the loation of the type onit. We all this loation
the reinterpretation site, and the additional semanti material, the
reinterpretation operator.
We now desribe Egg's approah to modeling phenomena of this
type by exploiting the underspeiation mehanisms introdued in
the previous setion. Then we show that the general relaxation oper-
ation underlyingthisapproah maygive riseto overgeneration, whih
introduesthe problemthat leadsthewayfortherest of thispaper.
3.1. Underspeified Reinterpretation
Reently,Egg(2000)hasproposedtodesribesentenesrequiringrein-
terpretation in an underspeied way, thereby avoiding onits. His
main idea is to derive suÆientlyrelaxed semanti representations in
whihgaps, modeledbydominaneedges,are left openat all possible
reinterpretationsites.Theatualreinterpretationstepsimplyisfurther
speialization, i.e instantiation of these relaxation gaps; the approah
assumesthatsuitablereinterpretationoperatorsan bedeterminedby
someindependentproess.Forillustration,Egg'ssemantionstrution
appliedto(1)derivesarelaxedsemantirepresentationthatessentially
looks as in Figure 5. The reinterpretation site, where relaxation took
plae, ishighlightedinthisgure bythedashed box.
By introduing a dominane edge at the reinterpretation site, the
semanti representationismadelessspei.Figure5an beseen asa
desriptionof(7)aswellas(6), sinethegap in(5)an be eliminated
byidentifyingthe twonodesat its ends.
One advantage of Egg's approah to reinterpretation is that it is
ompatiblewithanunderspeiedtreatmente.g.ofsopeinastraight-
forward way. Consider the following example whih ontains a sope
ambiguityinaddition to atype onit.
a maaboss lam
every
of var var
lam
&
driver var
outbak
X
beparked X l
var lam
Figure 6. RelaxedunderspeiedrepresentationofExample(8).
(8) Every driverof a maabossis parkedout bak.
The (relaxed) semanti representation of (8) is presented inFigure 6.
In reading the onstraint graph in Figure 6, it is rst of all helpful
to identifyits variousfragments, mostnotably theontributionsof \a
maaboss",\everydriver",\of",and\beparkedoutbak".Thesope
ambiguitybetween\amaaboss"and\everydriver"ismodeledbyre-
quiringthatbothfragmentsmustdominatethefragmentorresponding
to \of", but leaving theirrelative positionopen. Reinterpretation has
to oere \every driver of a maa boss" into their vehiles. We an
do this by lling the appropriate reinterpretation operator into the
relaxation gap,i.e.thegapprovidedbythedominaneedgeX
X l
in
the desriptionof theverb semantis. An appropriatereinterpretation
operatorlinkingthedriversto theirars is giveninFigure 7.
3.2. A Problem?
There are four dominaneedges in the desriptionin Figure 6. Three
of them, namely the ones onneting \a maa boss", \every driver",
\of", and \be parked out bak", were introdued in order to provide
for a orret modeling of sope ambiguities,i.e. they allow to arrange
thesemantimaterialthatwasintroduedbysemanti onstrutionin
dierent ways into trees. The fourth dominane edge, betweenX and
X l
,wasintroduedbyrelaxationtomakespaeforthereinterpretation
operator. It isimportantthat only reinterpretation operators that are
newly added to the desriptionshould appear inside the gap. In par-
tiular, we do not want solutions where material that was introdued
X
lam
lam
9
&
of var
var
^
ar var
var var
X l
Figure 7. ThereinterpretationoperatorforFigure6.
bysemanti onstrutionappears intherelaxation gap asiftherewas
a sope ambiguity.
ThisannothappeninFigure6{wesaythatthisdesriptionissafe
{,butisthisageneralfeatureofrelaxation?Unfortunatelynot,asthe
abstrat examplesin Figure 8 show. Here a relaxation of a onstraint
graph has a solution where material already present in the original
onstraint appears at the position of the gap. We all this kind of
solutionunintended.
Theproblemisthatdesriptionsmayontainmaterialthatisfreely
oatingaround(likee.g.thefragmentsontainingthenodelabeledwith
f in Figure 8). In a solution, this material an be mapped to almost
anypositionwherethespeiationoftheonstraintleavesagapopen.
So, it ould end up in a relaxation gap, whih it should not. This is
thepotentialovergenerationproblemofrelaxationwementionedinthe
introdution: relaxationsan have unintendedsolutions inaddition to
those solutionswereallywant;that is,notallrelaxations aresafe.
There areseveral oneivableways out of thisproblem.One would
beto expliitlyexlude unintendedsolutions byadding rst-orderfor-
mulasduringrelaxationwhihstate thatthevariablesoftheunrelaxed
onstraint must not be mapped into a region of a tree model that
llsarelaxation gap.This ideahasthedisadvantagethatemployinga
moreexpressivedesriptionlanguage(rst-orderinsteadofonjuntive
onstraints) would inrease the omplexity of omputation involving
underspeiedrepresentations.Note, however, that a similaridea has
already beenproposedin(Rambow etal.,1995).
Another idea might be that unintended solutions typially violate
type and sort restritions, and they ouldbe lteredout by exluding
ill-typed solutions. However, this will not always work; and more im-
f
g
h
a
relax
=) f
g
h
a
solve
=) g
f
h
a
f
g
a
relax
=)
f
g
a
solve
=) g
f
a
Figure 8. Unsafeonstraintsandtheirrelaxations.
portantly,asthepreviousdisussionhasshown,typeandsortonits
do ourinnatural languageand do notneessarily lead to exlusion,
butratherto a proess of reinterpretation repairingtheonit.
The solution proposed inthis artilerelies on the observation that
those CLLS onstraints that atuallyarise as underspeiedsemanti
representations have partiular properties whih ensure that their re-
laxationissafeanyway.Thatis,althoughingeneralrelaxationmaygive
riseto unintendedsolutions,itneverdoeswhenusedforunderspeied
reinterpretation.
4. Relaxation
Nowthatwehavea denitionof ourunderspeiationformalismand
an intuition of what \relaxation" is supposed to mean, we make this
notionformally preise. We will denethe oneptsof safe relaxation
and of open onstraints. These onepts are both onneted to the
notion of an \intended solution", whih we used informally in the
previous setion. We will prove in Setion 7 that the relaxations we
use inunderspeiedreinterpretationare allsafe and open.
4.1. Constraint Relaxation
Intuitively, relaxation of a onstraint means to split a single node of
the orresponding onstraint graph in two, onneting the two new
nodeswitha dominaneedge.In apturing thisidea formally,itturns
that works on every possible onstraint. Instead, we dene what it
means for a onstraint to be the relaxation of another onstraint. In
Setion 7, we then need to show that the \relaxed" onstraints that
the syntax/semantis interfae produes are really relaxations of the
\unrelaxed" versions.
Denition 4.1 (Relaxation). Let ';' 0
beonstraintsand X a vari-
ablein '.' 0
isalled a relaxationof 'atX withX l
i ' 0
j=X
X l
,
Var(' 0
)=Var(') ℄ fX l
g and
9X l
(' 0
^X l
=X)j=j':
Thisdenitionapturestheideaofarelaxationbeauseitsaysthat
there issomevariabilityforwhat isinbetweenX andX l
inasolution
of ' 0
;butassumingthat theyareequal, every solutionof' 0
mustalso
bea solutionof '.
However, relaxations inthis sensean misbehave in various patho-
logial ways. We have seen in the previous setion how an unsafe
relaxation an allow for material to disappearinto the newly opened
gap. Another problem is that by the above denition, ' is its own
relaxation { whih is ertainly ounterintuitive. Below, we dene two
properties of relaxations { safety and openness { whih exlude these
two typesof problems.
4.2. Safety and Openness
As we have seen in the previous setion, safety is theproperty whih
exludesmaterialspeiedbyaonstraintfrombeingmappedintothe
relaxation gap. We an denesafetyasfollows:
Denition 4.2 (Safety). Let ' 0
be a relaxation of ' at X with X l
.
' 0
is a safe relaxation of 'at X i
' 0
j=
^
Y2Var(') (Y
X_Y ?X_X l
Y):
Another interesting propertyof a onstraint is to be open between
two variables: This means that the tree struture between the deno-
tations of the two variables X and X l
is not onstrained. Thus, one
an freely ll in arbitrary material at the relaxations site while still
satisfying the onstraint. In order to formalize this idea, we need the
notionof aprojetion.
Denition 4.3 (Projetion). Let M=(V;E;L;) bea -struture
withnodesu
u l
inV.LetV u
u l
bethesetofnodesofMsituatedstritly
u l
. . u
projet
=)
at u;u
l
. u l
Figure 9. Projetionatu;u l
.
between u and u l
:
V u
u l
= fv 2V ju
v in Mand not u l
v in Mg
We dene the projetion p u
u l
(M) of Mat u;u l
illustrated in Figure 9
by applyingthe following onseutive steps toM:
1. Remove all edges whosesoure isin V u
u l
.
2. Remove all -binding pairs that involve a node in V u
u l
.
3. Replae the edge (r;u),if it exists, by (r;u l
).
4. Remove all nodesin V u
u l
.
Note that p u
u l
(M) isindeed a -struturesineu
u l
.Givena vari-
able assignment into M and variables X;X l
with (X)
(X l
)
we dene its projetion p X
X l
() to be the variable assignment into
p (X)
(X l
)
(M) whih maps, for all Y 2Vars ,
p X
X l
()(Y)= 8
>
<
>
:
undened if (Y)2V (X)
(X l
)
nf(X)g
(X l
) if (Y)=(X)
(Y) otherwise
Theprojetion p X
X l
(M;) of a pair (M;) satisfying(X)
(X l
) is
the pair (p (X)
(X l
) (M);p
X
X l
()).
Denition 4.4 (Openness). Aonstraint 'is open betweenX and
X l
i 'j=X
X l
and for eah pair (M;) where assigns into M:
ifp X
X l
(M;)j='then M;j=':
Clearly, it is important for a relaxation to be open, but not all
relaxations are. X: a, for instane, is its own relaxation at X with X l
and is not open between X and X l
. There is however a ertain lass
of safe relaxations, whih all are open, as expressed by the following
proposition.
Proposition 4.5 (Openness of Safety). Let ' be a safe relaxation
at X with X l
suh that no onstraints in 'mention X and X l
, exept
for X
X l
and X l
: f(:::). Then ' isopen between X and X l
.
Proof. Let(M;)beatupleonsisting ofa-strutureandavariable
assignment, and let p X
X l
(M;) be a solution of '. We have to show
that (M;) also is a solution of '. Beause ' is safe, we know that
and p X
X l
() oinide on Var('). By the denition of projetion all
atomi onstraints of ' exept for X
X l
are satised by M in the
same way they are satisedby p X
X l
(M). AndX
X l
is of oursealso
satisedby(M;).
4.3. Intended solutions
An equivalentharaterization ofsafetyandopennessanbe obtained
by speifying intended solutions of relaxed onstraints. The fat that
suh an equivalent denitionexists reinfores our ondene into the
notionof open and saferelaxations, even thoughit isnot essentialfor
theremainder ofthisartile.
Denition 4.6 (Intended Solutions). Let ' 0
be a relaxation of '
at X with X l
. A pair (M;) is alled an intended solution of ' 0
i
(X)
(X l
) and the projetion p X
X l
(M;) is a solution of '.
Lemma 4.7. Let' 0
be asafe relaxation of 'at X withX l
.If (M;)
is a solution of ' 0
then its projetion p X
X l
(M;) solves ' 0
aswell.
Proof. LetM=(V;E;L;)andM;j=' 0
.SineVar(' 0
)=Var(')℄
fX l
g, safety yieldsthat p X
l
X
() is dened forall variables inVars(' 0
).
Now we an verify thatthe projetion satiseseah atomi onstraint
in' 0
.Forinstane,let Y
Z be in' 0
then(Z) an bereahedform
(Y)viaedgesinM.Thus,p X
l
X
()(Z)anbereahedfromp X
l
X
()(Y)
inp (X
l
)
(X)
(M),i.e.p X
X l
(M;)j=Y
Z.Theargumentsforlabelingand
-bindingonstraints aresimilar.
Proposition 4.8. All solutions of a safe relaxation areintended.
a
b b
X
1 Y
1 F1
b b
X
2
? Y
2 F2
b b
X
3 Y
3 F3
a
Z1
G
1
Z2
G
2
a
Figure 10. Ahainoffragments.
Proof. Let ' 0
be a safe relaxation of ' at X with X l
and (M;) a
solution of ' 0
. By Lemma 4.7, the projetion p X
X l
(M;) solves ' 0
as
well. Trivially, it also solves X = X
l
and thus ' by the equivalene
9X l
(' 0
^X l
=X)j=j',i.e (M;) isan intended solution.
Proposition 4.9. Every intended solution of an open relaxation is
indeed a solution.
Proof. Let ' 0
be an open relaxation of ' at X with X l
. If (M;) is
an intended solutionof ' 0
thenits projetionp X
X
l
(M;) solves', and
thus9X l
(X=X l
^' 0
).Sinep X
X
l
()mapsX andX
l
tothesamevalue,
theprojetionp X
X
l
(M;)also solves' 0
.Theopennessof' 0
yieldsthat
(M;) solves' 0
aswell.
5. Chains of Fragments
Inthissetion,weintroduehains ofrigidfragments. Chainsarepro-
totypial substrutures of underspeied semanti representations in
CLLS.Itwillturnoutthattheserepresentationsmaybemoreomplex
thana single hain,butthat they an be overed byseveral hains.
A hain is intuitively a onstrution as in Figure 10, where rigid
fragments(drawnastriangles)areonnetedviadominaneonstraints
between unlabeled leaves of the upperfragments and therootsof the
lowerfragments.Bywayofexample,reonsiderFigure6whihontains
a hainof lengthtwo,whose upperfragmentsare those orresponding
toamaabossandeverydriver andwhoselowerfragmentorresponds
to thepreposition.
Now we make theinformal notions of fragments and hains, whih
we have used in explanations of onstraint graphs all along, preise.
Fragments in an overlap-free onstraint are sets of variables that are
onneted bylabelingonstraints.
Denition 5.1 (Conneted Variables). Connetednessin 'is the
smallest binary equivalene relation over Var(') whih ontains all
pairs (X;Y) suh that X: f(:::Y :::) in '.
Denition 5.2 (Fragments). Let ' bean overlap-free onstraint. A
fragment of ' is a subset F Var(') of variables that are pairwise
onneted in'.A variableX 2F isalled a labeledleafof a fragment
F if ' ontains X: a for some onstant a2 andan unlabeledleaf if
no labeling onstraintX: f(:::) oursin 'atall. A leafof F is either
a labeled or an unlabeled leaf of F.
We areusuallyinterestedinmaximalfragments,butwillallownon-
maximalonesaswell.Intheonstraintgraph,fragmentslooklikeparts
oftrees.Sointuitively,theyshouldhaveauniqueroot,andtheirleaves
shouldbepairwisedisjoint.Thisis indeedthease:
Lemma 5.3 (Treeness of Fragments). Let ' be an overlap-free
onstraint, and let F be a fragment of '. Then F has the following
properties:
1. Thereisauniquevariable Y 2F (whih weallits root)suh that
'j=Y
X for all X 2F.
2. For any two dierent leaves X;Y of F it holds 'j=X?Y atF:
We omit theproof,whih isstraightforward.
Denition 5.4 (Chains). Let'beanoverlap-freeonstraint,andlet
F = (F
1
;:::;F
n
) and G = (G
1
;:::;G
n 1
) be sequenes of fragments
in'.Weassumethatno variableappearsintwo dierentfragments;so
these fragmentsannot overlapproperly.Foralli,letX
i
;Y
i
bedierent
unlabeled leaves of F
i
, and let Z
i
bethe root of G
i
. Then the pair C =
(F;G) is alled a hain in ' of length n and with end points X
1 and
Y
n
i for all 1in 1:
'j=Y
i
Z
i
^X
i+1
Z
i
If a onstraint ontains a hain then it entails some very useful
relationshipsbetweenits variables.
Proposition 5.5 (Relations in Chains). If C is a hain in a on-
straint ' withend point X then all other variables Y of C satisfy:
'j=X?Y _Y
X:
Proof. The proof is by indution on the length n of the hain C. Let
fragmentsandvariablesofCbenamedasinDenition5.4.Theasen=
1 follows from the treeness of upperfragments (Lemma 5.3). Assume
n2 and let X=X
1
w.l.o.g. Sine Y
1
Z
1
^X
2
Z
1
2'it follows
that ' j= Y
1
X
2 _X
2
Y
1
. Let U
1 2 F
1
and U
2 2 F
2
be the roots
of theirfragments. As fragments of hainsannot overlap properly,it
follows that ' j= Y
1
U
2 _X
2
U
1
. For the rst ase, we onsider
' ^ Y
1
U
2
: If Y 2 F
2 [G
1
then 'j= Y
1
Y ^ X
1
?Y
2
and we are
done. Otherwise, we anel out G
1
and F
2
and apply the indution
hypothesis. Fortheseond ase, we onsider'j=X
2
U
1
.Again, the
ase Y 2 F
2 [G
1
are obvious. Otherwise, we anel out F
1
and G
1 .
The indution hypothesis yields ' j= X
2
?Y _Y
X
2
. Thus, ' j=
X?Y _Y
X.
Proposition 5.6 (Disjointness of End Points). If X;Y are the
end pointsof a hain in ', then 'j=X?Y.
Proof. Proposition5.5appliedtwie (one foreahendpoint)proves:
'j=(X?Y _Y
X)^(Y?X_X
Y)
The disjointness of end points holds obviously in all ase exept for
'^Y
X^X
Y whih isimpossibleas'j=X6=Y.
6. Semanti Constrution
Now, we dene a toy grammar and a syntax/semantis interfaethat
produesunderspeiedsemantirepresentationsinCLLS. Theover-
age ofthegrammarisobviouslylimited.However,thestrutureof the
semanti representations derived byit areprototypial for thekindof
struturesonewillseeinrepresentationsofnaturallanguagesemantis.
As a matter of fat, we have implemented an HPSG grammar with a
muhwideroveragewhihproduesthesametypeofonstraints.The
syntax/semantis interfae produes onstraints that may be relaxed
at every variable whose label arries the semantis of a verb. We will
prove in the next setion that these relaxations are always open and
safe.
We leave itasa (nontrivial) open problemto generalize ourresults
further, to lasses of grammars and syntax/semantis interfaes. The
mainproblemindoingsoistondtherightabstrat propertiesof the
syntax/semantis interfaewhih ensuresafety ofrelaxations.
(a1) S! NP VP
(a2) NP ! DetN
(a3) N! N
(a4) N! NPP
(a5) PP!PNP
(a6) VP! VP Adv
(a7) VP! IV
(a8) VP! TV NP
(a9) VP! CV to VP
(a10) ! W if(W;) 2Lex
Figure 11. Thegrammar
6.1. The Grammar
The grammar fragment we onsider is displayed in Figure 11 (where
IV = intransitive verb; TV = transitive verb; CV = (subjet) on-
trol verb). Lex is a relation between words W and lexial ategories
2fDet;N;IV;TV;CV;Advgwhihrepresentsthelexion.Theoverage
of thisgrammar islimited,butitshouldbe asimplematter to extend
our resultsto alarger grammar that overs onstrutionslike relative
lauses,sententialomplementverbs,andditransitiveverbs.Ofourse,
anyserious NLPsystem would employsome uniationgrammarfor-
malism, whih would then also allow to take are of aspets suh as
agreement whih we haveignored ompletely.
6.2. The Syntax/SemantisInterfae
The syntax/semantis interfae of the grammar assoiates with eah
nonterminalnodeoftheparsetree avariableX
andasubonstraint
that talks about this variable. The ontributions of all these nodes
arethenonjoined, anda suÆientnumberof inequalityonstraintsis
added to make theresult overlap-free, i.e. to prevent identiation of
labelednodesinthesolution.
Therules bywhihthe subonstraintsareintroduedarepresented
inFigure12.Wetake[
:
1
2
℄to meanthatthenode inthesyntax
treeislabeledwithategory,anditstwodaughternodes1and2are
labeledwith
1 and
2
,respetively.Theonstraintintroduedbysuh
a rule then imposes a CLLS onstraint on the variables X
;X
1
;X
2
whiharedistinguishedvariablesinthesubonstraintsassoiatedwith
;1,and 2 respetively.
Some other variables inthe onstraints arementionedexpliitly as
well, e.g. X sope
in rule (b1). This is to make their spei funtions
[
:S NPVP℄
(b1)
)
X
X
1
lam X sope
1
X2
[:NPDetN℄
(b2)
)
X
X
1 X
2
;X restr
[
:N N℄
(b3)
) X
;X
1
[
:N NPP℄
(b4)
)
lam X
&
X
2
X
1
var
[
:PP PNP℄
(b5)
)
X
2
lam
X
X
1
var X arg2
1 var X
arg1
1
[:VPVPAdv℄
(b6)
)
X
X2 X1
[
:VP IV℄
(b7)
)
X
X u
1
var X arg1
1
[:VPTVNP℄
(b8)
)
X
2
lam X sope
2
X
X
1
varX arg2
1 varX
arg1
1
[:VPCVtoVP℄
(b9)
)
X
X
1
X
2 varX
arg1
1
[
: W℄
(b10)
) Æ(W) X
where2fDet;N;Adv;IV;TV;CVg,(W;)2Lex,
andÆ(W)isthesemantiontentofW
[:W℄
(b10 0
)
)
X
Æ(W) X l
where2fIV;TV;CVg,(W;)2Lex,andÆ(W)
isthesemantiontentofW
Figure 12. Thesyntax/semantisinterfae
expliit and failitate later referene. The variable X sope
introdued
byrule(b1)isintuitivelythesopeofthequantierrepresentedbythe
NP.
Weexploitthistoaddappropriate-bindingonstraintsthatannot
be speied loally in Figure 12 without luttering up the notation.
Instead, we assume information about subjets and objets of verbs,
whih will be available in any more serious grammar formalism. For
the objetvariablesintroduedintherules (b7) to (b9), supposethat
isaVP node intheparse tree and 0
istheNPnode thatrepresents
thesubjet; thenweadd thefollowing-bindingonstraint:
(X arg1
)=X sope
0
Similarly for rule(b5), if 0
is the NP node modied by the PP at
then we addthe following-bindingonstraint:
(X arg1
)=X restr
0
S
NP
Det
every N
N
driver PP
P
of NP
Det
a N
N
maaboss VP
isparkedoutbak
a maaboss I lam
II
every
III
of var var
lam
&
driver var
IV
outbak
X
beparkedX l
V var lam
(b1)
(b2)
(b10)
(b4)
Figure 13. Everydriverofamaabossisparkedoutbak.
Finally, notehow relaxation is ompiled into thesyntax/semantis
interfae. Rules (b10) and (b10') are the semanti onstrution rules
forthesyntatirulessubsumedunder(a10). Forategoriesotherthan
verbs, we an onlyapplytherule(b10), whih justontributesa vari-
able with the appropriate label. For verbs, however, we have a hoie
betweenappliationof(b10)and(b10');(b10')introduesadominane
onstraint forpotentialreinterpretation.Itistheserelaxationsthatwe
must prove open and safe. This hoie an be made nondeterministi-
ally.In\real"semantionstrution,wewillalwaysapply(b10'),that
is, we produe maximally relaxed onstraints; but we still need (b10)
sowean formulate theresultsinSetion 7.
Beforewe ometo theproofsthatthisspei typeof reinterpreta-
tion hasall thepleasant propertiesdisussedabove,wego throughan
exampleto illustratehow semanti onstrutionoperates.
To thisend, we briey return to Example (8), whose (relaxed) un-
derspeiedsemanti representationwe showed inFigure 6.Figure 13
relates the semanti representation to the parse tree our grammar
assigns to this sentene. The dotted arrows go from node in the
parsetreetotheorrespondingnodeX
inthesemantirepresentation,
indiating whih partof theonstraint is thesemanti ontributionof
node.Thedottedarrowsarefurthermorelabeledwiththerulesofthe
syntax/semantis interfae that were used. For the top most NP node
e.g., rule (b2) introdues onlya labeling onstraint. Its rst daughter
labeledwithDetonlyaddsthenodelabelevery(rule(b10)).Itsseond
daughter, labeled with N, then introdues a bigger fragment, the root
of whih islabeledbylam(rule(b4)).
7. Corretness of Underspeied Reinterpretation
In this setion, we put all the piees we have assembled throughout
the paper together. We identify a lass of onstraints with partiular
struturalpropertiesthateÆientlysupportsprovingsafetyofaertain
relaxation. This struture depends heavily on the notion of hains,
whihwe presentedin Setion5.
Toillustratetheuseofoverings,we applythem rstto twoparti-
ular relaxed semanti representations, showing that these relaxations
are safe. Then we generalize these results to any output of the syn-
tax/semantisinterfaeand demonstrate astrategy forprovingthata
syntax/semantis interfaenever produesunsafe relaxations.
7.1. Covering Constraints
Westart bydeningwhat itmeansfora onstraint to be overed.
Denition 7.1 (Covering). Let ' be an overlap-free onstraint, let
O be a set of fragments in '. We all O a overing of ' for X;Y
i Var(') = S
fFjF 2 Og and for eah fragment F 2 O one of the
following holds:
1. eitherY dominates the root U of F,i.e. 'j=Y
U,
2. orthereisa hain C withfragments fromO inludingF suh that
someend point Z of C dominates X, i.e. 'j=Z
X,
3. orthereisa hain C in 'with fragmentsof O then oneof the end
points dominates X and the other one the root of F.
Now, we prove a property of overings, whih make them a very
interestingand onvenient strutureforourpurposes.
Proposition 7.2 (Safety of Coverings). Let ' be an overlap-free
onstraint withovering O for X;Y. Then, for all Z 2Var('):
'j=Z
X_Z?X_Y
Z :
Proof. Every variable Z 2Var(') belongsto a fragment F 2O,suh
that one of the onditions of Denition 7.1 is satised. Let U be the
root of F.
1. 'j=Y
U and therefore holdsY
Z aswell.
2. Z belongsto a hain C and an endpointof C dominatesX.So we
knowfrom Proposition5.5, that 'j=X?Z_Z
X:
I
a town lam
II
every
IV
of var var
lam
&
VIP var
V
Y
expet Y l
VI
var
VII
X
on X l
VIII var
var lam
III
every guestlist lam
Figure 14. EveryVIPof atownexpetstobe oneveryguestlist.
3. Z is dominated by one end point of a hain and X by the other.
From Proposition 5.6 we know that these two end point must be
disjoint and therefore X?Z also holds.
We now illustratebymeansof twoexamples, how overingsan be
usedtoshowthesafetyofarelaxation.Westartwiththeexampleofthe
previous setion (Figure 13). To show that this onstraint is atually
safe, wehaveto provethateveryvariableintheonstraintmusteither
dominate X,bedisjointfrom X,orbedominatedbyX l
.Tothisend,
we have to nda overing of the onstraint for X, X l
, as dened in
Denition7.1. This willthengive usthedesiredresult byProposition
7.2. Fragment V,ontaining onlyX l
iseasy{ itisifoursedominated
byitself. Fragments I,II, and III form a hain of lengthtwo and an
end point of this hain dominates X. This leaves only fragment IV
whih an be seen as a hains of length one, and sine X is a leaf of
thisfragment, X isof oursedominatedbyan endpoint.
Admittedly,thestrutureofthisexampleisverysimple:itismoreor
less one hain with an extra fragment dangling from one of the outer
onnetion points. As an example for a more omplex struture the
grammar fragment we presentedan handleontrol verbs (rules (a9),
(b9)). Considerthefollowingexample:
(9) Every VIP ofa townexpets to be on every guest list.
I II III
IV V
VI VII
VIII relaxation site)
relaxation site)
Figure 15. ShemativiewoftheExample.
Wewanttobeableto reinterpretthissentenebeauseonlythenames
of the VIPs will be on the guest lists. The (maximally relaxed) un-
derspeied semanti representation derived by the syntax/semantis
interfaeisshown inFigure 14.To portraythestruturemore learly,
wehavegivenashemati representationinFigure15.Notethatthere
are two relaxation gaps in this example. To show that the onstraint
is safe we have to prove that every variable in the onstraint must
either dominate X, be disjoint from X, or be dominated by X l
and
that the same holds with respet to Y and Y l
. Again we will use
appropriateoverings from whihwean drawthe desiredonlusions
byProposition7.2.
FortherelaxationgapbetweennodesX andX l
,wewillemployone
hainoflengthtwo,withupperfragmentsI;II andlowerfragmentIV.
Furthermore we needthreehainsof lengthone onsistingof fragment
III,V,andVII respetively.Notethattheone onsistingoffragment
V also does the overing for fragment VI. X l
again dominates itself,
sothat we do nothaveto worryaboutfragment VIII.
7.2. Strutureof the Constraints
The onstraints that the syntax/semantis interfae an generate are
of a speiform, speiedbythe followingtheorem.
Theorem 7.3 (Struture of the Constraints). Let ' 0
be a on-
straint that we have obtained from the syntax/semantis interfae by
applyingtheonstrutionrule(b10') foraertain verbnode,andthat
' is the onstraint that we obtain from the same sentene by applying
a
NP NP
PP :::
PP
| {z }
subjet
NP NP
PP :::
PP
| {z }
objet
reinterpretationsites
verb
z }| {
b
b CV
.
.
.
b
b CV
b
b TV
a
Figure 16. Outputofthesyntax/semantisinterfae:shematiview
the same onstrution rules everywhere, exept for the node , where
we apply (b10). Then, there isa overing O of ' 0
for X
, X
l
.
For the most part, the proof is only a preise formulation of the
fat that the struture of a generated onstraint generally looks as in
Figure 16.Wewillnotgo into mostpartsof thisproof,butone of the
mostinterestingpartsis theproof thatthesemanti ontributionof a
nounphrase isa hain whoselength is thenumberof NP nodesin the
parse tree ofthat nounphrase.
Proposition 7.4. Let T be the parse tree from whih we onstruted
' 0
.Furthermore,lettbea subtreeof T whoserootislabeled withNP,let
n bethe number of NP nodesin t, and let '
t
bethe onjuntion of the
onstraints orresponding to the nodes of t. Then there is a singleton
overing fCg of '
t
, where C is a hain of length n.
Proof. By indutionovern.
n=1:Inthisase,thastheformNP(Det(W
1
)N(N(W
2
))),whereW
1
and W
2
are words.As we an easily seefrom the rules (b2) and (b3),
the resulting onstraint '
t
is a single fragment and the laim is true
with Cbeing ahainof length1.
n 1!n: Let t 0
be the largest proper subtree of t whose root is
labeled with NP and let '
t 0
be the ontribution of t 0
. Note that t =
NP(Det(W
0
)N(N(W
1
)PP(P(W
2 )t
0
))),whereW
1
andW
2
areagainwords,
and t 0
ontainsn 1NP nodes.
BytheindutionhypothesisthereisasingletonoveringfC 0
gof'
t 0,
suhthat C 0
=((F 0
1
;:::;F 0
n 1 );(G
0
1
;::: ;G 0
n 2 )).
Aording to thesyntax/semantisinterfae,weobtain'
t
from'
t 0
byappliationsof therules(b2), (b4), and(b5). Theserulesintrodue
two new fragments; let F
0
be thefragment onsisting of theontribu-
tionsof rules(b2) and(b4), and letG
0
bethe fragmentintroduedby
rule (b5). Furthermore, rule (b5) extends fragment F 0
1
to a fragment
F
1
whih ontains a new leaf, namely the sope. Finally, there are
dominane onstraints, demanding that the root of G
0
be dominated
bythenew leafof fragment F
1
aswellasa leaf offragment F
0 .
Thismeansthat(F
0
;F
1
;F 0
2
;:::;F 0
n 1
)and(G
0
;G 0
1
;:::;G 0
n 2 )form
ahainoflengthnin'ontainingallvariablesof'
t
.Hene,thishain
formsa singletonoveringof '
t .
7.3. Safe, Open Relaxation
Now, the proof that all relaxed onstraints derived by our syn-
tax/semantisinterfaearesafeandopenrelaxationsoftheirunrelaxed
ounterparts, beomesquitesimple.
Assumethat'and' 0
areasdenedinTheorem7.3.Firstofall,' 0
isreallyarelaxationof'atX
withX l
.Thetwoonstraintsareequal,
exept for theonstraintsfor the variables X
and X l
. By adding an
equality onstraint for these two variables to ' 0
, we learly obtain a
onstraint whihisequivalentto '.
Safetyoftherelaxation followsbyProposition7.2diretlyfrom the
fatthat' 0
hasa overingforX
;X l
.Opennessof' 0
betweenX
;X l
follows fromthefatthat ' 0
issafeat X
withX l
byProposition4.5,
sineweknowthattheonlyonstraintsonerningX
andX l
thatthe
syntax/semantisinterfaeaddsarethedominaneonstraint between
X
and X l
anda labelingonstraint forX l
.
8. Conlusion
Typeandsortonitsinsemantishavetoberesolvedbyreinterpreta-
tion inthepresene of underspeiation.This artileinvestigates the
formalfoundationsofunderspeiedreinterpretationintheframework
CLLS. Itsbasioperationistherelaxation operation.It makesunder-
speiedsemantirepresentationsevenlessspei,therebymakingthe
introdutionofadditionalmaterial,suh asreinterpretation operators,
possible.
We identieda strutural danger of overgeneration inherent to the
twowellnesspropertiesofrelaxation,whihensurethatstruturalover-
generationannotarise:openness (arbitraryreinterpretationoperators
an be lledinto therelaxation site) and safety (only reinterpretation
operators an slipinto relaxationgaps).Wethen introduedhains of
fragments in tree desriptions and applied these strutures to prove
safetyandopennessforall CLLSrelaxeddesriptionsproduedbythe
syntax/semantis interfaeof a prototypial toy grammar. Thisresult
manifestsourbelievethat whilerelaxation ingeneralmayauseprob-
lems,noneofthese problemsourinits appliationto underspeied
reinterpretation.
9. Aknowledgements
We thank our reviewers for their detailed and helpful omments. We
furtherthank allmembersoftheCHORUS 1
projetoftheSFB378at
theUniversitatdesSaarlandesforthestimulatingresearhenvironment
they managedto reate. This work was supported bythe SFB 378 of
theDeutsheForshungsgemeinshaftandtheDAADProopeprojet.
Notes
1
PapersoftheCHORUSprojetareavailableatthewebpagesoftheSFB378:
http://www.oli.uni-sb.de/sfb3 78
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