The paper introduces a generalisation of the grammar formalism of Topological Dependency Grammar (TDG) (Duchier and Debusmann 2001).  The result, eXtensible Dependency Grammar (XDG), is a description language for sets of labeled directed graphs. Lexicalisation turns XDG into a powerful dependency-based meta grammar formalism.  As an instance of XTDG, we present Semantic Topological Dependency Grammar (STDG), a new dependency-based grammar formalism with a syntax-semantics interface to underspecified semantics

Debusmann, Ralph

Universaar

Dependency grammar as graph description

Acronym

Dependency Grammar as Graph DescriptionRalph DebusmannProgramming Systems LabUniversita¨t des Saarlandes, Geb. 45Postfach 15 11 5066041 Saarbru¨cken, Germanyrade@ps.uni-sb.deAbstractThe paper introduces a generalisa-tion of Topological Dependency Gram-mar (TDG) (Duchier and Debusmann,2001). The result, Extensible Depen-dency Grammar (XDG), is a descrip-tion language for sets of labeled directedgraphs. Lexicalisation turns XDG intoa powerful meta grammar formalism.XDG can be instantiated to yield spe-cific grammar formalisms based on de-pendency grammar. We present oneof these instances, Semantic Topolog-ical Dependency Grammar (STDG), anew grammar formalism with a syntax-semantics interface to underspecified se-mantics.1 Introduction(Duchier and Debusmann, 2001) introduced thegrammar formalism of Topological DependencyGrammar (TDG). The most distinguishing featureof TDG is its ability to specify non-trivial restric-tions on word order. These restrictions emergefrom the interaction of simple declarative princi-ples, and lead to an elegant account of many noto-rious word order phenomena in German.TDG concentrates only on syntax and does notoffer a syntax-semantics interface. In this paper,we introduce Semantic Topological DependencyGrammar (STDG), which extends TDG with aninterface to underspecified semantics. As the tar-get semantics formalism, we employ the Con-straint Language for Lambda Structures (CLLS)(Egg et al., 1998).Before we extend TDG, we take a step back-wards: we generalise TDG to Extensible De-pendency Grammar (XDG), a graph descriptionlanguage over sets of labeled directed graphs.Only by lexicalisation, i.e. the addition of lexi-calised feature structures, does XDG become ameta grammar formalism for dependency gram-mar. TDG is one of the possible instances of XDG.XDG can be likened to Head-driven PhraseStructure Grammar (HPSG) (Pollard and Sag,1994): both XDG and HPSG are meta grammarformalisms which need to be instantiated with par-ticular principles and parameters to obtain specificgrammar formalisms. The fundamental differenceis that whereas HPSG is based on typed featurestructures, XDG is based on graph descriptions.The purpose of XDG is twofold. On the onehand, XDG can be utilised for research on existingdependency-based grammar formalisms. On theother, XDG serves as a launchpad for the develop-ment of new grammar formalisms like STDG.The structure of this paper is as follows. Weintroduce XDG in section 2. In section 3, we de-velop the new STDG as an instance of XDG. Webriefly touch the issue of complexity in section 4,before section 5 concludes.2 Extensible Dependency Grammar(XDG)XDG describes a set of graph dimensionsd1, . . . , dm, where each graph dimension d corre-sponds to a graph Gd(V,Ed). All graphs share thesame set V = {v1, . . . , vn} of nodes, but have dif-ferent edges Ed ⊆ V ×Ld×V , and different edgelabels Ld. Feature structures can be attached toeach node, where features are functions V → Rto an arbitrary codomain R.The well-formedness conditions of an XDGanalysis are stipulated by parametrised principles.Each dimension d uses a set of principles Pd, stip-ulating restrictions on the licensed graphs on d.Principles can also pose restrictions on any num-ber of dimensions simultaneously. An XDG anal-ysis is well-formed only if all used principles aresatisfied on all dimensions.2.1 PrinciplesPrinciples are taken from a shared principle li-brary. In the following, we present some of theprinciples from the current XDG principle library.We describe each principle using three boxes: thetop one shows how to invoke the principle, themiddle one explains the formal parameters, andthe bottom one explains what the principle means.Note that we omit a number of principles (pro-jectivity, climbing, and barriers, linking) for lackof space. Most of these are already explained in(Duchier and Debusmann, 2001).2.1.1 Treedag(G)G: a graph dimensionG is a tree.2.1.2 Directed acyclic graphdag(G)G: a graph dimensionG is a directed acyclic graph.2.1.3 Outout(Gd, f)Gd: a graph dimension, f : V → (Ld → 2 ):an out specificationThe outgoing edges of each node in Gd mustsatisfy in label and number the stipulation oftheir out specification.The out principle restricts the number of outgo-ing edges with a certain label. In (1) below, theout specification of v1 requires that the node has1) no outgoing edges labeled l1, 2) one outgoingedge labeled l2, and 3) zero, one, two or three out-going edges labeled l3. Thus, the configuration in(1) satisfies the out principle: there is no outgoingedge labeled l1, one labeled l2, and three labeledl3.(1)v1l2l3l3l3l3: {0,1,2,3}l2: {1}l1: {0}f :2.1.4 Inin(Gd, f)Gd: a graph dimension, f : V → (Ld → 2 ):an in specificationThe incoming edges of each node in Gd mustsatisfy in label and number the stipulation oftheir in specification.The in principle is the same as the out principle,but for incoming instead of outgoing edges.2.1.5 Covariancecovariance(Gd1, Gd2 , f)f : V → 2Ld1 : covariant labels specificationEach edge (v1, l, v2) in Gd1 , where l iscovariant on v1, is only licensed if v1 isabove v2 in Gd2 .In (2) below, the edge (v1, l, v2) in Gd1 is li-censed because l is a covariant label on v1, andv1 is above v2 in Gd2 (as indicated by the dashededge).(2)v12vGd1v12vGd2lf : { l }2.1.6 Contravariancecontravariance(Gd1 , Gd2 , f)f : V → 2Ld1 : contravariant labels specifica-tionEach edge (v1, l, v2) in Gd1 , where l iscontravariant on v1, is only licensed if v2 isabove v1 in Gd2 .The contravariance principle is the converse ofthe covariance principle.2.1.7 Linkinglinking(Gd1 , Gd2 , f1, f2)Gd1 , Gd2 : graph dimensions, f1 : V →(Ld1 → 2Ld2 ): realisation specification, f2 :V → (Ld2 → 2Ld2 ): substitution specifica-tionAn edge (v1, l, v2) in Gd1 is only licensed ifeither:1. there is a corresponding edge (v1, l′, v2)in Gd2 and v1 realises l by l′, or2. there is an edge (v3, l′′, v2) in Gd2 , v3substitutes l′ by l′′ and v1 realises l by l′.Below, (3) shows a configuration licensed bythe first clause of the linking principle: edge(v1, l, v2) in Gd1 is licensed because there is a cor-responding edge (v1, l′, v2) in Gd2 and the realisa-tion specification of v1 specifies that the node canrealise l as l′, i.e. l′ ∈ f1(v1)(l).(4) displays a configuration licensed by the sec-ond clause: edge (v1, l, v2) in Gd1 is licensed be-cause there is an edge (v3, l′′, v2) in Gd2 , v3 sub-stitutes l′ by l′′ (i.e. l′′ ∈ f2(v3)(l′)) and v1 realisesl by l′.(3)v12vl’v12v1f : l : { l’ }Gd2Gd1l(4)v12vf1: l : { l’ }Gd12vv3f2: l’ : { l’’ }Gd2l’’l2.2 LexicalisationBy lexicalisation, we can turn the graph descrip-tion language described above into a powerfuldependency-based meta grammar formalism. Tothis end, we assume a 1:1-correspondence be-tween nodes and words, and assign to each nodeone of the possible lexical entries for the corre-sponding word.The result of lexicalisation is that we can usethe lexicon to specify all the features referred toin the principles. For instance lexicalising the outspecification of the out principle gives us valency.Lexicalising the realisation and substitution speci-fications of the linking principle allow us to modellexicalised mappings from thematic roles to gram-matical functions which realise them.3 Syntax-semantics interfaceIn this section, we introduce the new grammarformalism of Semantic Topological DependencyGrammar (STDG). STDG has a syntax-semanticsinterface to underspecified semantics in CLLS.Note that CLLS is just one of many possiblechoices for the target semantics formalism.3.1 CLLSCLLS is based on dominance constraints (Marcuset al., 1983). CLLS structures describe λ-terms. In(6) below, we show a CLLS structure describingthe strong reading of sentence (5):(5) A woman, every man seems to love.(6)@@ lam@@ lam@seem @@ varvarloveevery manwomanaHere, the @ symbol stands for functional appli-cation, lam for a lamda binder, var for a variable,and the dashed arrows represent lambda bindings.3.2 STDGSTDG employs four graph dimensions: GID, GLP,GTH, GDE. As in TDG, the ID (Immediate Domi-nance) dimension models syntactic dependencies,and its edge labels are grammatical functions likesubj (subject), obj (object) and vinf (verb infini-tive). The LP (Linear Precedence) dimension mod-els word order, and its edge labels are topologi-cal fields (linear positions) like topf (topicalisationfield), subjf (subject field) and vcf (verbal com-plement field). We use the same principles as in(Duchier and Debusmann, 2001).We introduce the TH (THematic) dimension tomodel semantic argument structure. Its edge la-bels are thematic roles like act (actor), pat (pa-tient) and prop (proposition). We use the DE(DErivation) dimension to model the derivation ofa CLLS structure, with edge labels like s (scope),and r (restriction).On the TH dimension, we invoke thefollowing principles and parameters:dag(GTH), in(GTH, inTH), out(GTH, outTH),and linking(GTH, GID, real, subs). The dagprinciple ensures that every analysis on the THdimension is a directed acyclic graph. It cannotbe a tree since we also want to properly handlecontrol constructions like Mary persuades Peterto sleep., where the controlled noun Peter hasmore than one mother in the argument structure(here, it is the patient of persuades and the actorof sleep). Moreover, the in principle assignspossible thematic roles to words, the out principlemodels valency, and the linking principle ensuresthat thematic roles are realised by the appropriategrammatical functions on the ID dimension (e.g.actors are realised by subjects).On the DE dimension, we usethe following principles and pa-rameters: tree(GDE), in(GDE, inDE),out(GDE, outDE), covariance(GDE, GID, co),and contravariance(GDE, GID, contra).Below, we give an example STDG analysis ofthe strong reading of sentence (5).1 (7) displaysthe ID and LP dimensions, and (8) the TH and DEdimensions.(7)A woman, every man seems to love.objsubj vinfIDA woman, every man seems to love.topfsubjf vcfLP(8)A woman, every man seems to love.propactpatTHevery man seems to love.sssA woman,DEThe TH analysis states that to love is a proposi-tion of seems, and that every man is the actor and1Note that for simplicity, we treat a woman, every manand to love each as a word.a woman the patient of to love. The linking princi-ple establishes the correct mappings of of actor tosubject, and patient to object.The DE analysis states that every man is inthe scope of a woman, seems is in the scope ofevery man, and to love in the scope of seems.The covariance principle establishes that the edge(seems, s, love) is covariant: we state in the lex-icon that seems is covariant on s (scope) la-bels, and hence the edge corresponds to the edge(seems, vinf, love) on the ID dimension whichhas the same direction. Conversely, we use thecontravariance principle to ensure that the edge(every man, s, seems) is contravariant: in the lexi-con, we state that every man is contravariant on s,and hence we license the analysis where the edgecorresponds to the edge (seems, subj, every man)on the ID dimension which has the opposite direc-tion.3.3 From STDG to CLLSTo build the CLLS representation of the STDGanalysis, we assume a lexicalised mapping fromwords in the sentence to CLLS fragments, as illus-trated in (9) below:2(9)a woman@@lamevery man@@lamseem@seems@@ varlove varto love.A woman, every manWe use 1) the DE analysis to plug these frag-ments together, and 2) the TH analysis to recoverthe lambda bindings. We show an example of this2Note that we also have to keep track of positions withinthe CLLS fragments: the position of restriction and scope,the position of lambda binders and of variables. We omit thisfor lack of space.in (10) and (11):(10)to love.propseemsevery manA woman,patactTHto love.sssseemsevery manA woman,DE(11)a woman@@lamevery man@@lamseem@@@ varlove var(10) repeats the TH and DE analyses of sentence(5). The DE analysis determines how to plug thefour fragments together: every man is in the scopeof a woman, seems is in the scope of every man,and to love in the scope of seems. The TH analy-sis determines the lambda bindings: every man isthe actor and a woman the patient of to love. Theedge (seems, prop, to love) does not correspond toa lambda binding. We show the resulting CLLSanalysis in (11).Note that in the example shown, we derive afully specified CLLS representation of the seman-tics from a full STDG parse. We can howeverobtain underspecified CLLS representations frompartial STDG parses. E.g. we can decide not toenumerate the solutions on the DE dimension toget a CLLS analysis underspecified with respectto scope.4 ComplexityDue to its generality, the complexity of XDG it-self cannot be determined. This is not a prob-lem: we have intentionally developed XDG as ameta grammar formalism, and the goal is to findtractable instances of XDG for which complexityresults can be established and efficient parsers canbe found.TDG seems to be one of these tractable in-stances. (Koller and Striegnitz, 2002) haveshown that the parsing complexity of TDG is NP-complete. But parsing times of the constraint-based TDG parser, and also of the new, gener-alised XDG parser (using a TDG instance) sug-gest that parsing is polynomial in the average-case.So far, we have only confirmed this for small testgrammars, but we are currently working on gettingresults for bigger grammars.Denys Duchier, Alexander Koller, MarcoKuhlmann, Stefan Thater and me (p.c.) have es-tablished an equivalence between a restricted vari-ant of TDG and Tree Insertion Grammars (TIG)(Schabes and Waters, 1994). TIG is itself arestricted variant of Tree Adjoining Grammars(TAG) (Joshi, 1987).We have also found out that TDG and TAG arenot equivalent. It will be interesting to see whetherwe can find an instance of XDG (probably similarto TDG) which falls into the same class of mildlycontext-sensitive languages such as TAG and oth-ers (e.g. CCG (Steedman, 2000)).As of yet, we have no complexity results forSTDG.5 ConclusionsWe generalised TDG to get Extensible Depen-dency Grammar (XDG), a graph description lan-guage that can be turned into a powerful metagrammar formalism by lexicalisation. We usedXDG as a launchpad for the new grammar formal-ism of Semantic Topological Dependency Gram-mar (STDG). STDG is an instance of XDG thatallows compositional semantics construction.XDG is increasingly used by other linguists:(Duchier and Kruijff, 2003) create an interfaceto information structure, and (Korthals, 2003) in-duces word order constraints from the GermanTIGER corpus. (Dienes et al., 2003) use statis-tical information to guide the XDG parser usingA* search.In the future, our plans include the integration ofpreferences (e.g. PP attachment). We are also con-tinuing our search for equivalences between in-stances of XDG and existing grammar formalismssuch as TAG, and we are working on creating big-ger grammars.ReferencesPeter Dienes, Alexander Koller, and Marco Kuhlmann.2003. Statistical A* Dependency Parsing.Denys Duchier and Ralph Debusmann. 2001. Topo-logical dependency trees: A constraint-based ac-count of linear precedence. In ACL 2001 Proceed-ings, Toulouse/FRA.Denys Duchier and Geert-Jan M. Kruijff. 2003. Infor-mation structure in topological dependency gram-mar. In Proceedings of EACL 2003.Markus Egg, Joachim Niehren, Peter Ruhrberg, andFeiyu Xu. 1998. Constraints over lambda-structures in semantic underspecification. In Pro-ceedings of COLING/ACL 1998, pages 353–359,Montreal/CAN.Aravind K. Joshi. 1987. An introduction to tree-adjoining grammars. In Alexis Manaster-Ramer, ed-itor, Mathematics of Language, pages 87–115. JohnBenjamins, Amsterdam/NL.Alexander Koller and Kristina Striegnitz. 2002. Gen-eration as dependency parsing. In Proceedings ofACL 2002.Christian Korthals. 2003. Unsupervised learning ofword order rules. Master’s thesis, Saarland Univer-sity.Mitchell P. Marcus, Donald Hindle, and Margaret M.Fleck. 1983. D-theory: Talking about talking abouttrees. In Proceedings of ACL 1983, pages 129–136.Carl Pollard and Ivan Sag. 1994. Head-Driven PhraseStructure Grammar. University of Chicago Press,Chicago.Yves Schabes and Richard C. Waters. 1994. Treeinsertion grammar: A cubic-time parsable formal-ism that lexicalizes context-free grammar withoutchanging the trees produced. Technical report, Mit-subishi Electric Research Laboratories.Mark Steedman. 2000. The Syntactic Process. MITPress.

Scientific documents from the Saarland University,

Dependency Grammar as Graph Description
Ralph Debusmann
Programming Systems Lab
Universita¨t des Saarlandes, Geb. 45
Postfach 15 11 50
66041 Saarbru¨cken, Germany
rade@ps.uni-sb.de
Abstract
The paper introduces a generalisa-
tion of Topological Dependency Gram-
mar (TDG) (Duchier and Debusmann,
2001). The result, Extensible Depen-
dency Grammar (XDG), is a descrip-
tion language for sets of labeled directed
graphs. Lexicalisation turns XDG into
a powerful meta grammar formalism.
XDG can be instantiated to yield spe-
cific grammar formalisms based on de-
pendency grammar. We present one
of these instances, Semantic Topolog-
ical Dependency Grammar (STDG), a
new grammar formalism with a syntax-
semantics interface to underspecified se-
mantics.
1 Introduction
(Duchier and Debusmann, 2001) introduced the
grammar formalism of Topological Dependency
Grammar (TDG). The most distinguishing feature
of TDG is its ability to specify non-trivial restric-
tions on word order. These restrictions emerge
from the interaction of simple declarative princi-
ples, and lead to an elegant account of many noto-
rious word order phenomena in German.
TDG concentrates only on syntax and does not
offer a syntax-semantics interface. In this paper,
we introduce Semantic Topological Dependency
Grammar (STDG), which extends TDG with an
interface to underspecified semantics. As the tar-
get semantics formalism, we employ the Con-
straint Language for Lambda Structures (CLLS)
(Egg et al., 1998).
Before we extend TDG, we take a step back-
wards: we generalise TDG to Extensible De-
pendency Grammar (XDG), a graph description
language over sets of labeled directed graphs.
Only by lexicalisation, i.e. the addition of lexi-
calised feature structures, does XDG become a
meta grammar formalism for dependency gram-
mar. TDG is one of the possible instances of XDG.
XDG can be likened to Head-driven Phrase
Structure Grammar (HPSG) (Pollard and Sag,
1994): both XDG and HPSG are meta grammar
formalisms which need to be instantiated with par-
ticular principles and parameters to obtain specific
grammar formalisms. The fundamental difference
is that whereas HPSG is based on typed feature
structures, XDG is based on graph descriptions.
The purpose of XDG is twofold. On the one
hand, XDG can be utilised for research on existing
dependency-based grammar formalisms. On the
other, XDG serves as a launchpad for the develop-
ment of new grammar formalisms like STDG.
The structure of this paper is as follows. We
introduce XDG in section 2. In section 3, we de-
velop the new STDG as an instance of XDG. We
briefly touch the issue of complexity in section 4,
before section 5 concludes.
2 Extensible Dependency Grammar
(XDG)
XDG describes a set of graph dimensions
d1, . . . , dm, where each graph dimension d corre-
sponds to a graph Gd(V,Ed). All graphs share the
same set V = {v1, . . . , vn} of nodes, but have dif-
ferent edges Ed ⊆ V ×Ld×V , and different edge
labels Ld. Feature structures can be attached to
each node, where features are functions V → R
to an arbitrary codomain R.
The well-formedness conditions of an XDG
analysis are stipulated by parametrised principles.
Each dimension d uses a set of principles Pd, stip-
ulating restrictions on the licensed graphs on d.
Principles can also pose restrictions on any num-
ber of dimensions simultaneously. An XDG anal-
ysis is well-formed only if all used principles are
satisfied on all dimensions.
2.1 Principles
Principles are taken from a shared principle li-
brary. In the following, we present some of the
principles from the current XDG principle library.
We describe each principle using three boxes: the
top one shows how to invoke the principle, the
middle one explains the formal parameters, and
the bottom one explains what the principle means.
Note that we omit a number of principles (pro-
jectivity, climbing, and barriers, linking) for lack
of space. Most of these are already explained in
(Duchier and Debusmann, 2001).
2.1.1 Tree
dag(G)
G: a graph dimension
G is a tree.
2.1.2 Directed acyclic graph
dag(G)
G: a graph dimension
G is a directed acyclic graph.
2.1.3 Out
out(Gd, f)
Gd: a graph dimension, f : V → (Ld → 2
 
):
an out specification
The outgoing edges of each node in Gd must
satisfy in label and number the stipulation of
their out specification.
The out principle restricts the number of outgo-
ing edges with a certain label. In (1) below, the
out specification of v1 requires that the node has
1) no outgoing edges labeled l1, 2) one outgoing
edge labeled l2, and 3) zero, one, two or three out-
going edges labeled l3. Thus, the configuration in
(1) satisfies the out principle: there is no outgoing
edge labeled l1, one labeled l2, and three labeled
l3.
(1)
v1
l2
l3
l3
l3
l3: {0,1,2,3}
l2: {1}
l1: {0}
f :
2.1.4 In
in(Gd, f)
Gd: a graph dimension, f : V → (Ld → 2
 
):
an in specification
The incoming edges of each node in Gd must
satisfy in label and number the stipulation of
their in specification.
The in principle is the same as the out principle,
but for incoming instead of outgoing edges.
2.1.5 Covariance
covariance(Gd1, Gd2 , f)
f : V → 2Ld1 : covariant labels specification
Each edge (v1, l, v2) in Gd1 , where l is
covariant on v1, is only licensed if v1 is
above v2 in Gd2 .
In (2) below, the edge (v1, l, v2) in Gd1 is li-
censed because l is a covariant label on v1, and
v1 is above v2 in Gd2 (as indicated by the dashed
edge).
(2)
v1
2v
Gd1
v1
2v
Gd2
l
f : { l }
2.1.6 Contravariance
contravariance(Gd1 , Gd2 , f)
f : V → 2Ld1 : contravariant labels specifica-
tion
Each edge (v1, l, v2) in Gd1 , where l is
contravariant on v1, is only licensed if v2 is
above v1 in Gd2 .
The contravariance principle is the converse of
the covariance principle.
2.1.7 Linking
linking(Gd1 , Gd2 , f1, f2)
Gd1 , Gd2 : graph dimensions, f1 : V →
(Ld1 → 2
Ld2 ): realisation specification, f2 :
V → (Ld2 → 2
Ld2 ): substitution specifica-
tion
An edge (v1, l, v2) in Gd1 is only licensed if
either:
1. there is a corresponding edge (v1, l′, v2)
in Gd2 and v1 realises l by l′, or
2. there is an edge (v3, l′′, v2) in Gd2 , v3
substitutes l′ by l′′ and v1 realises l by l′.
Below, (3) shows a configuration licensed by
the first clause of the linking principle: edge
(v1, l, v2) in Gd1 is licensed because there is a cor-
responding edge (v1, l′, v2) in Gd2 and the realisa-
tion specification of v1 specifies that the node can
realise l as l′, i.e. l′ ∈ f1(v1)(l).
(4) displays a configuration licensed by the sec-
ond clause: edge (v1, l, v2) in Gd1 is licensed be-
cause there is an edge (v3, l′′, v2) in Gd2 , v3 sub-
stitutes l′ by l′′ (i.e. l′′ ∈ f2(v3)(l′)) and v1 realises
l by l′.
(3)
v1
2v
l’v1
2v
1f : l : { l’ }
Gd2Gd1
l
(4)
v1
2v
f1: l : { l’ }
Gd1
2v
v3
f2: l’ : { l’’ }
Gd2
l’’l
2.2 Lexicalisation
By lexicalisation, we can turn the graph descrip-
tion language described above into a powerful
dependency-based meta grammar formalism. To
this end, we assume a 1:1-correspondence be-
tween nodes and words, and assign to each node
one of the possible lexical entries for the corre-
sponding word.
The result of lexicalisation is that we can use
the lexicon to specify all the features referred to
in the principles. For instance lexicalising the out
specification of the out principle gives us valency.
Lexicalising the realisation and substitution speci-
fications of the linking principle allow us to model
lexicalised mappings from thematic roles to gram-
matical functions which realise them.
3 Syntax-semantics interface
In this section, we introduce the new grammar
formalism of Semantic Topological Dependency
Grammar (STDG). STDG has a syntax-semantics
interface to underspecified semantics in CLLS.
Note that CLLS is just one of many possible
choices for the target semantics formalism.
3.1 CLLS
CLLS is based on dominance constraints (Marcus
et al., 1983). CLLS structures describe λ-terms. In
(6) below, we show a CLLS structure describing
the strong reading of sentence (5):
(5) A woman, every man seems to love.
(6)
@
@ lam
@
@ lam
@
seem @
@ var
varlove
every man
womana
Here, the @ symbol stands for functional appli-
cation, lam for a lamda binder, var for a variable,
and the dashed arrows represent lambda bindings.
3.2 STDG
STDG employs four graph dimensions: GID, GLP,
GTH, GDE. As in TDG, the ID (Immediate Domi-
nance) dimension models syntactic dependencies,
and its edge labels are grammatical functions like
subj (subject), obj (object) and vinf (verb infini-
tive). The LP (Linear Precedence) dimension mod-
els word order, and its edge labels are topologi-
cal fields (linear positions) like topf (topicalisation
field), subjf (subject field) and vcf (verbal com-
plement field). We use the same principles as in
(Duchier and Debusmann, 2001).
We introduce the TH (THematic) dimension to
model semantic argument structure. Its edge la-
bels are thematic roles like act (actor), pat (pa-
tient) and prop (proposition). We use the DE
(DErivation) dimension to model the derivation of
a CLLS structure, with edge labels like s (scope),
and r (restriction).
On the TH dimension, we invoke the
following principles and parameters:
dag(GTH), in(GTH, inTH), out(GTH, outTH),
and linking(GTH, GID, real, subs). The dag
principle ensures that every analysis on the TH
dimension is a directed acyclic graph. It cannot
be a tree since we also want to properly handle
control constructions like Mary persuades Peter
to sleep., where the controlled noun Peter has
more than one mother in the argument structure
(here, it is the patient of persuades and the actor
of sleep). Moreover, the in principle assigns
possible thematic roles to words, the out principle
models valency, and the linking principle ensures
that thematic roles are realised by the appropriate
grammatical functions on the ID dimension (e.g.
actors are realised by subjects).
On the DE dimension, we use
the following principles and pa-
rameters: tree(GDE), in(GDE, inDE),
out(GDE, outDE), covariance(GDE, GID, co),
and contravariance(GDE, GID, contra).
Below, we give an example STDG analysis of
the strong reading of sentence (5).1 (7) displays
the ID and LP dimensions, and (8) the TH and DE
dimensions.
(7)
A woman, every man seems to love.
obj
subj vinf
ID
A woman, every man seems to love.
topf
subjf vcf
LP
(8)
A woman, every man seems to love.
prop
act
pat
TH
every man seems to love.
s
s
s
A woman,
DE
The TH analysis states that to love is a proposi-
tion of seems, and that every man is the actor and
1Note that for simplicity, we treat a woman, every man
and to love each as a word.
a woman the patient of to love. The linking princi-
ple establishes the correct mappings of of actor to
subject, and patient to object.
The DE analysis states that every man is in
the scope of a woman, seems is in the scope of
every man, and to love in the scope of seems.
The covariance principle establishes that the edge
(seems, s, love) is covariant: we state in the lex-
icon that seems is covariant on s (scope) la-
bels, and hence the edge corresponds to the edge
(seems, vinf, love) on the ID dimension which
has the same direction. Conversely, we use the
contravariance principle to ensure that the edge
(every man, s, seems) is contravariant: in the lexi-
con, we state that every man is contravariant on s,
and hence we license the analysis where the edge
corresponds to the edge (seems, subj, every man)
on the ID dimension which has the opposite direc-
tion.
3.3 From STDG to CLLS
To build the CLLS representation of the STDG
analysis, we assume a lexicalised mapping from
words in the sentence to CLLS fragments, as illus-
trated in (9) below:2
(9)
a woman
@
@
lam
every man
@
@
lam
seem
@
seems
@
@ var
love var
to love.
A woman, every man
We use 1) the DE analysis to plug these frag-
ments together, and 2) the TH analysis to recover
the lambda bindings. We show an example of this
2Note that we also have to keep track of positions within
the CLLS fragments: the position of restriction and scope,
the position of lambda binders and of variables. We omit this
for lack of space.
in (10) and (11):
(10)
to love.
prop
seemsevery manA woman,
pat
actTH
to love.
s
s
s
seemsevery manA woman,
DE
(11)
a woman
@
@
lam
every man
@
@
lam
seem
@
@
@ var
love var
(10) repeats the TH and DE analyses of sentence
(5). The DE analysis determines how to plug the
four fragments together: every man is in the scope
of a woman, seems is in the scope of every man,
and to love in the scope of seems. The TH analy-
sis determines the lambda bindings: every man is
the actor and a woman the patient of to love. The
edge (seems, prop, to love) does not correspond to
a lambda binding. We show the resulting CLLS
analysis in (11).
Note that in the example shown, we derive a
fully specified CLLS representation of the seman-
tics from a full STDG parse. We can however
obtain underspecified CLLS representations from
partial STDG parses. E.g. we can decide not to
enumerate the solutions on the DE dimension to
get a CLLS analysis underspecified with respect
to scope.
4 Complexity
Due to its generality, the complexity of XDG it-
self cannot be determined. This is not a prob-
lem: we have intentionally developed XDG as a
meta grammar formalism, and the goal is to find
tractable instances of XDG for which complexity
results can be established and efficient parsers can
be found.
TDG seems to be one of these tractable in-
stances. (Koller and Striegnitz, 2002) have
shown that the parsing complexity of TDG is NP-
complete. But parsing times of the constraint-
based TDG parser, and also of the new, gener-
alised XDG parser (using a TDG instance) sug-
gest that parsing is polynomial in the average-case.
So far, we have only confirmed this for small test
grammars, but we are currently working on getting
results for bigger grammars.
Denys Duchier, Alexander Koller, Marco
Kuhlmann, Stefan Thater and me (p.c.) have es-
tablished an equivalence between a restricted vari-
ant of TDG and Tree Insertion Grammars (TIG)
(Schabes and Waters, 1994). TIG is itself a
restricted variant of Tree Adjoining Grammars
(TAG) (Joshi, 1987).
We have also found out that TDG and TAG are
not equivalent. It will be interesting to see whether
we can find an instance of XDG (probably similar
to TDG) which falls into the same class of mildly
context-sensitive languages such as TAG and oth-
ers (e.g. CCG (Steedman, 2000)).
As of yet, we have no complexity results for
STDG.
5 Conclusions
We generalised TDG to get Extensible Depen-
dency Grammar (XDG), a graph description lan-
guage that can be turned into a powerful meta
grammar formalism by lexicalisation. We used
XDG as a launchpad for the new grammar formal-
ism of Semantic Topological Dependency Gram-
mar (STDG). STDG is an instance of XDG that
allows compositional semantics construction.
XDG is increasingly used by other linguists:
(Duchier and Kruijff, 2003) create an interface
to information structure, and (Korthals, 2003) in-
duces word order constraints from the German
TIGER corpus. (Dienes et al., 2003) use statis-
tical information to guide the XDG parser using
A* search.
In the future, our plans include the integration of
preferences (e.g. PP attachment). We are also con-
tinuing our search for equivalences between in-
stances of XDG and existing grammar formalisms
such as TAG, and we are working on creating big-
ger grammars.
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