Tree-Local Multicomponent Tree-Adjoining
Grammars with Shared Nodes
Laura Kallmeyer
∗
TALaNa/Lattice, Universit´eParis7
This article addresses the problem that the expressive power of tree-adjoining grammars (TAGs)
is too limited to deal with certain syntactic phenomena, in particular, with scrambling in free-
word-order languages. The TAG variants proposed so far in order to account for scrambling are
not entirely satisfying. Therefore, the article introduces an alternative extension of TAG that
is based on the notion of node sharing, so-called (restricted) tree-local multicomponent TAG
with shared nodes (RSN-MCTAG). The analysis of some German scrambling data is sketched
in order to show that this TAG extension can deal with scrambling. Then it is shown that
for RSN-MCTAGs of a specific type, equivalent simple range concatenation grammars can
be constructed. As a consequence, these RSN-MCTAGs are mildly context-sensitive and in
particular polynomially parsable. These specific RSN-MCTAGs probably can deal not with all
scrambling phenomena, but with an arbitrarily large subset.
1. Introduction: LTAG and Scrambling
1.1 Lexicalized Tree-Adjoining Grammars
Tree-adjoining grammar (TAG) is a tree-rewriting formalism originally defined by Joshi,
Levy, and Takahashi (1975). A TAG (see Joshi and Schabes 1997 for an introduction)
consists of a finite set of trees (elementary trees). The nodes of these trees are labeled
with nonterminals and terminals (terminals label only leaf nodes). Starting from the
elementary trees, larger trees are derived using composition operations of substitution
(replacing a leaf with a new tree) and adjunction (replacing an internal node with a new
tree). In the case of an adjunction, the tree being adjoined has exactly one leaf node that
is marked as the foot node (marked with an asterisk). Such a tree is called an auxiliary
tree. When such a tree is adjoined to a node µ, in the resulting tree, the subtree with root
node µ from the old tree is put below the foot node of the new auxiliary tree. Elementary
trees that are not auxiliary trees are called initial trees. Each derivation starts with an
initial tree. In the final derived tree, all leaves must have terminal labels.
1
∗ UFR de Linguistique, Case 7003, 2 Place Jussiec, 75005 Paris. E-mail: Laura.Kallmeyer@linguist.jussieu.fr.
1 Additionally, TAG allows for each internal node to specify the set of auxiliary trees that can be adjoined
using so-called adjunction constraints and, furthermore, to specify whether adjunction at that node is
obligatory. This is an important feature of TAG, since it influences the generative capacity of the
formalism: {a
n
b
n
c
n
d
n
|n ≥ 0}, for example, is a language that can be generated by a TAG with adjunction
constraints but not by a TAG without adjunction constraints (Joshi 1985). For this article, however,
adjunction constraints do not play any important role.
Submission received: 1st September 2003; Revised submission received: 4th May 2004; Accepted for
publication: 17th June 2004
© 2005 Association for Computational Linguistics
Computational Linguistics Volume 31, Number 2
Figure 1
TAG derivation for John always laughs.
Figure 1 shows a sample TAG derivation. Here, the three elementary trees for laughs,
John, and always are combined: Starting from the elementary tree for laughs, the tree for
John is substituted for the noun phrase (NP) leaf and the tree for always is adjoined at
the verb phrase (VP) node.
TAG derivations are represented by derivation trees that record the history of how
the elementary trees are put together. A derivation tree is the result of carrying out
substitutions and adjunctions. Each edge in the derivation tree stands for an adjunction
or a substitution. The edges are labeled with Gorn addresses of the nodes where the
substitutions and adjunctions have taken place: The root has the address epsilon1,andthejth
child of the node with address p has address pj. In Figure 1, for example, the derivation
tree indicates that the elementary tree for John is substituted for the node at address 1
and always is adjoined at node address 2.
What we have sketched so far are the mathematical aspects of the TAG formal-
ism. For natural languages, TAGs with specific properties are used. These properties
are not part of the formalism itself, but they are additional linguistic principles that
are respected when a TAG is constructed for a natural language. First, a TAG for
natural languages is lexicalized (Schabes 1990), which means that each elementary
tree has a lexical anchor (usually unique, but in some cases, there is more than one
anchor). Second, the elementary trees of a lexicalized TAG (LTAG) represent extended
projections of lexical items (the anchors) and encapsulate all syntactic arguments of
the lexical anchor; that is, they contain slots (nonterminal leaves) for all arguments.
Furthermore, elementary trees are minimal in the sense that only the arguments of the
anchor are encapsulated; all recursion is factored away. This amounts to the condition
on elementary tree minimality (CETM) from Frank (1992) (see also Frank [2002] for
further discussions of the linguistic principles underlying TAG).
2
The tree for laughs in
Figure 1, for example, contains only a nonterminal leaf for the subject NP (a substitution
node), and there is no slot for a VP adjunct. The adverb always is added by adjunction
at an internal node.
Because of these principles, in linguistic applications, combining two elementary
trees by substitution or adjunction corresponds to the application of a predicate to
an argument. The derivation tree then reflects the predicate-argument structure of the
sentence. This is why most approaches to semantics in TAG use the derivation tree as an
interface between syntax and semantics (see, e.g., Candito and Kahane 1998; Joshi and
Vijay-Shanker 1999; Kallmeyer and Joshi 2003). In this article, we are not particularly
concerned with semantics, but one of the goals of the article is to obtain analyses with
derivation trees representing the correct predicate-argument dependencies.
2 This minimality is actually the reason that the substitution operation is needed; formally TAGs without
substitution and TAGs as introduced above have the same weak and strong generative capacity.
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Kallmeyer Multicomponent TAGs with Shared Nodes
An extension of TAG that has been shown to be useful for several linguistic ap-
plications is multicomponent TAG (MCTAG) (Joshi 1987; Weir 1988). Instead of single
elementary trees, an MCTAG has sets of elementary trees. In each derivation step, one
of these sets is chosen, and all trees from the set are added simultaneously. Depending
on the nodes to which the different trees from the set attach, different kinds of MCTAGs
are distinguished: If all nodes are required to be part of the same elementary tree, the
MCTAG is called tree-local; if all nodes are required to be part of the same tree set, the
grammar is set-local; and otherwise the grammar is nonlocal.
1.2 Scrambling in TAG
Roughly, scrambling can be described as the permutation of elements (arguments and
adjuncts) of a sentence (we use the term scrambling in a purely descriptive sense without
implying any theory involving actual movement). A special case of scrambling is so-
called long-distance scrambling, in which arguments or adjuncts of an embedded
infinitive are “moved” out of the embedded VP. This occurs, for instance, in languages
such as German, Hindi, Japanese, and Korean. As an example of long-distance scram-
bling in German, consider example (1):
(1) ...dass [es]
1
der Mechaniker [t
1
zu reparieren] verspricht
...thatit themechanic torepair promises
‘...thatthemechanicpromisestorepairit’
In example (1), the accusative NP es is an argument of the embedded infinitive zu
reparieren, but it precedes der Mechaniker, the subject of the main verb verspricht,and
it is not part of the embedded VP.
It has been argued (see Rambow 1994a) that in German, there is no bound on the
number of scrambled elements and no bound on the depth of scrambling (i.e., in terms
of movement, the number of VP borders crossed by the moved element).
TAGs are not powerful enough to describe scrambling in German in an adequate
way (Becker, Joshi, and Rambow 1991). By this we mean that a TAG analysis of
scrambling respecting the CETM and therefore giving the correct predicate-argument
structure (i.e., an analysis with each argument attaching to the verb it depends on) is
not possible.
Let us consider the TAG analysis of example (1) in order to see why scram-
bling poses a problem for TAG. If we leave aside the complementizer dass,standard
TAG elementary trees for verspricht and reparieren in the style of the XTAG grammar
(XTAG Research Group 1998) might look as shown in Figure 2. In the derivation, the
Figure 2
Standard TAG combination of der Mechaniker, zu reparieren,andverspricht in example (1).
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Computational Linguistics Volume 31, Number 2
verspricht-tree adjoins to the root node of the reparieren-tree, and the nominative NP der
Mechaniker is substituted for the subject node in the verspricht-tree. This leads to the tree
on the right in Figure 2.
When es is added, there is a problem: It should be added to reparieren, since it is
one of its arguments. But at the same time, it should precede der Mechaniker;thatis,
it must be adjoined either to the root or to the NP
nom
node. The root node belongs to
verspricht,andtheNP
nom
node belongs to der Mechaniker. Consequently, an adjunction
to one of them would not give the desired predicate-argument structure. If one wanted
to analyze only example (1), one could add a tree to the grammar for reparieren with a
scrambled NP that allows adjunction of verspricht between the NP and the verb. But as
soon as there are several scrambled elements that are arguments of different verbs, this
no longer works.
This example has given an idea of why scrambling is problematic for TAG. How-
ever, adopting specific elementary trees, it is possible to deal with a part of the difficult
scrambling data: It has been shown (see Joshi, Becker, and Rambow 2000) that TAG can
describe scrambling up to depth two (two crossed VP borders). But this is not sufficient.
Even though examples of scrambling of depth greater than two are rare, they can occur.
An example is example (2), taken from Kulick (2000):
(2) ...dass [den K¨uhlschrank]
1
niemand [[[t
1
zu reparieren] zu versuchen]
...thattherefrigerator nobody torepair totry
zu versprechen] bereit ist
to promise willing is
‘...thatnobodyiswilling to promise to try to repair the refrigerator’
Consequently, TAG is not powerful enough to account for scrambling.
3
Becker, Rambow, and Niv (1992) argue that even linear context-free rewriting sys-
tems (LCFRSs) (Weir 1988) are not powerful enough to describe scrambling. (LCFRSs
are weakly equivalent to set-local MCTAGs and therefore more powerful than TAGs.)
Although we think that the language Becker, Rambow, and Niv define as a kind of test
language for scrambling is not exactly what one needs (see section 2.3), we still suspect
that they are right in claiming that LCFRSs cannot describe scrambling.
1.3 TAG Variants Proposed for Scrambling
The problem with long-distance scrambling and TAG is that the trees representing
the syntax of scrambled German subordinate clauses do not have the simple nested
structure that ordinary TAG generates. The CETM requires that (positions for) all of
the arguments of the lexical anchor of an elementary tree be included in that tree. But
in a scrambled tree, the arguments of several verbs are interleaved freely. All TAG
extensions that have been proposed to accommodate this interleaving involve factoring
the elementary structures into multiple components and inserting these components at
multiple positions in the course of the derivation.
One of the first proposals made was an analysis of German scrambling data using
nonlocal MCTAG with additional dominance constraints (Becker, Joshi, and Rambow
1991). However, the formal properties of nonlocal MCTAG are not well understood, and
3 See also Gerdes (2002) for a discussion of the limitation of TAG with respect to scrambling in German.
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Kallmeyer Multicomponent TAGs with Shared Nodes
it is assumed that the formalism is not polynomially parsable. Therefore this approach
is no longer pursued, but it has influenced the different subsequent proposals.
An alternative formalism for scrambling is V-TAG (Rambow 1994a, 1994b; Rambow
and Lee 1994), a formalism that has nicer formal properties than nonlocal MCTAG.
V-TAG also uses multicomponent sets (vectors) for scrambled elements; in this it is
a variant of MCTAG. Additionally, there are dominance links among the trees of the
same vector. In contrast to MCTAG, the trees of a vector in V-TAG are not required to be
added simultaneously. The lexicalized V-TAGs that are of interest for natural languages
are polynomially parsable. Rambow (1994a) proposes detailed analyses of a large range
of different word order phenomena in German using V-TAG and thereby shows the
linguistic usefulness of V-TAG.
Even though V-TAG does not pose the problems of nonlocal MCTAG in terms
of parsing complexity, it is still a nonlocal formalism in the sense that, as long as
the dominance links are respected, arbitrary nodes can be chosen to attach the single
components of a vector. Therefore, in order to formulate certain locality restrictions
(e.g., for wh-movement and also for scrambling), one needs an additional means of
putting constraints on what can interleave with the different trees of a vector, or in
other words, constraints on how far a dominance link can be stretched. V-TAG allows
us to put integrity constraints on certain nodes that disallow the occurrence of these
nodes between two trees linked by a dominance link. This has the effect of making these
nodes act as barriers. With integrity constraints, constructions involving long-distance
movements can be correctly analyzed. But the explicit marking of barriers is somewhat
against the original appealing TAG idea that such constraints result from imposition
of the CETM, according to which the position of the moved element and the verb it
depends on must be in the same elementary structure, and from the further combination
possibilities of this structure. In other words, in local formalisms with an extended
domain of locality such as TAG or tree-local and set-local MCTAG, such constraints
result from the form of the elementary structures and from the locality of the derivation
operation. That is, they follow from general properties of the grammar, and they need
not be stated explicitly. This is one of the aspects that make TAG so attractive from a
linguistic point of view, and it gets lost in nonlocal TAG variants.
D-tree substitution grammars (DSGs) (Rambow, Vijay-Shanker, and Weir 2001) are
another TAG variant one could use for scrambling. DSGs are a description-based for-
malism; that is, the objects a DSG deals with are tree descriptions. A problem with DSG
is that the expressive power of the formalism is probably too limited to deal with all
natural language phenomena: According to Rambow, Vijay-Shanker, and Weir (2001)
it “does not appear to be possible for DSG to generate the copy language” (page 101).
This means that the formalism is probably not able to describe cross-serial dependencies
in Swiss German. Furthermore, DSG is nonlocal and therefore, as in the case of V-TAG,
additional constraints (path constraints) have to be placed on material interleaving with
the different parts of an elementary structure.
Another TAG variant using tree descriptions is local tree description grammar
(TDG) (Kallmeyer 2001). Local TDG can be used for scrambling in a way similar to DSG
or V-TAG. The languages generated by local TDGs are semilinear. However, the formal-
ism allows one to generate tree descriptions with underspecified dominance relations,
and the process of resolving the remaining dominance links is nonlocal. Therefore one
may have the same problem as in the case of DSG and V-TAG. Furthermore, so far it
has not been shown that the formalism is polynomially parsable, and it is not clear
whether such parsing is possible without any additional constraint or limitation on the
underspecified tree descriptions.
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Computational Linguistics Volume 31, Number 2
A further TAG variant proposed in order to deal with scrambling is segmented tree-
adjoining grammar (SegTAG) (Kulick 2000). SegTAG uses an operation on trees called
segmented adjunction that consists partly of a standard TAG adjunction and partly of
a kind of tree merging or tree unification. In this operation, two different things get
mixed up, the more or less resource-sensitive adjoining operation of standard TAG, in
which subtrees cannot be identified,
4
and the completely different unification operation.
Perhaps using tree descriptions instead of trees, a more coherent definition of SegTAG
can be achieved. But we will not pursue this here.
The formal properties of SegTAG are not clear. Kulick (2000) suggests that SegTAGs
are probably in the class of LCFRSs, but there is no actual proof of this. However, if
SegTAG is in LCFRS, the generative power of the formalism is probably too limited
to deal with scrambling in a general way. But it seems that the limit imposed by the
grammar on the complexity of the scrambling data is fixed but arbitrarily high. (With
increasing complexity, the elementary trees, however, get larger and larger.) This means
that one can probably define a SegTAG that can analyze scrambling up to some com-
plexity level n for any n ∈ IN. (A definition of what a complexity level is, is not given;
it is perhaps the depth of scrambling.) In this sense, a general treatment of scrambling
might be possible. We follow a similar approach in this article by proposing a mildly
context-sensitive formalism that can deal with scrambling up to some fixed complexity
limit n that can be chosen arbitrarily high.
All these TAG variants are interesting with respect to scrambling, and they give
a great deal of insight into what kind of structures are needed for scrambling. But as
explained above, none of them is entirely satisfying. The most convincing one is V-
TAG, since this formalism can deal with scrambling, lexicalized V-TAG is polynomially
parsable, and the set of languages V-TAG generates contains the set of all tree-adjoining
languages (TALs) (in particular, the copy language). Furthermore, a large range of word
order phenomena has been treated with V-TAG, and thereby the usefulness of V-TAG
for linguistic applications has been shown. But as already mentioned, V-TAG has the
inconvenience of being a nonlocal formalism. For the reasons explained above, it is
desireable to find a local TAG extension for scrambling (as opposed to the nonlocality
of derivations in V-TAG, DSG, and nonlocal MCTAG) such that locality constraints
for movements follow only from the form of the elementary structures and from the
local character of derivations. This article proposes a local TAG variant that can deal
with scrambling (at least with an arbitrarily large set of scrambling phenomena), that
is polynomially parsable, and that properly extends TAG in the sense that the set of all
TALs is a proper subset of the languages it generates.
In section 2, tree-local MCTAG with shared nodes (SN-MCTAG) and in particular
restricted SN-MCTAG (RSN-MCTAG) are introduced, formalisms that extend TAG in
the sense mentioned above. Section 3 shows linguistic applications of RSN-MCTAG, in
particular, an analysis of scrambling. In section 4, a relation between RSN-MCTAG and
range concatenation grammar (RCG) (Boullier 1999, 2000) is established. This relation
allows us to show that certain subclasses of RSN-MCTAG are mildly context-sensitive
and therefore in particular polynomially parsable. These subclasses do not cover all
cases of long-distance scrambling but, in contrast to TAG, they cover an arbitrarily large
4 More precisely, only the root of the new elementary tree and eventually (i.e., in the case of an adjunction)
the foot node get identified with the node the new tree attaches to. But there is no unification of whole
subtrees. Consequently, every edge occurring in the derived tree comes from exactly one edge in an
elementary tree, and every edge from the elementary trees used in the derivation occurs exactly once in
the derived tree. In this sense the operation is resource-sensitive.
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Kallmeyer Multicomponent TAGs with Shared Nodes
set, providing scrambling analyses that respect the CETM. This means that the limit they
impose on the complexity of the scrambling data one can analyze is variable. Based on
empirical studies, it can be chosen sufficiently great such that the grammar covers all
scrambling cases that one assumes to occur.
2. The Formalism
An informal introduction of (restricted) tree-local MCTAG with shared nodes can also
be found in Kallmeyer and Yoon (2004).
2.1 Motivation: The Idea of Shared Nodes
Let us consider again example (1) in order to illustrate the general idea of shared nodes.
In standard TAG, nodes to which new elementary trees are adjoined or substituted
disappear; that is, they are replaced by the new elementary tree. For example, after
having performed the derivation steps shown in Figure 2, the root node of the reparieren
tree does not exist any longer. It is replaced by the verspricht tree, and its daughters have
become daughters of the foot node of the verspricht tree. That is, the root node of the
derived tree is considered to belong only to the verspricht tree. Therefore, an adjunction
at that node is an adjunction at the verspricht tree.
However, this standard TAG view is not completely justified: In the derived tree,
the root node and the lower S node might as well be considered to belong to reparieren,
since they are results of identifying the root node of reparieren with the root and the foot
node of verspricht.
5
Therefore, we propose that the two nodes in question belong to both
verspricht and reparieren. In other words, these nodes are shared by the two elementary
trees. Consequently, they can be used to add new elementary trees to verspricht and (in
contrast to standard TAG) also to reparieren.
In the following, we use an MCTAG, and we assume tree-locality; that is, the nodes
to which the trees of such a set are added must all belong to the same elementary tree.
Standard tree-local MCTAGs are weakly and even strongly equivalent to TAGs, but they
allow us to generate a richer set of derivation structures. In combination with shared
nodes, tree-local multicomponent derivation extends the weak generative power of the
grammar (see Figure 4 for a sample tree-local MCTAG with shared nodes that generates
a language that is not a tree-adjoining language).
6
Let us go back to example (1). Assume the tree set in Figure 3 for the scrambled NP
es. If the idea of shared nodes is adopted, this tree set can be added to reparieren using
the root of the derived tree for adjunction of the first tree and the NP
acc
substitution
node for substitution of the second tree. The operation is tree-local, since both nodes are
part of the reparieren tree.
5 Actually, in a feature-structure based TAG (FTAG) (Vijay-Shanker and Joshi 1988), the top feature
structure of the root of the derived tree is the unification of the top of the root of verspricht and the top of
the root of reparieren. The bottom feature structure of the lower S node is the unification of the bottom of
the foot of verspricht and the bottom of the root of reparieren. In this sense, the root of the reparieren tree
gets split into two parts. The upper part merges with the root node of the verspricht tree, and the lower
part merges with the foot node of the verspricht tree.
6 In a way, the idea of node sharing is already present in description-based definitions of TAG-related
formalisms (see Vijay-Shanker 1992; Rogers 1994; Kallmeyer 2001). This is why these formalisms are
monotonic with respect to the node properties described in the tree descriptions. However, the possibility
of exploiting this in order to obtain multiple adjunctions combined with multicomponent tree
descriptions has not been pursued so far.
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Computational Linguistics Volume 31, Number 2
Figure 3
Derivation of (1) dass es der Mechaniker zu reparieren verspricht (‘that the mechanic promises to
repair it’) using shared nodes.
The notion of shared nodes means in particular that a node can be used for more
than one adjunction. (E.g., in Figure 3, two trees were adjoined at the root of the
reparieren tree.) A similar idea has led to the definition of extended derivation in Schabes
and Shieber (1994). For certain auxiliary trees, Schabes and Shieber allow more than one
adjunction at the same node. However, the definition of the derived tree in Schabes and
Shieber (1994) is such that if first β
1
and then β
2
are adjoined at some node µ (i.e., in
the derivation tree there are edges from some γ to β
1
and β
2
, both with the position
p of the node µ in γ), then first the whole tree derived from β
1
is added to position p,
and afterwards the whole tree derived from β
2
is added to position p. In other words,
before β
2
is adjoined, all the trees to be added by adjunction or substitution to β
1
must
be added. This is different in the case of shared nodes: After β
1
and then β
2
have been
adjoined, the root node of β
2
in the derived tree is shared by β
1
and β
2
and consequently
can be used for adjunctions at β
1
.
7
In other words, trees to be adjoined at the roots of
β
1
and β
2
can be adjoined in any order. This is important for obtaining all the possible
permutations of scrambled elements.
2.2 Formal Definition of Tree-Local MCTAG with Shared Nodes
As already mentioned, the idea of tree-local MCTAG with shared nodes is the following:
In the case of a substitution of an elementary tree α into an elementary tree γ,in
the resulting tree, the root node of the subtree α is considered to be part of α and
of γ. Similarly, when an elementary tree β is adjoined at a node that is part of the
elementary trees γ
1
,...,γ
n
, then in the resulting tree, the root and foot node of β are
both considered to be part of γ
1
,...,γ
n
and β. Consequently, if an elementary tree γ
prime
is added to an elementary tree γ, and if there is then a sequence of adjunctions at root
7 In this case, one obtains crossed dotted edges in the SN-derivation structure defined later (see Figure 14
for an example).
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Kallmeyer Multicomponent TAGs with Shared Nodes
or foot nodes starting from γ
prime
, then each of these adjunctions can be considered an
adjunction at γ, since it takes place at a node shared by γ,γ
prime
, and all the subsequently
adjoined trees.
Therefore, one way to define SN-MCTAG refers to the standard TAG derivation tree
in the following way. Define the grammar as an MCTAG and then allow only derivation
trees that satisfy the following tree-locality condition: For each instance {γ
1
,...,γ
k
} of
an elementary tree set in the derivation tree, there is a γ such that each of the γ
i
is either
a daughter of γ or is linked to one of the daughters of γ by a chain of adjunctions at root
or foot nodes.
As an example, consider the derivation tree for (1) in Figure 3. It shows that the trees
used in the derivation are the reparieren tree, the verspricht tree, the Mechaniker tree, and
the two trees es and epsilon1-es from the tree set in Figure 3. epsilon1-es is substituted into reparieren
at position 21, and verspricht is adjoined to reparieren at position epsilon1. Then, Mechaniker is
substituted into verspricht at position 1, and es is adjoined to verspricht at position epsilon1.The
derivation is tree-local in the node-sharing sense, since for the tree set {epsilon1-es, es}, epsilon1-es
is a daughter of reparieren in the derivation tree and es is linked to reparieren byafirst
adjunction of verspricht to reparieren and a further adjunction of es to the root of verspricht.
In the following, we adopt this way of viewing derivations in SN-MCTAG as
specific multicomponent TAG derivations; that is, we define SN-MCTAG as a variant
of MCTAG. This avoids formalizing a notion of shared nodes, even though this was the
starting motivation for the formalism.
We assume a definition of TAG as a tuple G =〈I, A, N, T〉 with I being the set of
initial trees, A the set of auxiliary trees, and N and T the nonterminal and terminal
node labels, respectively (see, for example, Vijay-Shanker [1987] for a formal definition
of TAG). Now we formally introduce multicomponent tree-adjoining grammars (Joshi
1987; Weir 1988):
Definition 1
A multicomponent tree-adjoining grammar is a tuple G =〈I, A, N, T,A〉 such that
a114
G
TAG
:=〈I, A, N, T〉 is a TAG;
a114
A ⊆ P(I ∪ A)isasetofsubsetsofI ∪ A, the set of elementary tree sets.
8
γ ⇒ γ
prime
is a multicomponent derivation step in G iff there is an instance{γ
1
,...,γ
n
}
of an elementary tree set in A and there are pairwise different node addresses p
1
,..., p
n
such that γ
prime
= γ[p
1
,γ
1
]...[p
n
,γ
n
], where γ[p
1
,γ
1
]...[p
n
,γ
n
] is the result of adding the
γ
i
(1 ≤ i ≤ n) at node positions p
i
in γ.
9
As in TAG, a derivation starts from an initial tree, and in the end, in the final derived
tree, there must not be any obligatory adjunction constraint, and all leaves must be
labeled by a terminal or by the empty word.
In each MCTAG derivation step, an elementary tree set is chosen, and the trees from
this set are added to the already derived tree. Since they are added to pairwise different
8 P(X) is the set of subsets of some set X.
9 As usual (see Vijay-Shanker 1987; Weir 1988), γ[p,γ
prime
] is defined as follows: If γ
prime
is (derived from) an
initial tree and the node at position p in γ is a substitution node, then γ[p,γ
prime
] is the tree one obtains by
substitution of γ
prime
into γ at node position p.Ifγ
prime
is (derived from) an auxiliary tree and the node at
position p in γ is an internal node, then γ[p,γ
prime
] is the tree one obtains by adjunction of γ
prime
to γ at node
position p.Otherwiseγ[p,γ
prime
] is undefined.
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Computational Linguistics Volume 31, Number 2
nodes, one can just as well add them one after the other; that is, each multicomponent
derivation in an MCTAG G =〈I, A, N, T,A〉 corresponds to a derivation in the TAG
G
TAG
:=〈I, A, N, T〉. Let us define the TAG derivation tree of such a multicomponent
derivation as the corresponding derivation tree in G
TAG
. We can then define tree-local,
set-local, and nonlocal MCTAG and also the different variants of SN-MCTAG this article
deals with by putting different constraints on this derivation tree.
10
Note that for each
operation γ[p,γ
i
],thenodeaddressp in the derived tree γ points at a node that is
at some address p
prime
in some elementary tree γ
prime
that was already added (γ
prime
and p
prime
are
unique). In the TAG derivation tree, there will be in this case an edge from γ
prime
to γ
i
with
position p
prime
.
A TAG derivation tree can be considered a tuple of nodes and edges. As usual
in finite trees, the edges are directed from the mother node to the daughter. Linear
precedence is not needed in a derivation tree, since it does not influence the result of the
derivation. So a derivation tree is a tuple 〈N,E〉,withN being a finite set of instances of
elementary trees and with E ⊂ N ×N × IN
∗
, where IN
∗
is the set of Gorn addresses. We
define the parent relation as the relation between mothers and daughters in a derivation
tree, the dominance relation as the reflexive transitive closure of the parent relation,
and the node-sharing relation as the relation between nodes that either are mother and
daughter or are linked first by a substitution/adjunction and then a chain of adjunctions
at root or foot nodes:
Definition 2
Let D =〈N,E〉 be a derivation tree in a TAG.
a114
P
D
:={〈n
1
, n
2
〉|n
1
, n
2
∈ N, and there is a p ∈ IN
∗
such that 〈n
1
, n
2
, p〉∈E}
is the parent relation in D.
a114
D
D
:={〈n
1
, n
2
〉|n
1
, n
2
∈ N, and either n
1
= n
2
, or there is a n
3
such that
〈n
1
, n
3
〉∈P
D
and 〈n
3
, n
2
〉∈D
D
} is the dominance relation in D.
a114
SN
D
:={〈n
1
, n
2
〉| either 〈n
1
, n
2
〉∈P
D
or there are t
1
,..., t
k
∈ N, such that
〈n
1
, t
1
〉∈P
D
, n
2
= t
k
and for all j,1≤ j ≤ k − 1: 〈t
i
, t
i+1
, p
prime
〉∈E with either
p
prime
= epsilon1 or t
i
being an auxiliary tree with foot node address p
prime
} is the
node-sharing relation in D.
A node-sharing relation 〈γ
1
,γ
2
〉 that corresponds to an actual parent relation is
called a primary node-sharing relation, and γ
2
is called a primary SN-daughter of γ
1
,
whereas any other node-sharing relation 〈γ
1
,γ
2
〉 is called secondary and in this case γ
2
is called a secondary SN-daughter of γ
1
.
The TAG derivation trees for MCTAG derivations have certain properties resulting
from the requirement that the elements of instances of elementary tree sets must be
added simultaneously to the already derived tree: First, if an elementary tree set is used,
then all trees from this set must occur in the derivation tree. Secondly, one tree from an
elementary tree set cannot be substituted or adjoined into another tree from the same
set, and, thirdly, two tree sets cannot be interleaved. For nonlocal MCTAG, these are all
constraints the TAG derivation tree needs to satisfy.
10 This TAG derivation tree is not the MCTAG derivation tree defined in Weir (1988). The nodes of Weir’s
MCTAG derivation trees are labeled by sequences of elementary trees (i.e., by elementary tree sets), and
each edge stands for simultaneous adjunctions/substitutions of all elements of such a set.
196
Kallmeyer Multicomponent TAGs with Shared Nodes
Lemma 1
Let G =〈I, A, N, T,A〉 be an MCTAG, G
TAG
:=〈I, A, N, T〉.LetD =〈N,E〉 be a derivation
tree in G
TAG
with the corresponding derived tree t being in L(G
TAG
).
D is a possible TAG derivation tree in G with t ∈ L(G)iffD is such that
a114
(MC1) The root of D is an instance of an initial tree α ∈ I and all other
nodes are instances of trees from tree sets in A such that for all instances Γ
of elementary tree sets from A and for all γ
1
,γ
2
∈Γ,ifγ
1
∈ N, then
γ
2
∈ N.
a114
(MC2) For all instances Γ of elementary tree sets from A and for all
γ
1
,γ
2
∈Γ, γ
1
negationslash= γ
2
: 〈γ
1
,γ
2
〉 negationslash∈ D
D
.
a114
(MC3) For all pairwise different instances Γ
1
,Γ
2
,...,Γ
n
, n ≥ 2, of
elementary tree sets from A, there are no γ
(i)
1
,γ
(i)
2
∈Γ
i
,1≥ i ≥ n, such that
〈γ
(1)
1
,γ
(n)
2
〉∈D
D
and 〈γ
(i)
1
,γ
(i−1)
2
〉∈D
D
for 2 ≥ i ≥ n.
The proof of this lemma is given in the appendix. The lemma gives us a way
to characterize nonlocal MCTAG via the properties of the TAG derivation trees the
grammar licenses. With this characterization we get rid of the original simultaneity
requirement: The corresponding properties are now captured in the three constraints
(MC1)–(MC3). But since these constraints need to hold only for the TAG derivation
trees that correspond to derived trees in the tree language, subderivation trees need not
satisfy them. In other words, γ
1
and γ
2
from the same elementary tree set can be added
at different moments of the derivation as long as the final complete TAG derivation tree
satisfies (MC1)–(MC3).
We now define tree-local, set-local, SN-tree-local, and SN-set-local TAG derivation
trees by imposing further conditions. Basically, the difference between the first two and
their SN variants is that in the first two, the definition refers to the parent relation,
whereas in the second two, it refers to the node-sharing relation.
Definition 3
Let G =〈I, A, N, T,A〉 be an MCTAG. Let D =〈N,E〉 be a TAG derivation tree for some
t ∈ L(〈I, A, N, T〉).
a114
D is a multicomponent derivation tree iff it satisfies (MC1)–(MC3).
a114
D is tree-local iff for all instances {γ
1
,...,γ
n
} of elementary tree sets
with γ
1
,...,γ
n
∈ N, there is one γ such that 〈γ,γ
1
〉,...,〈γ,γ
n
〉∈P
D
.
a114
D is set-local iff for all instances {γ
1
,...,γ
n
} of elementary tree sets with
γ
1
,...,γ
n
∈ N, there is an instance Γ of an elementary tree set such that
for all 1 ≤ i ≤ n, there is a t
i
∈Γ with 〈t
i
,γ
i
〉∈P
D
.
a114
D is SN-tree-local iff for all instances {γ
1
,...,γ
n
} of elementary tree
sets with γ
1
,...,γ
n
∈ N, there is one γ such that 〈γ,γ
1
〉,...,〈γ,γ
n
〉
∈ SN
D
.
a114
D is SN-set-local iff for all instances {γ
1
,...,γ
n
} of elementary tree sets
with γ
1
,...,γ
n
∈ N, there is an instance Γ of an elementary tree set such
that for all 1 ≤ i ≤ n, there is a t
i
∈Γ with 〈t
i
,γ
i
〉∈SN
D
.
197
Computational Linguistics Volume 31, Number 2
Figure 4
SN-MCTAG for {w
3
|w ∈ T
∗
}.
The formalism we are proposing for scrambling is MCTAG with SN-tree-local TAG
derivation trees. We call these grammars tree-local MCTAGs with shared nodes:
Definition 4
Let G be an MCTAG. G is a tree-local MCTAG with shared nodes iff the set of trees
generated by G, L
T
(G), is defined as the set of those trees that can be derived with an
SN-tree-local multicomponent TAG derivation tree in G.
As usual, the string language L
S
(G) is then defined as the set of strings yielded by the
trees in L
T
(G).
All tree-adjoining languages can be generated by SN-MCTAGs, since a TAG cor-
responds to an MCTAG with unary multicomponent sets. For such an MCTAG, each
TAG derivation tree is trivially SN-tree-local. In other words, in this case the tree sets
are the same, whether the grammar is considered a TAG, a tree-local MCTAG, or an
SN-MCTAG.
11
In particular, all TAG analyses proposed so far can be maintained, since
each TAG is trivially also an instance of SN-MCTAG.
SN-MCTAG is a proper extension of TAG (and of tree-local MCTAG) in the sense
that there are languages that can be generated by an SN-MCTAG but not by a TAG.
As an example, consider Figure 4, which shows an SN-MCTAG for {www|w ∈ T
∗
}.
12
Similar to the grammar in Figure 4, for all copy languages {w
n
|w ∈ T
∗
} for some n ∈ IN ,
an SN-MCTAG can be found. Other languages that can be generated by SN-MCTAG
and that are not TALs are the counting languages {a
n
1
...a
n
k
|n ≥ 1} for any k > 4(for
k ≤ 4, these languages are tree-adjoining languages).
There are two crucial differences between V-TAG and SN-MCTAG: First, in V-
TAG, the adjunctions of auxiliary trees from the same set need not be simultaneous.
In this respect, V-TAG differs not only from SN-MCTAG, but from any of the different
11 However, viewing a TAG as an SN-MCTAG allows us to obtain a richer set of SN-derivation structures,
as introduced in the next section. This is exploited in Kallmeyer (2002) for semantics.
12 The subscript NA in the figure stands for null adjunction; that is, it disallows adjunctions at the node in
question.
198
Kallmeyer Multicomponent TAGs with Shared Nodes
MCTAGs mentioned above. Secondly, V-TAG is nonlocal in the sense of nonlocal MC-
TAG, whereas SN-MCTAG is local, even though the locality is not based on the parent
relation in the derivation tree, as is the case in standard local MCTAG, but on the SN-
dominance relation in the derivation tree. As a consequence of the locality, we do not
need dominance links (i.e., dominance constraints that have to be satisfied by the de-
rived tree) in SN-MCTAG, in contrast to other TAG variants for scrambling. The locality
condition put on the derivation sufficiently constrains the possibilities for attaching
the trees from elementary tree sets: Different trees from a tree set attach to different
nodes of the same elementary tree. Consequently, the dominance relations among these
different nodes determine the dominance relations among the different trees from the
tree set. Therefore extra dominance links are not necessary. This is different for nonlocal
TAG variants such as V-TAG or DSG, in which one can in principle attach the different
components of an elementary structure at arbitrary nodes in the derived tree.
2.3 SN-MCTAG and Scrambling: Formal Considerations
Figure 5 shows an SN-MCTAG generating a language that cannot even be generated by
linear context-free rewriting systems (see Becker, Rambow, and Niv [1992] for a proof),
and therefore not by set-local MCTAG. This example, however, concerns neither weak
nor strong generative capacity, but something that Becker, Rambow, and Niv (1992) call
derivational capacity: the derivation of n
k
v
k
must be such that the π(i)th n and the ith v
come from the same elementary tree set in the grammar.
The grammar in Figure 5 works in the following way: Each derivation starts with α.
Then a first instance of the tree set (yielding n
1
and v
1
) is added to the N and V nodes in
α. For each further instance of the tree set (yielding n
i
and v
i
), β
v
is adjoined to the root
node of the β
v
tree of v
i−1
. Therefore all β
v
adjunctions except the first are occurring
at root nodes, and consequently all β
v
are (primary or secondary) SN-daughters of α.
The β
n
tree of n
i
can be adjoined to any of the root or foot nodes of the β
n
that have
already been added, since in this way all adjunctions of β
n
except the first one occur
at root or foot nodes, and therefore all these β
n
are SN-daughters of α. This allows us
to place n
i
at any position in the string already containing {n
1
,..., n
i−1
}, and thereby
any permutation of the ns can be obtained. Since all nodes in the derivation tree are SN-
daughters of α, the derivation is SN-tree-local. Note that in the grammar in Figure 5,
there is no NA constraint on the foot node of the first auxiliary tree in the tree set. This
is crucial for allowing all permutations of the n
1
,..., n
k
. In this respect, the elementary
trees differ from what is usually done in TAG.
Becker, Rambow, and Niv (1992) argue that a formalism that cannot generate the
language in Figure 5 is not able to analyze scrambling in an adequate way. We think,
Figure 5
SN-MCTAG for {n
π(k)
...n
π(1)
v
k
...v
1
|k ≥ 0, n
i
= n, v
i
= v,andn
i
and v
i
are in the same
elementary tree set and they were added in the ith derivation step for all i,1≤ i ≤ k,andπ is a
permutation of (1,..., k)}.
199
Computational Linguistics Volume 31, Number 2
Figure 6
Predicate argument structure for SCR.
however, that this language is not exactly what one needs for scrambling. The assump-
tion underlying the language in Figure 5 is that n
i
is an argument of v
i
. But in this case,
instead of adding n
i
and v
i
at the same time, n
i
should be added to v
i
. If one makes
the additional assumption that argument NPs are added by substitution, then one can
require that the argument NPs have already been substituted (this is what Joshi, Becker,
and Rambow [2000] call the weak co-occurrence constraint), that is, that the tree for
v
i
contain n
i
. In this case, the language in Figure 5 is an appropriate test language for
scrambling. But we do not want to make this assumption.
Furthermore, there are more predicate-argument dependencies: v
i
is also an argu-
ment of v
i−1
for i ≤ 2. This is what Joshi, Becker, and Rambow (2000) call the strong
co-occurrence constraint. In other words, the dependency tree should be as in Figure 6.
Additionally to the permutation of the n
1
,..., n
k
,alsothev
i
can be moved leftward,
as long as they do not permute among themselves. Consequently, for scrambling data
(without extraposition), one rather wants to generate the following language: SCR :=
{w = π(n
1
...n
k
v
1
...v
k
)|k ≥ 1, n
i
= n, v
i
= v, for all 1 ≤ i ≤ k,andπ is a permutation of
n
1
...n
k
v
1
...v
k
such that n
i
precedes v
i
in w for all 1 ≤ i ≤ k and v
i
precedes v
i−1
in w
for all 1 < i ≤ k} with the derivation structure in Figure 6. An SN-MCTAG generating
this language is shown in Figure 7.
The SN-MCTAG in Figure 7 yields the following derivations: Either start with α
2
,in
which case an instance of {β
n
,α
n
} must be added and nv is obtained with n depending
on v, or start with α
1
for v
1
, in which case, for all v except the leftmost one, the set
{β
v1
,α
v
1
} is added, for the leftmost v,aset{β
v2
,α
v2
} is added, and for all the ns, sets
{β
n
,α
n
} are added. These sets can be added in any order; the auxiliary tree is always
adjoined to the root node of the already derived tree that is shared by all auxiliary trees
that have been used so far and by the first α
1
. The initial tree is primarily substituted
Figure 7
SN-MCTAG for SCR.
200
Kallmeyer Multicomponent TAGs with Shared Nodes
into the argument slot it fills. So the only condition for adding such a tree set is that
the verb it depends on has already been added, since the tree of this verb provides the
substitution node for the initial tree. Therefore, since the lexical material is always left
of the foot node, one obtains that v
i
precedes v
i−1
for all 1 < i ≤ k and n
i
precedes v
i
for
all 1 ≤ i ≤ k.
Note that in Figure 7, for a scrambled n
i
, the substitution node is filled with an
empty node, while the n is adjoined higher at a node that is not yet available in the
elementary structure of v
i
. So the combination of n
i
and v
i
cannot be precompiled here.
2.4 Restricted SN-MCTAG
When the formal properties of SN-MCTAG are examined, it becomes clear that the for-
malism is hard to compare to other local TAG-related formalisms, since in the derivation
tree, arbitrarily many trees can be secondary SN-daughters of a single elementary tree,
such that these secondary links are considered to be adjunctions to that tree. This means
that these secondary links are relevant for the SN-tree-locality of the derivation. An
example is the grammar in Figure 5, in which in each derivation step, the relevant node-
sharing relations are the links between α and the two auxiliary trees of the new set.
This means that for a word of length k, there are k SN-daughters of α that are relevant
for the SN-tree-locality of the derivation. The grammar in Figure 5 indicates that this
property of SN-MCTAG is at least partly responsible for the fact that SN-MCTAG
allows us to generate languages that are not even mildly context-sensitive (i.e., that
are not in the class of languages that can be generated by LCFRS). However, it would
be desirable to stay inside the class of mildly context-sensitive languages. Therefore, in
the following, we define a restricted version, RSN-MCTAG, that limits the number of
relevant secondary SN-daughters of an elementary tree. The restriction is obtained as
follows: We require that in each derivation step, among the SN-relations between the old
γ and the new set Γ, there be at least one primary SN-relation. The number of primary
SN-daughters of a specific elementary tree is limited, since the primary SN-daughters
correspond to substitutions/adjunctions at pairwise different nodes and the number of
nodes in an elementary tree is limited. Consequently, the number of relevant secondary
SN-daughters for a node is limited as well.
An example of a derivation satisfying the new constraint is that in Figure 3, in which
es is a secondary SN-daughter of reparieren, while the second element of the tree set, epsilon1-es,
is a primary SN-daughter of reparieren.
Definition 5
a114
Let G =〈I, A, N, T,A〉 be an MCTAG. Let D =〈N,E〉 be the TAG derivation
tree of a tree t ∈ L
T
(〈I, A, N, T〉). D is RSN-tree-local iff for all instances
{γ
1
,...,γ
n
} of an elementary tree set with γ
1
,...,γ
n
∈ N, there is one γ
such that
1. 〈γ,γ
1
〉,...,〈γ,γ
n
〉∈SN
D
;
2. there is one i,1≤ i ≤ n,with〈γ,γ
i
〉∈P
D
.
a114
An MCTAG G is called a restricted SN-MCTAG iff the set of trees
generated by G, L
T
(G), is defined as the set of those trees that can be
derived with an RSN-tree-local multicomponent TAG derivation tree in G.
201
Computational Linguistics Volume 31, Number 2
The first condition of the definition says that the grammar is SN-tree-local, and the
second condition ensures that at least one of the relevant SN-daughters of γ is a primary
SN-daughter, that is, an actual daughter of γ.
As for SN-MCTAG, all tree-adjoining languages can also be generated by RSN-
MCTAGs. The sample grammars in Figures 4 and 5 are not RSN-MCTAGs. We suspect
that there is no RSN-MCTAG that generates the language in Figure 5. But the grammar
in Figure 7 for the language SCR is an RSN-MCTAG.
It can be shown that for the TAG derivation trees of an RSN-MCTAG, the following
holds: For each instance of an elementary tree set Γ,theγ to which all elements of Γ
are linked by node-sharing relations with at least one primary link is unique (which is
not necessarily the case for general SN-MCTAG). This is formulated in the following
lemma:
Lemma 2
Let G =〈I, A, N, T,A〉 be an RSN-MCTAG. Let D =〈N,E〉 be a TAG derivation tree
in G.
Then for all instances {γ
1
,...,γ
n
} of elementary tree sets with γ
1
,...,γ
n
∈ N, there
is exactly one γ such that 〈γ,γ
1
〉,...,〈γ,γ
n
〉∈SN
D
, and there is one i,1≤ i ≤ n,with
〈γ,γ
i
〉∈P
D
.
For such an elementary tree set {γ
1
,...,γ
n
},withγ being the unique elementary
tree as described in the lemma, all 〈γ,γ
i
〉∈SN
D
\P
D
,1≤ i ≤ n, are called secondary
adjunction links in D. The proof of the lemma is given in the appendix.
Now we introduce the SN-derivation structure of a TAG derivation tree D in an
RSN-MCTAG. It consists of D enriched with additional links for the secondary adjunc-
tions. These links are equipped with the positions of the first substitutions/adjunctions
on the chain that corresponds to the secondary adjunctions.
Definition 6
Let G =〈I, A, N, T,A〉 be an RSN-MCTAG. Let D =〈N,E〉 be a TAG derivation
tree in G.TheSN-derivation structure of D, D
SN
, is then D
SN
:=〈N,E
prime
〉,
with E ⊆ E
prime
.
a114
For all secondary adjunction links 〈γ
1
,γ
2
〉 in D with γ
prime
and p such that
〈γ
1
,γ
prime
, p〉∈E and 〈γ
prime
,γ
2
〉∈D
D
: 〈γ
1
,γ
2
, p〉∈E
prime
.
a114
These are all elements of E
prime
.
All e ∈ E are called primary edges in D
SN
,andalle ∈ E
prime
\E are called secondary
edges in D
SN
.
With the notion of the SN-derivation structure, we can formulate the limitation on
the maximal number of secondary adjunctions to an elementary tree that we mentioned
at the beginning of this section:
Lemma 3
Let G =〈I, A, N, T,A〉 be an RSN-MCTAG. Then there is a constant c such that for all
TAG derivation trees D in G with SN-derivation structure D
SN
:=〈N,E〉, the following
holds:
202
Kallmeyer Multicomponent TAGs with Shared Nodes
There is no n ∈ N such that there exist m ≥ c + 1 pairwise different n
1
,..., n
m
such
that for all i,1≤ i ≤ m, there is a p such that
a114
either 〈n, n
i
, p〉 is a secondary edge in D
SN
;
a114
or 〈n, n
i
, p〉 is a primary edge in D
SN
, and there are no n
prime
and p
prime
such that
〈n
prime
, n
i
, p
prime
〉 is a secondary edge in D
SN
.
That this lemma holds is nearly immediate: Each secondary adjunction must be as-
sociated with a primary adjunction or substitution into the same tree instance. There
are at most k primary adjunctions or substitutions into any tree instance if k is the
maximal number of nodes per elementary tree. Consequently there are at most k × n
secondary adjunctions per node if n + 1 is the maximal number of trees per elementary
tree set.
In linguistic applications, the SN-derivation structure is intended to reflect the
predicate-argument dependencies of a sentence in the following way: For each tree in
the SN-derivation structure, if this tree is secondarily adjoined to some other tree γ,
then it depends on γ. Otherwise it depends on its mother node in the TAG derivation
tree. In this way, the grammar for SCR in Figure 7 yields the desired dependency
structure.
3. Linguistic Applications
3.1 Scrambling with RSN-MCTAG
In this section, we present a small German grammar that allows us to analyze some
cases of scrambling. The aim is not an exhaustive treatment of the phenomenon, but
just to show that in principle, an analysis of scrambling in German is possible using
RSN-MCTAG. The data to which we restrict ourselves are word order variations of
example (3) without extraposition, that is, under the assumption that the order of the
verbs is zu reparieren zu versuchen verspricht:
(3) ...dass er dem Kunden das Fahrrad zu reparieren
...thathe
nom
the customer
dat
the bike
acc
to repair
zu versuchen verspricht
to try promises
‘...thathepromisesthecustomertotrytorepairthebike’
The elementary trees and tree sets for example (3) are shown in Figure 8. In contrast
to standard TAG practices, which are often guided by technical considerations, we
represent all arguments of a verb (including an embedded VP) by substitution nodes.
For those parts that might be scrambled, there is a single elementary tree (for the case
without scrambling) and a tree set used for scrambling. The tree set contains an auxiliary
tree that can be primarily or secondarily adjoined to some root node and a tree with the
empty word that is intended to fill the argument position. In order to avoid spurious
ambiguities, we assume that whenever a derivation using the single elementary tree is
possible, this is chosen.
A scrambled element always adjoins to a VP node, and the scrambled element is
to the left of the foot node. Therefore it precedes everything that is below or on the
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Computational Linguistics Volume 31, Number 2
Figure 8
Elementary trees for scrambling.
right of the VP node to which it adjoins. Consequently, given the form of the verbal
elementary trees in Figure 8, in which the verb is always below or to the right of all VP
nodes allowing adjunction, the order xvfor an x being a nominal or a verbal argument
of v is always respected.
For an element (a lexical item), the tree set for scrambling is used whenever one of
the following three cases holds:
a114
The element is scrambled.
a114
Scrambling of depth more than one out of the element takes place.
a114
The element intervenes between some element A (on its right) and some
element B (on its left) scrambled out of A, and the element itself does not
belong to A.
In other words, the fact that the set for scrambling is used for some element does
not necessarily mean that this element is scrambled. It just means that one of the
three cases above holds, that is, that some scrambling around this element takes
place.
One could actually do without the single trees and always use the tree sets. In this
case, even if no scrambling took place, all argument slots would be filled by empty
words, and all lexical material would be adjoined to the root node of the derived tree.
At first glance, this seems rather odd. But if one does not consider the substitution
nodes argument slots but rather some kind of subcategorization features marking which
arguments need to be added, an analysis using only the tree sets makes sense. However,
for this article, we keep the single trees.
For example (3), a derivation without secondary adjunctions and using only the
single trees is possible. Let us consider the following word orders as examples of how
secondary adjunction is used for scrambling:
204
Kallmeyer Multicomponent TAGs with Shared Nodes
(4) ...dass er
1
[[das Fahrrad zu reparieren] zu versuchen]
2
t
1
dem Kunden t
2
...thathe thebike torepair totry thecustomer
verspricht
promises
(5) ...dass er [das Fahrrad zu reparieren]
1
dem Kunden [t
1
zu versuchen]
...thathethebike torepair thecustomer totry
verspricht
promises
(6) ...dass [das Fahrrad]
1
er
2
[[t
1
zu reparieren] zu versuchen]
3
t
2
dem Kunden t
3
...thatthebike he torepair totry thecustomer
verspricht
promises
In example (4), the versuchen-VP and er are scrambled.
13
Consequently, for versuchen and
er, the sets with two trees are used, whereas for all the other elements, the single trees
can be used. In example (5), the reparieren-VP is scrambled out of the versuchen-VP, with
dem Kunden intervening between the two. Therefore, the tree sets are used for reparieren
and dem Kunden. For versuchen, the single tree can be used, since the scrambling out of
versuchen is of depth one. In example (6), we have the same scrambling as in example (4),
and additionally, das Fahrrad is scrambled out of the reparieren-VP and the versuchen-VP
(depth two). Consequently, in this case one needs tree sets for Fahrrad, er, versuchen,and
reparieren.
Let us consider the analysis of example (4): Starting with verspricht, the single tree
for dem Kunden and the tree set for versuchen (with adjunction of the auxiliary tree at
the root) are added. This leads to the first tree in Figure 9. The VP nodes in boldface
type in the figure are shared by versuchen and verspricht; that is, they can be used for
further adjunction at the verspricht tree. (Of course, only the root node can be used for
adjunction, since the other nodes have NA constraints.) It does not matter in which order
er and zu reparieren are added. For er, the tree set is used. The auxiliary tree is secondarily
adjoined to the root node, and the initial tree is substituted for the NP
nom
node in the
verspricht tree. This leads to the second tree in Figure 9. For reparieren and das Fahrrad,
the single trees are added below the VP substitution node in the versuchen tree. The
corresponding SN-derivation structure (see Figure 9) contains the desired predicate-
argument dependencies. The TAG derivation tree is RSN-tree-local.
Next, let us consider example (5). Here, the single trees for er and versuchen are
added to verspricht. This leads to the first tree in Figure 10. The VP node in boldface type
in the figure belongs to verspricht and versuchen. It is next used for secondary adjunction
of dem Kunden to the verspricht tree. The initial tree is substituted at the NP
dat
slot. This
leads to the second tree. Here, the bold VP node belongs to verspricht, versuchen,and
Kunde. It is next used for secondary adjunction of the auxiliary tree of reparieren to
versuchen, while the initial tree is substituted for the VP leaf in the versuchen tree. This
13 Actually, er here is not really scrambled, but since in our formalism, scrambled elements attach at the left
of a VP, any other element even more to the left is treated as if it is scrambled (even if it depends on the
matrix verb).
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Computational Linguistics Volume 31, Number 2
Figure 9
Analysis of example (4).
Figure 10
Analysis of example (5).
leads to the third tree. After that, one needs only to add the single tree for das Fahrrad
to reparieren. Note that this is a derivation in which the foot node of the elementary tree
containing the lexical material does not dominate the tree with the empty word.
Now let us consider the derivation of example (6). Here, only for dem Kunden,the
single tree is added by substitution. In all other cases, the tree set is used with (primary
206
Kallmeyer Multicomponent TAGs with Shared Nodes
or secondary) adjunction at the root node of the already derived tree. This root node
consequently belongs to all verbs that have already occurred in the derivation and can
therefore be used to add arguments to any of them.
We leave it to the reader to verify that all word orders can be generated. This kind
of analysis also works for more than two embeddings.
Since all scrambled elements attach to a VP node in the elementary tree of the verb
they depend on, they cannot attach to the VP of a higher finite verb that embeds the
sentence in which the scrambling occurs. In this way, a barrier effect is obtained without
establishing any explicit barrier, as is done in V-TAG. Instead, this locality of scrambling
is a consequence of the form of the elementary trees and of the locality of the derivations.
Concerning adjunct scrambling, each adjunct has a single auxiliary tree as in stan-
dard TAG and additionally a set of two auxiliary trees, a lower auxiliary tree with an
empty word and a higher auxiliary tree with the adjunct. This is shown in Figure 11.
The internal VP node of the higher tree in the tree set serves as an adjunction site for the
lower parts of other adjuncts. Similarly, the elementary trees of verbs need an extra VP
node in order to adjoin adverbs.
For more analyses of scrambling, including scrambling in combination with extra-
position and topicalization, and also for an extension of the analysis presented here to
Korean data, see Kallmeyer and Yoon (2004).
3.2 Raising Verbs and Subject-Auxiliary Inversion
Other phenomena often mentioned in the TAG literature (see, e.g., Rambow, Vijay-
Shanker, and Weir 1995; Kulick 2000; Dras, Chiang, and Schuler 2004) as being
problematic for TAG and tree-local MCTAG are sentences with raising verbs and
subject-auxiliary inversion, as in examples (7) and (8):
(7) Does Gabriel seem to be likely to eat gnocchi?
(8) What does John seem to be certain to like?
The standard TAG analyses of examples (7) and (8) (see Figure 12 for the analysis of
example (8)) start with the eat and like tree, respectively, adjoin an auxiliary tree for
likely and certain, respectively, and then add the trees for does and seem, respectively. If
we assume that these trees are in the same elementary tree set, then this last derivation
step is nonlocal, since the does tree adjoins to eat and like, respectively, while the seem
tree adjoins to likely and certain, respectively. Though different from scrambling, this
problem seems to be of a similar nature, and formalisms that have been proposed for
scrambling have also been used to treat these examples (see Kulick 2000).
RSN-MCTAG allows us to analyze examples (7) and (8) in a way that puts does and
seem into a single elementary tree set: After having adjoined to be likely and to be certain,
Figure 11
Trees for adjuncts.
207
Computational Linguistics Volume 31, Number 2
Figure 12
Derivation for (8).
respectively, the root nodes of the adjoined trees are considered still to be part of the
elementary trees of eat and like, respectively. These elementary trees can then be used to
add the elementary tree set for does and seem: Both auxiliary trees are adjoined to these
trees. Figure 12 shows the corresponding SN-derivation structure.
4. RSN-MCTAG and Range Concatenation Grammar
In the following, we show that for each RSN-MCTAG of a certain type (i.e., with
an additional restriction), a weakly equivalent simple range concatenation grammar
(Boullier 1999, 2000) can be constructed. It has been shown that RCGs generate ex-
actly the class of all polynomially parsable languages (Bertsch and Nederhof 2001;
appendix A). Furthermore, as shown in Boullier (1998b), simple RCGs in particular
are even weakly equivalent to linear context-free rewriting systems (Weir 1988). As a
consequence, one obtains that the languages generated by simple RSN-MCTAGs are
mildly context-sensitive. This last property was introduced in Joshi (1985). It includes
formalisms that are polynomially parsable, are semilinear, and allow only a limited
number of crossing dependencies. (We do not give formal definitions of mild context-
sensitivity and of LCFRS, since we do not need these definitions in this article.)
Concerning RSN-MCTAGs in general, that is, without any further restriction, we are
almost sure that they are not mildly context-sensitive. Perhaps they can even generate
languages that are not in the class of languages generated by RCGs.
4.1 Range Concatenation Grammars
This section defines range concatenation grammars.
14
Definition 7
A range concatenation grammar is a tuple G =〈N, T, V, S, P〉 such that
a114
N is a finite set of predicates, each with a fixed arity;
a114
T and V are disjoint finite sets of terminals and of variables;
a114
S ∈ N is the start predicate, a predicate of arity 1;
a114
P is a finite set clauses of the form A
0
(x
01
,..., x
0a
0
) → epsilon1,
or A
0
(x
01
,..., x
0a
0
) → A
1
(x
11
,..., x
1a
1
)...A
n
(x
n1
,..., x
na
n
),
with n ≥ 1andA
i
∈ N, x
ij
∈ (T ∪ V)
∗
and a
i
being the arity of A
i
.
14 Since throughout the article, we use only positive RCGs; whenever we say “RCG,” we actually mean
“positive RCG.”
208
Kallmeyer Multicomponent TAGs with Shared Nodes
When applying a clause with respect to a string w = t
1
···t
n
, the arguments of the
predicates in the clause are instantiated with substrings of w, more precisely, with the
corresponding ranges. A range 〈i, j〉 with 0 ≤ i < j ≤ n corresponds to the substring
between positions i and j, that is, to the substring t
i+1
···t
j
.Ifi = j, then 〈i, j〉 corresponds
to the empty string epsilon1.Ifi > j, then 〈i, j〉 is undefined.
Definition 8
For a given clause, an instantiation with respect to a string w = t
1
...t
n
consists of
a function f : {t
prime
|t
prime
is an occurrence of some t ∈ T in the clause}∪V →{〈i, j〉|i ≤ j, i,
j ∈ IN} such that
a114
for all occurrences t
prime
of a t ∈ T in the clause: f (t
prime
):=〈i, i + 1〉 for some
i,0≤ i < n, such that t
i
= t;
a114
for all v ∈ V: f (v) =〈j, k〉 for some 0 ≤ j ≤ k ≤ n;
a114
if consecutive variables and occurrences of terminals in an argument in the
clause are mapped to 〈i
1
, j
1
〉,...,〈i
k
, j
k
〉 for some k, then j
m
= i
m+1
for
1 ≤ m < k. By definition, we then state that f maps the whole argument to
〈i
1
, j
k
〉.
The derivation relation is defined as follows. For a predicate A of arity k, a clause
A(...) → ..., and ranges 〈i
1
, j
1
〉,...,〈i
k
, j
k
〉 with respect to a given w: If there is an instan-
tiation of this clause with left-hand side A(〈i
1
, j
1
〉,...,〈i
k
, j
k
〉), then A(〈i
1
, j
1
〉,...,〈i
i
, j
k
〉)
can be replaced with the right-hand side of this instantiation.
The language of an RCG G is the set of strings that can be reduced to the empty
word, that is, {w|S(〈0,|w|〉)
∗
⇒ epsilon1 with respect to w}.
15
An RCG with maximal predicate
arity n is called an RCG of arity n.
For illustration, let us consider a sample RCG: The RCG with N ={S, A, B},
T ={a, b}, V ={X, Y, Z}, start predicate S, and clauses S(XYZ) → A(X, Z) B(Y),
A(aX, aY) → A(X, Y), B(bX) → B(X), A(epsilon1,epsilon1) → epsilon1, B(epsilon1) → epsilon1 has the string language
{a
n
b
k
a
n
|k, n ∈ IN}. Consider the reduction of w = aabaa:
We start from S(〈0, 5〉). First we can apply the following clause instantiation:
S(XY Z) → A(X, Z) B(Y)
〈0,2〉〈2,3〉〈3,5〉〈0,2〉〈3,5〉〈2,3〉
aa b aa aa aa b
With this instantiation, S(〈0, 5〉) ⇒ A(〈0, 2〉,〈3, 5〉)B(〈2, 3〉). Then
B(bX) → B(X)
〈2,3〉〈3,3〉〈3,3〉
b epsilon1epsilon1
15 |w| is the length of the word w; that is, the range 〈0,|w|〉 with respect to w corresponds to the whole
word w.
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Computational Linguistics Volume 31, Number 2
and B(epsilon1) → epsilon1 lead to A(〈0, 2〉,〈3, 5〉)B(〈2, 3〉) ⇒ A(〈0, 2〉,〈3, 5〉)B(〈3, 3〉) ⇒ A(〈0, 2〉,
〈3, 5〉). Next,
A(aX a Y) → A(X, Y)
〈0,1〉〈1,2〉〈3,4〉〈4,5〉〈1,2〉〈4,5〉
aaaa aa
leads to A(〈0, 2〉,〈3, 5〉) ⇒ A(〈1, 2〉,〈4, 5〉). Then
A(aX a Y) → A(X, Y)
〈1,2〉〈2,2〉〈4,5〉〈5,5〉〈2,2〉〈5,5〉
a epsilon1 a epsilon1epsilon1epsilon1
and A(epsilon1,epsilon1) → epsilon1 lead to A(〈1, 2〉,〈4, 5〉) ⇒ A(〈2, 2〉,〈5, 5〉) ⇒ epsilon1.
AnRCGissaidtobenoncombinatorial if each of the arguments in the right-hand
sides of the clauses are single variables. It is said to be linear if no variable appears more
than once in the left-hand sides of the clauses and no variable appears more than once
in the right-hand side of the clauses. It is said to be nonerasing if for each clause, each
variable occurring in the left-hand side occurs also in the right-hand side and vice versa.
It is said to be simple if it is noncombinatorial, linear, and nonerasing.
Simple RCGs and LCFRSs are equivalent (Boullier 1998b).
4.2 Relation between RSN-MCTAG and Simple RCG
The goal of this section is to construct an equivalent simple RCG for a given RSN-
MCTAG. In order to be able to perform this construction, in the following we further
constrain the formalism of RSN-MCTAG by defining RSN-MCTAG of a specific arity
n. For this version of RSN-MCTAG, the construction of an equivalent simple RCG is
possible.
First, let us sketch the general idea of the transformation from TAG to RCG (see
Boullier 1998a). The RCG contains predicates 〈α〉(X)and〈β〉(L, R) for initial and aux-
iliary trees, respectively. X covers the yield of α and all trees added to α,andL and R
cover those parts of the yield of β (including all trees added to β) that are to the left
and the right of the foot node of β. The clauses in the RCG reduce the argument(s)
of these predicates by identifying those parts that come from the elementary tree α/β
itself and those parts that come from one of the elementary trees added by substitution
or adjunction. A sample TAG with an equivalent RCG is shown in Figure 13.
For the construction of an equivalent RCG from a given RSN-MCTAG, we follow
the same ideas while considering a secondary adjunction of β at some γ as adjunction
at γ and not as adjunction at the elementary tree that is the mother node of β in the
TAG derivation tree. There are two main differences between RSN-MCTAG and TAG
that influence the construction of an equivalent RCG.
First, more than one tree can be added to a node. Therefore we allow predicates
of the form 〈αβ
1
...β
k
〉 and 〈β
0
β
1
...β
k
〉. The first means that at the node in question,
first α was added by substitution, and then β
1
...β
k
(in this order) were secondarily
adjoined. The second means that at the node in question (an internal node), first β
0
was
primarily adjoined, and then β
1
...β
k
were secondarily adjoined. Since the number of
secondary adjunctions at a node is limited by some constant depending on the grammar
210
Kallmeyer Multicomponent TAGs with Shared Nodes
Figure 13
A sample TAG and an equivalent RCG.
(see Lemma 3), k is limited as well, and therefore this extension with respect to TAG adds
only a finite number of predicates.
Second, the contribution of an elementary tree α/β including the trees added
to it can be separated into arbitrarily many parts. Since each of the arguments
of the predicates in the RCG has to cover a true substring of the input string,
one needs predicates of arbitrary arities, namely, 〈α...〉(L
n
,..., L
1
, X, R
1
,..., R
n
)and
〈β...〉(L
n
,..., L
1
, L
0
, R
0
, R
1
,..., R
n
), for the case where n auxiliary trees were added at
the root of α/β that were actually secondarily adjoined at some higher tree such that
these n trees separate the contribution of α/β into 2n + 1/2n + 2 parts, respectively.
This extension is problematic, since it leads to an RCG with predicates of arbitrary arity:
a dynamic RCG (Boullier 2001), a variant of RCG that is not polynomially parsable and
that we therefore want to avoid. For this reason, we need an additional constraint on
theRSN-MCTAGsweemploy.
An example in which the contribution of an elementary tree is separated into
three different parts is example (9), analyzed with the RSN-MCTAG in section 3.1
(see Figure 14). In the derived tree, the VP das Fahrrad zu reparieren zu versuchen (the
broken triangles), which is the contribution of versuchen, is separated into three parts,
Figure 14
Analysis of example (9).
211
Computational Linguistics Volume 31, Number 2
since reparieren secondarily adjoins at versuchen and das Fahrrad secondarily adjoins at
reparieren.
(9) ...dass[dasFahrrad]
1
er [t
1
zu reparieren]
2
dem Kunden [t
2
zu versuchen]
...that thebike he torepair thecustomer totry
verspricht
promises
The crucial point in example (9) is that in the SN-derivation structure (see Figure 14),
there are two crossings of secondary edges inside one group of secondary links. This
means that the contribution of versuchen is interrupted twice by arguments of verspricht
(by Kunde and er). In order to avoid predicates of arbitrary arity, we therefore limit
the number of crossings of secondary links. We define the arity of an RSN-MCTAG
depending on the maximal number of crossings that are allowed.
First, we define special subgraphs of the SN-derivation structure, secondary
groups. These are subgraphs consisting of a chain of one primary substitu-
tion/adjunction and subsequent adjunctions at root or foot nodes such that there are
secondary adjunctions along the whole chain. For example, the nodes verspricht, zu
versuchen, Kunde, zu reparieren, er,andFahrrad in the SN-derivation structure in Figure 14
form such a group. For an SN-derivation structure of a certain arity, the number of
crossings of secondary edges inside a single secondary group is then limited: For an
SN-derivation structure of arity n, the number of crossings of secondary edges per
secondary group is limited to
n
2
− 1. In other words, if i is the maximal number of
crossings, then 2(i + 1) is the arity of the grammar. Of course, the arity is chosen such
that an equivalent RCG of the same arity can be constructed. TAG, for example, is a
grammar with 0 crossings, that is, an arity 2(0 + 1) = 2 if the grammar is viewed as an
SN-MCTAG, and the corresponding RCG is actually of arity 2.
Definition 9
Let D
SN
=〈N,E〉 be a SN-derivation structure.
1. 〈N
prime
,E
prime
〉 is a secondary group in D
SN
iff
a114
N
prime
={n
0
, n
1
,..., n
k
}⊆N for some k > 1 such that there are
primary edges 〈n
i
, n
i+1
, p
i
〉 for 0 ≤ i < k with p
i
∈ IN ;
a114
E
prime
⊆ E such that for all n, n
prime
∈ N, p ∈ IN w i t h 〈n, n
prime
, p〉∈E:if
n, n
prime
∈ N
prime
, then 〈n, n
prime
, p〉∈E
prime
;
a114
for all i,0< i < k, there are i
1
, i
2
with i
1
≤ i ≤ i
2
, i
1
negationslash= i
2
, such that
〈n
i
1
, n
i
2
, p〉∈E
prime
is a secondary edge in D for some p ∈ IN .
2. D
SN
is of arity n iff for each secondary group 〈N
prime
,E
prime
〉 in D
SN
with primary
edges 〈n
0
, n
1
, p
0
〉, 〈n
1
, n
2
, p
1
〉, ..., 〈n
k−1
, n
k
, p
k−1
〉 as above, there are at most
i ≤
n
2
− 1 pairwise different sets of the form {j
0
, j
1
, j
2
, j
3
} such that
j
0
< j
1
< j
2
< j
3
and there are secondary edges 〈n
j
0
, n
j
2
, p
1
〉 and 〈n
j
1
, n
j
3
, p
2
〉
for some p
1
, p
2
∈ IN .
Definition 10
Let G be an MCTAG, n ≥ 1. G is a restricted tree-local MCTAG with shared nodes of
arity n iff the set of trees generated by G, L
T
(G), is defined as the set of those trees that
can be derived in G with an RSN-tree-local multicomponent TAG derivation tree such
that the corresponding SN-derivation structure is of arity n.
212
Kallmeyer Multicomponent TAGs with Shared Nodes
Figure 15
Sample RSN-MCTAG of arity four.
Consider a simple example of a construction of an equivalent RCG for a given
RSN-MCTAG. We choose an RSN-MCTAG of arity four, and we see that the arity
of the corresponding RCG is four as well. The RSN-MCTAG is shown in Figure 15.
Whether this grammar is considered to be a general RSN-MCTAG or an RSN-MCTAG
of arity four does not matter in this case, since even in the general case, all possible
SN-derivation structures are of arity four. However, in the case of other RSN-MCTAGs,
the restriction to a certain arity might exclude certain TAG derivation trees and thereby
decrease the language generated by the grammar.
The language generated by the RSN-MCTAG in Figure 15 is {er zu kommen (zu
versuchen)
∗
verspricht, zu kommen (zu versuchen)
+
er (zu versuchen)
∗
verspricht}.TheSN-
derivation structures corresponding to the different strings are shown in Figure 15. The
last one contains one crossing of secondary links; that is, the RSN-MCTAG is of arity
four.
Now let us look at the corresponding RCG. Since the arity of the RSN-MCTAG is
four, the predicates of the corresponding RCG are of arity three (for initial trees) and
four (for auxiliary trees).
The contribution of α
1
is never separated into parts, therefore the first and the third
arguments of the predicate 〈α
1
〉 are always epsilon1. Looking at the SN-derivation structures
in Figure 15, we have three different possibilities for 〈α
1
〉:
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Computational Linguistics Volume 31, Number 2
〈α
1
〉(epsilon1,LNVversprichtR,epsilon1) →〈β
2
〉(epsilon1,L,R,epsilon1)〈α
β
2
〉(epsilon1,N,epsilon1)〈α
2
〉(epsilon1,V,epsilon1)|
〈β
1
β
2
〉(epsilon1,L,R,epsilon1)〈α
β
2
〉(epsilon1,N,epsilon1)〈α
β
1
〉(epsilon1,V,epsilon1)|
〈β
2
β
1
〉(epsilon1,L,R,epsilon1)〈α
β
2
〉(epsilon1,N,epsilon1)〈α
β
1
〉(epsilon1,V,epsilon1)
The interesting part of the grammar is the clauses for 〈β
1
β
2
〉, where two trees were
added to the same node and further adjunctions at the root of β
1
are possible. The point
is that the part covered by β
1
and the trees added to it can be separated into different
substrings. This leads to
〈β
1
β
2
〉(epsilon1,L
1
L
2
L
3
,R
3
R
2
R
1
,epsilon1) →〈β
1
〉(L
1
,L
3
,R
3
,R
1
)〈β
2
〉(epsilon1,L
2
,R
2
,epsilon1)
〈β
1
β
2
〉(L
1
,L
2
L
3
,R
3
R
2
,R
1
) →〈β
1
〉(L
1
,L
3
,R
3
,R
1
)〈β
2
〉(epsilon1,L
2
,R
2
,epsilon1)
〈β
1
β
2
〉(L
1
L
2
,L
3
,R
3
,R
2
R
1
) →〈β
1
〉(L
1
,L
3
,R
3
,R
1
)〈β
2
〉(epsilon1,L
2
,R
2
,epsilon1)
Concerning 〈β
2
β
1
〉, the contribution of β
2
cannot be separated into different parts,
since nothing can be adjoined to β
2
. Consequently
〈β
2
β
1
〉(epsilon1,L
1
L
2
,R
2
R
1
,epsilon1) →〈β
2
〉(epsilon1,L
2
,R
2
,epsilon1)〈β
1
〉(epsilon1,L
1
,R
1
,epsilon1)
〈β
2
β
1
〉(L
1
,L
2
,R
2
,R
1
) →〈β
2
〉(epsilon1,L
2
,R
2
,epsilon1)〈β
1
〉(epsilon1,L
1
,R
1
,epsilon1)
Concerning 〈β
1
〉(L
1
, L
3
, R
3
, R
1
), either (L
1
, R
1
) cover something adjoined to β
1
(this
can only be β
1
), or (L
1
, R
1
) cover something adjoined to something ...adjoined to β
1
.
In this case, (L
3
, R
3
) cover β
1
and β
1
adjoined to its root, respectively. This leads to
〈β
1
〉(L
1
,L
2
V zu versuchen,R
2
,R
1
) →〈β
1
〉(L
1
,L
2
,R
2
,R
1
)〈α
β
1
〉(epsilon1,V,epsilon1)
If the inner parts are empty, the outer parts are moved:
〈β
1
〉(L,epsilon1,epsilon1,R) →〈β
1
〉(epsilon1,L,R,epsilon1)
Furthermore, there is a clause for 〈β
1
〉 without adjunction at the root:
〈β
1
〉(epsilon1,V zu versuchen,epsilon1,epsilon1) →〈α
2
〉(epsilon1,V,epsilon1)
For elementary trees where nothing needs to be added, simple epsilon1-clauses are introduced:
〈α
2
〉(epsilon1, zukommen,epsilon1) →epsilon1 〈α
β
1
〉(epsilon1,epsilon1,epsilon1) →epsilon1
〈β
2
〉(epsilon1,er,epsilon1,epsilon1) →epsilon1 〈α
β
2
〉(epsilon1,epsilon1,epsilon1) →epsilon1
In order to see how the RCG simulates the RSN-MCTAG, let us consider the deriva-
tion for zu kommen zu versuchen er zu versuchen verspricht:
〈α
1
〉(epsilon1,zukommenzuversuchener zuversuchenverspricht,epsilon1)
⇒〈β
1
β
2
〉(epsilon1,zukommenzuversuchener zuversuchen,epsilon1,epsilon1)〈α
β
2
〉(epsilon1,epsilon1,epsilon1)〈α
β
1
〉(epsilon1,epsilon1,epsilon1)
∗
⇒〈β
1
〉(zukommenzuversuchen, zuversuchen,epsilon1,epsilon1)〈β
2
〉(epsilon1,er,epsilon1,epsilon1)
∗
⇒〈β
1
〉(zukommenzuversuchen,epsilon1,epsilon1,epsilon1)〈α
β
1
〉(epsilon1,epsilon1,epsilon1)
∗
⇒〈β
1
〉(epsilon1,zukommenzuversuchen,epsilon1,epsilon1)
⇒〈α
2
〉(epsilon1,zukommen,epsilon1) ⇒epsilon1
214
Kallmeyer Multicomponent TAGs with Shared Nodes
This example should give an idea of how an equivalent RCG for a given RSN-MCTAG
of arity n can be constructed.
As already mentioned, in an RSN-MCTAG, the number of substitutions and
(primary or secondary) adjunctions that can occur at each node is limited (see Lemma 3).
Therefore, the number of predicates needed in the corresponding RCG is limited as well.
Furthermore, in an RSN-MCTAG of arity n, the contribution of an elementary tree is
separated into at most n parts. This still needs to be shown:
Lemma 4
Let G be an RSN-MCTAG of arity n. Then for all w in the string language of G and for
all elementary trees γ used to derive w in G, the contribution of γ, that is, the yield of γ
and everything added to γ, is separated into at most n parts.
The proof is given in the appendix.
Theorem 1
For each RSN-MCTAG G of arity n, a simple RCG G
prime
of arity n can be constructed such
that L(G) = L(G
prime
).
The construction algorithm and a sketch of the proof are presented in the appendix. As
a consequence of this theorem, the following corollary holds:
Corollary
For a given n, the string languages generated by RSN-MCTAGs of arity n are mildly
context-sensitive, and they are in particular polynomially parsable.
Since we have shown that for RSN-MCTAG with a fixed arity, one obtains gram-
mars that are LCFRSs, we know that we can even construct a weakly equivalent set-
local MCTAG. This set-local MCTAG, however, does not present an alternative to the
RSN-MCTAG with fixed arity: It is very large, containing a large number of elementary
trees per tree set (the number depends on the arity of the grammar) and, furthermore, a
large number of trees without lexical material and a large number of internal nodes that
are needed only to provide adjunction sites.
An example is the set-local MCTAG in Figure 16. It is weakly equivalent to the
RSN-MCTAG of arity four in Figure 15, and it even gives the correct dependency
structure. The verspricht tree contains several VP nodes that are needed in order to
provide adjunction sites for the different parts of er and versuchen.Theversuchen tree
set needs an extra auxiliary tree that provides an additional VP node for adjunction
and has to be separated from the tree containing versuchen, since the contribution
of versuchen might be separated into different parts. Of course this little grammar
is still simple, since there are almost no possibilities of adjoining different trees at
the same node or of separating the contribution of one lexical item into different
parts.
As we have seen in Lemma 4, the linguistic signification of restricting the arity
of the grammar to some n is that the lexical material containing a verb, all its argu-
ments (including arguments and adjuncts of these arguments, etc.), and all its adjuncts
cannot be separated into more than n discontinuous substrings in the whole sentence.
For example, an RSN-MCTAG of arity two with elementary tree sets similar to those
proposed above for scrambling would not be able to analyze example (9). However,
RSN-MCTAGs of arity n for some sufficiently large fixed n can perhaps even describe
215
Computational Linguistics Volume 31, Number 2
Figure 16
Equivalent set-local MCTAG for the RSN-MCTAG from Figure 15.
all cases of scrambling: See again the analysis of example (9) in Figure 14. Here, the
contribution of versuchen and its arguments is split only by other elements secondarily
adjoined to verspricht. If only a limited number of such secondary adjunctions were
possible (this is the case), and if none of these other secondarily adjoined elements
allowed for further secondary adjunctions at its root or foot node (this still needs to
be investigated), then the number of crossings might be limited. We leave this issue for
further research.
Even if RSN-MCTAG with a fixed arity could not analyze all scrambling data, based
on empirical studies, n could be chosen sufficiently great such that the grammar would
cover all scrambling cases that one assumes to occur.
16
The important point is that the
complexity limit given by the fixed n is variable; that is, an arbitrary n can be chosen.
This is different from TAG, for example, in which the limit is fixed (assuming, of course,
that we desire only analyses respecting the CETM). In this sense one can say that RSN-
MCTAG can analyze scrambling in general.
5. Conclusion and Future Work
This article addresses the problem of scrambling in tree-adjoining grammar, a formalism
known not to be powerful enough to treat scrambling phenomena. In order to keep
the advantages of TAG while being able to analyze scrambling, a local TAG variant is
proposed that is based on the notion of node sharing, so-called (restricted) tree-local
multicomponent TAG with node sharing. RSN-MCTAG is a true extension of TAG in
the sense that the formalism can generate all tree-adjoining languages. The analysis of
some German scrambling data is sketched in order to show that this TAG extension can
treat scrambling.
Then, RSN-MCTAGs of specific arities are defined, and it is shown that for each
RSN-MCTAG of a fixed arity n, an equivalent simple RCG of arity n can be constructed.
Simple RCGs are mildly context-sensitive and in particular polynomially parsable and
therefore, this also holds for RSN-MCTAGs of a fixed arity. RSN-MCTAGs of arity n
perhaps cannot analyze all scrambling phenomena but, if the n is appropriately chosen,
it can analyze an arbitrarily large set.
16 Joshi, Becker, and Rambow (2000) even argue that there might be a competence limit regarding the
complexity of scrambling data. However, we do not discuss this issue here.
216
Kallmeyer Multicomponent TAGs with Shared Nodes
The scrambling data analyzed in section 3 present just a small part of the possible
scrambling configurations. As already noted, this article does not present an exhaustive
treatment of the phenomenon. Even though the examples we looked at indicate that
RSN-MCTAGs are able to deal with scrambling, an exhaustive analysis of a larger
amount of data still needs to be done, in particular, of scrambling in combination
with other “movements” that cause word order variations, such as topicalization or
extraposition. A first proposal in this direction can be found in Kallmeyer and Yoon
(2004), but this proposal does not cover all phenomena one needs to take into account.
So this is still an important issue for further research.
A formal issue one would like to see investigated more in detail is the relations
between the different types of MCTAG. We have shown that the languages of RSN-
MCTAG with fixed arity are in the class of set-local MCTALs. Furthermore, general
SN-MCTAG can generate languages that cannot be generated by set-local MCTAG.
However, this leaves open many interesting questions concerning the relations be-
tween set-local MCTAG and non-local MCTAG and the different formalisms defined
in this article, namely, RSN-MCTAG of fixed arity and RSN-MCTAG and MCTAG with
SN-tree-local and SN-set-local derivations. We plan to address these questions in the
future.
Appendix: Proofs
Proof of Lemma 1
Let G =〈I, A, N, T,A〉 be an MCTAG, G
TAG
:=〈I, A, N, T〉.LetD =〈N,E〉 be a derivation
tree in G
TAG
with corresponding derived tree t ∈ L(G
TAG
).
1. First show ⇒ of the iff: Let D be a TAG derivation tree with t ∈ L(G). It is
immediate that the root of D is an instance of an initial tree and that all other nodes
are elements of instances of elementary tree sets.
Assume that
a114
either there is an instance Γ of elementary tree sets from A such that there
are γ
1
,γ
2
∈Γ,withγ
1
∈ N and γ
2
negationslash∈ N,andγ
1
is not the root of D. ⇒ it is
not possible that Γ has been used in one of the multicomponent derivation
steps in the course of the derivation of t. Contradiction.
a114
or there is an instance Γ of an elementary tree set such that there are
γ
1
,γ
2
∈Γ, γ
1
negationslash= γ
2
,with〈γ
1
,γ
2
〉∈D
D
⇒γ
2
has been added to a tree
derived from γ
1
. Contradiction to condition that all elements of Γ must
have been added simultaneously.
a114
or there are pairwise different instances Γ
1
,Γ
2
,...,Γ
n
of elementary tree
sets from A such that there are γ
(i)
1
,γ
(i)
2
∈Γ
i
,1≥ i ≥ n,with
〈γ
(1)
1
,γ
(n)
2
〉∈D
D
and 〈γ
(i)
1
,γ
(i−1)
2
〉∈D
D
for 2 ≥ i ≥ n. ⇒γ
(i)
1
was added
before γ
(i−1)
2
for 2 ≤ i ≤ n, and since all elements from Γ
i
must be added
simutaneously for 1 ≤ i ≤ n, Γ
n
was added before Γ
1
. ⇒〈γ
(1)
1
,γ
(n)
2
〉 negationslash∈ D
D
.
Contradiction.
Consequently, D satisfies (MC1)–(MC3).
2. Then show ⇐ of the iff: Let D be a derivation tree in G
TAG
satisfying (MC1)–
(MC3).
217
Computational Linguistics Volume 31, Number 2
There are different orderings of the derivation steps in D possible: Let the node
positions on the derived tree be pairs 〈γ, p〉,withγ being an instance of an elementary
tree and p being a position in γ. Every top-down order read off D (no matter whether
[partly] depth first or not and whether left to right or right to left) is a possible deriva-
tion order in G
TAG
for the derivation tree D, since in order to perform the derivation
step ...[〈γ
1
, p〉,γ
2
] corresponding to an edge 〈γ
1
,γ
2
, p〉 in D, one needs only to ensure
that γ
1
(i.e., the mother node of γ
2
) has already been added.
Because of (MC1), the root of D is an initial tree, and the set of all other nodes in D
can be partitioned into pairwise different instances of elementary tree sets.
To show: There is a top-down traversal of D such that the traversal starts with an
initial tree and then there is always one instance Γ of an elementary set whose members
are visited next in any order (i.e., simultaneously).
The top-down traversal has to start with the root node (i.e., an initial tree α). Assume
that at some point of the traversal, the choice of a new instance of an elementary set to
be visited next is not possible. ⇒ for each set Γ that has not been visited yet, there is
at least one γ ∈Γ whose mother node has not been visited yet (otherwise Γ could be
visited next).
Pick an unvisitedΓ
1
with at least one γ
(1)
1
∈Γ
1
whose mother node has been visited.
Assume γ
(1)
2
∈Γ
1
with mother not yet visited. Suppose γ
(2)
1
to be the highest unvisited
node dominating γ
(1)
2
. Since γ
(2)
1
negationslash= γ
(1)
2
and 〈γ
(2)
1
,γ
(1)
2
〉∈D
D
,(with(MC2))γ
(2)
1
∈Γ
2
negationslash=
Γ
1
.
Then there is a γ
(2)
2
∈Γ
2
with unvisited mother such that
(a) either 〈γ
(1)
1
,γ
(2)
2
〉∈D
D
. Contradiction to (MC3) with n = 2.
γ
(1)
1
γ
(2)
1
γ
(2)
2
γ
(1)
2
(b) or 〈γ
(1)
1
,γ
(2)
2
〉 /∈ D
D
. Because of (MC2), 〈γ
(2)
1
,γ
(2)
2
〉 /∈ D
D
.Letγ
(3)
1
∈Γ
3
be the
highest unvisited node dominating γ
(2)
2
. Because of (MC2), Γ
3
negationslash=Γ
2
,and
because of (MC3), Γ
3
negationslash=Γ
1
.
γ
(1)
1
γ
(2)
1
γ
(3)
1
γ
(1)
2
γ
(2)
2
In the (b) case, there is a γ
(3)
2
∈Γ
3
with unvisited mother node. Because of (MC2)
and (MC3), 〈γ
(1)
1
,γ
(3)
2
〉 /∈ D
D
, 〈γ
(2)
1
,γ
(3)
2
〉 /∈ D
D
,and〈γ
(3)
1
,γ
(3)
2
〉 /∈ D
D
. Then there is a high-
est unvisited node γ
(4)
1
∈Γ
4
dominating γ
(3)
2
with Γ
4
negationslash=Γ
3
,Γ
4
negationslash=Γ
2
,andΓ
4
negationslash=Γ
1
. And
there is a γ
(4)
2
∈Γ
4
with unvisited mother node.
In general, for each of the Γ
n
,1≤ n,withγ
(n)
1
,γ
(n)
2
as above, the situation is
as follows: Γ
n
negationslash=Γ
i
for 1 ≤ i < n (otherwise contradiction to (MC2) or (MC3)) and
〈γ
(i)
1
,γ
(n)
2
〉 /∈ D
D
for i ≤ n (otherwise contradiction to (MC2) for i = n or to (MC3)
218
Kallmeyer Multicomponent TAGs with Shared Nodes
for i negationslash= n). Consequently, there is always a new Γ
n+1
, with a new γ
(n+1)
1
being the
highest unvisited node dominating γ
(n)
2
and γ
(n+1)
2
being a node with unvisited
mother.
γ
(1)
1
γ
(2)
1
γ
(n)
1
γ
(n+1)
1
γ
(1)
2
... γ
(n−1)
2
γ
(n)
2
Contradiction to the finiteness of the number of nodes in D. ⇒ there is a top-down
traversal of D that corresponds to a multicomponent derivation in G in the sense that it
allows us to visit the instances of elementary tree sets one after the other. squaresolid
Proof of Lemma 2
Only the uniqueness needs to be shown.
Let G =〈I, A, N, T,A〉 be an RSN-MCTAG. Let D =〈N,E〉 be a TAG derivation tree
in G.
Assume that there is an instance {γ
1
,...,γ
n
} of an elementary tree set such that
there are γ,γ
prime
with γ negationslash= γ
prime
and 〈γ,γ
1
〉,...,〈γ,γ
n
〉,〈γ
prime
,γ
1
〉,...,〈γ
prime
,γ
n
〉∈SN
D
and there
are i, j,1≤ i, j ≤ n with 〈γ,γ
i
〉,〈γ
prime
,γ
j
〉∈P
D
.
⇒ since 〈γ
prime
,γ
j
〉∈P
D
and 〈γ,γ
j
〉∈D
D
with γ negationslash= γ
j
, there is a γ
primeprime
with 〈γ,γ
primeprime
〉∈P
D
and 〈γ
primeprime
,γ
prime
〉∈D
D
. Furthermore, 〈γ
i
,γ
primeprime
〉 /∈ D
D
(otherwise 〈γ
i
,γ
j
〉∈D
D
which would
contradict (MC2)) and 〈γ
primeprime
,γ
i
〉 /∈ D
D
(otherwise, since γ
i
negationslash= γ
primeprime
, 〈γ,γ
i
〉 /∈ P
D
). Conse-
quently 〈γ
prime
,γ
i
〉 /∈ D
D
. Contradiction to assumption. squaresolid
Proof of Lemma 4
Let G be an RSN-MCTAG of arity n, w ∈ L(G), such that the elementary tree γ was used
to derive w. Assume that the contribution of γ is separated into m > n parts.
Then the SN-derivation structure for this derivation is as shown in Figure 17.
Consequently, there are at least ceilingleft
m
2
− 1ceilingright crossings, and the arity of G is (ceilingleft
m
2
− 1ceilingright+
1)· 2. ⇒ if m is even, G is of arity m,andifm is odd, G is of arity m + 1. This is a
contradiction to the assumption that the arity of G is n < m. squaresolid
Proof of Theorem 1
For reasons of space, we do not give the whole proof of the theorem but re-
strict ourselves to the construction algorithm and a rough outline of the rest of the
proof.
Construction algorithm. Let G be an RSN-MCTAG of arity n.
Construction of a weakly equivalent RCG G
prime
:
The terminals and nonterminals will be implicitly defined by the clauses of the
grammar.
Predicates: Let k
1
be the maximal number of nodes in an elementary tree in G and
k
2
be the maximal number of trees in an elementary tree set. k := k
1
(k
2
− 1).
219
Computational Linguistics Volume 31, Number 2
Figure 17
SN-derivation structure for proof of Lemma 4.
a114
There is a unary predicate S.
a114
Each 〈α〉 and each 〈αβ
1
···β
l
〉,withl < k, α an initial tree, and β
1
,...,β
l
auxiliary trees is an n − 1-ary predicate.
a114
Each 〈β〉 and each 〈ββ
1
···β
l
〉 with l < k and β,β
1
,...,β
l
auxiliary trees is
an n-ary predicate.
Define the decoration string σ
γ
of an elementary tree γ as in Boullier (1999), except
that a root node µ has n variables, L
µ
1
...L
µ n
2
on the left and R
µ n
2
...R
µ
1
on the right.
Every other internal node µ has two variables L
µ
and R
µ
, and each substitution node
has one variable X. In a top-down, left-to-right traversal, the left variables are collected
during the top-down traversal, the terminals and variables of substitution nodes are
collected while visiting the leaves, and the right variables are collected during bottom-
up traversal.
Construction of the clauses:
In the following, P(epsilon1,...,epsilon1, x,epsilon1,...,epsilon1) signifies that x is the
n
2
th argument,
and P(epsilon1,...,epsilon1, x
1
, x
2
,epsilon1,...,epsilon1) signifies that x
1
is the
n
2
th and x
2
the (
n
2
+ 1)th
argument.
220
Kallmeyer Multicomponent TAGs with Shared Nodes
(1) Predicate S
For each initial α, there is a clause
S(X) →〈α〉(epsilon1,...,epsilon1,X,epsilon1,...,epsilon1)
(2) Predicates 〈γ〉
For each elementary γ: lhs := σ
γ
, rhs := epsilon1.
For each combination of substitutions and adjunctions at γ, with substitutions at all
substitution nodes and adjunctions at all internal nodes, with obligatory adjunction that
respects the conditions for restricted tree-local multicomponent derivation:
For all nodes µ in γ:
a114
If µ is an internal node that is not the root, and no adjunction takes place at
µ, then delete L
µ
and R
µ
in lhs.
a114
If µ is the root node and no adjunction takes place at µ, then delete
L
µ
1
···L
µ n
2
and R
µ n
2
···R
µ
1
in lhs.
a114
If only the initial tree α is substituted at µ,
rhs :=〈α〉(epsilon1,...,epsilon1, S
µ
,epsilon1,...,epsilon1)rhs.
a114
If α is substituted at µ and then β
1
,...,β
m
(in that order) are secondarily
adjoined at µ, then rhs :=〈αβ
1
···β
m
〉(epsilon1,...,epsilon1, S
µ
,epsilon1,...,epsilon1)rhs.
a114
If µ is an internal node that is not the root and β
1
,...,β
m
are adjoined (in
that order) at µ, then rhs :=〈β
1
···β
m
〉(epsilon1,...,epsilon1, L
µ
, R
µ
,epsilon1,...,epsilon1)rhs.
If there is no adjunction at the root of γ, there is a clause
〈γ〉(epsilon1,...,epsilon1,lhs,epsilon1,...,epsilon1) → rhs
If β
1
,...,β
m
are adjoined in this order at the root of γ,andifL
1
,..., L n
2
, R n
2
,..., R
1
are the parts of the root in lhs such that lhs = L
1
···L n
2
lhs
prime
R n
2
···R
1
, then there is a
clause
〈γ〉(L
1
,...,Ln
2
−1
,Ln
2
lhs
prime
Rn
2
,Rn
2
−1
,...,R
1
) →
〈β
1
···β
m
〉(L
1
,...,Ln
2
,Rn
2
,...,R
1
)rhs
Further, for each initial γ and for all i,1≤ i ≤
n
2
− 2 , there are clauses
〈γ〉(L
1
,...,L
i
,epsilon1,L
i+1
,...,Ln
2
−2
,X,Rn
2
−2
,...,R
i+1
,epsilon1,R
i
,...,R
1
) →
〈γ〉(epsilon1,L
1
,...,Ln
2
−2
,X,Rn
2
−2
,...,R
1
,epsilon1)
And for each auxiliary γ and for all i,1≤ i ≤
n
2
− 1 , there are clauses
〈γ〉(L
1
,...,L
i
,epsilon1,L
i+1
,...,Ln
2
−1
,Rn
2
−1
,...,R
i+1
,epsilon1,R
i
,...,R
1
) →
〈γ〉(epsilon1,L
1
,...,Ln
2
−1
,Rn
2
−1
,...,R
1
,epsilon1)
221
Computational Linguistics Volume 31, Number 2
(3) Predicates 〈γ
1
γ
2
···γ
m
〉
For each 〈γ
1
γ
2
···γ
m
〉 with m ≥ 2 occurring in the clauses constructed so far:
Define sets of variables
L := {L
1
(γ
1
),...,Ln
2
(γ
1
),L
1
(γ
2
),...,Ln
2
(γ
m
)}, and
R := {R
1
(γ
1
),...,Rn
2
(γ
1
),R
1
(γ
2
),...,Rn
2
(γ
m
)}, and
three other pairwise different variables X
1
,X
2
,X /∈L∪R.
Define for all x ∈ L
∗
: R(x):= y ∈ R
∗
such that if L
j
(γ
i
)isthekth letter of x, then R
j
(γ
i
)
is the (|x|−k + 1)th letter of y.
For all w ∈ L
∗
such that
(a) each L ∈ L occurs exactly once in w,
(b) for all 1 ≤ i
1
< i
2
≤ m, L n
2
(γ
i
1
)istotherightofL n
2
(γ
i
2
)inw,and
(c) for all 1 ≤ i ≤ m and 1 ≤ j
1
< j
2
≤
n
2
, L
j
1
(γ
i
)istotheleftofL
j
2
(γ
i
)inw,
and for all x
1
,..., x n
2
∈ L
∗
with x
1
···x n
2
= w,
there is the following clause:
If γ
1
is an initial tree, a clause with L n
2
(γ
1
) eliminated from the x
1
,..., x n
2
:
〈γ
1
···γ
m
〉(x
1
,...,xn
2
XR(xn
2
),...,R(x
1
))→
〈γ
1
〉(L
1
(γ
1
),...,Ln
2
−1
(γ
1
),X,Rn
2
−1
(γ
1
),...,R
1
(γ
1
))
〈γ
2
〉(L
1
(γ
2
),...,Ln
2
(γ
2
),Rn
2
(γ
2
),...,R
1
(γ
2
))
.
.
.
〈γ
m
〉(L
1
(γ
m
),...,Ln
2
(γ
m
),Rn
2
(γ
m
),...,R
1
(γ
m
))
If γ
1
is an auxiliary tree, a clause
〈γ
1
···γ
m
〉(x
1
,...,xn
2
,R(xn
2
),...,R(x
1
))→
〈γ
1
〉(L
1
(γ
1
),...,Ln
2
(γ
1
),Rn
2
(γ
1
),...,R
1
(γ
1
))
.
.
.
〈γ
m
〉(L
1
(γ
m
),...,Ln
2
(γ
m
),Rn
2
(γ
m
),...,R
1
(γ
m
))
(4) These are all clauses.
Sketch of Proof. We do not give the whole proof of the correctness of the construction,
but we sketch the principal steps:
Mainly, two lemmas, concerning, respectively, the clauses constructed under para-
graph 2 above and under paragraph 3 above, are shown:
Lemma 5
For each elementary tree γ in G with decoration string σ
γ
as defined above:
There is a γ
prime
derived from γ with yield w (if γ is an initial tree) or 〈w
l
, w
r
〉 (if γ is an
auxiliary tree with w
l
on the left and w
r
on the right of the foot node) such that
a114
all leaves in γ
prime
have terminal labels;
a114
there are no OA constraints in γ
prime
;
222
Kallmeyer Multicomponent TAGs with Shared Nodes
a114
for all node positions p in γ at which substitutions or adjunctions took
place, the elementary trees γ
(p)
1
,γ
(p)
1
,...,γ
(p)
m
(in that order) were
substituted/adjoined at the node at position p in γ (these are all trees
attached to this node).
iff
There is a 〈γ〉-clause in G
prime
corresponding to the attachments to γ in this derivation in
the way described in the construction, with σ
prime
being the decoration string of γ without
the symbols for the nodes to which nothing was attached such that
There is an instantiation f : {t
prime
|t
prime
is an occurrence of some t ∈ T in σ
prime
}∪V →
{〈i, j〉|0 ≤ i ≤ j ≤|w|} as defined in Definition 8, and the following hold for f :
a114
For each substitution node µ in γ with position p, the yield of the trees
γ
(p)
1
,γ
(p)
1
,...,γ
(p)
m
and everything added to them is the connected substring
f (S
µ
), even if the derivation of γ
prime
from γ is part of a larger derivation.
a114
For each internal node µ in γ with position p at which adjunctions took
place, the yield of the trees γ
(p)
1
,γ
(p)
1
,...,γ
(p)
m
consists of the two connected
substrings 〈 f (L
µ
), f (R
µ
)〉, even if the derivation of γ
prime
from γ is part of a
larger derivation.
a114
If adjunctions at the root took place, then the yield of the trees
γ
(p)
1
,γ
(p)
1
,...,γ
(p)
m
consists of the substrings 〈 f (L
1
)f (L
2
)···f (L n
2
),
f (R n
2
)···f (R
1
)〉, and this yield can be disconnected if the derivation of γ
prime
from γ is part of a larger derivation; it can be separated into disconnected
substrings f (L
1
), f (L
2
),..., f (L n
2
), f (R n
2
),..., f (R
1
).
Proof by induction on structure of γ.
Lemma 6
For all γ
1
,γ
2
,...,γ
m
:
There is a derivation in G of a tree with yield w in which γ
1
,γ
2
,...,γ
m
(in that
order) attach to some node µ in an elementary tree γ such that the yield of the trees
γ
1
,γ
2
,...,γ
m
is separated into at most n disconnected substrings (ranges) of w.Ifγ
1
is an auxiliary tree, it is separated into the substrings l
1
,..., l n
2
, r n
2
,..., r
1
; otherwise
(γ
1
initial), the substrings are l
1
,..., l n
2
−1
, x, r n
2
−1
,..., r
1
.
iff
There is a 〈γ
1
γ
2
···γ
m
〉-clause as described in paragraph 3 of the construction above
such that there is an instantiation f of the clause such that
a114
If γ
1
is initial, then f (x
1
) = l
1
,..., f (x n
2
−1
) = l n
2
−1
, f (x n
2
XR(x n
2
)) =
x, f (R(x n
2
−1
)) = r n
2
−1
,..., f (R(x
1
)) = r
1
,andifγ
1
is auxiliary, then
f (x
1
) = l
1
,..., f (x n
2
) = l n
2
1
, f (R(x n
2
)) = r n
2
,..., f (R(x
1
)) = r
1
.
a114
If γ
1
is initial, then its yield in w consists of the n − 1 substrings (ranges)
f (L
1
(γ
1
)),..., f (L n
2
−1
(γ
1
)), f (X), f (R n
2
−1
(γ
1
)),..., f (R
1
(γ
1
)).
a114
For all auxiliary γ
i
,1≤ i ≤ m, the yield of γ
i
in w consists of the n
substrings (ranges) f (L
1
(γ
i
)),..., f (L n
2
(γ
i
)), f (R n
2
(γ
i
)),..., f (R
1
(γ
i
)).
Proof by induction on m.
The whole theorem can then be proven using these two lemmas.
223
Computational Linguistics Volume 31, Number 2
Acknowledgments
For valuable suggestions, helpful comments
and fruitful discussions of the subject of this
article, we would like to thank Anne Abeill´e,
Pierre Boullier, David Chiang, Eric de la
Clergerie, Chung-Hye Han, Aravind Joshi,
Tony Kroch, Seth Kulick, Maribel Romero
and SinWon Yoon. Furthermore, we are really
grateful to three anonymous reviewers who
gave many very helpful comments and whose
suggestions for improvements influenced
considerably the final form of the article.

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