Proceedings of the 8th International Workshop on Tree Adjoining Grammar and Related Formalisms, pages 121–126,
Sydney, July 2006. c©2006 Association for Computational Linguistics
Extended cross-serial dependencies in Tree Adjoining Grammars
Marco Kuhlmann and Mathias Möhl
Programming Systems Lab
Saarland University
Saarbrücken, Germany
{kuhlmann|mmohl}@ps.uni-sb.de
Abstract
The ability to represent cross-serial depen-
dencies is one of the central features of
Tree Adjoining Grammar (TAG). The class
of dependency structures representable by
lexicalized TAG derivations can be captured
by two graph-theoretic properties: a bound
on the gap degree of the structures, and a
constraint called well-nestedness. In this
paper, we compare formalisms from two
strands of extensions to TAG in the context
of the question, how they behave with re-
spect to these constraints. In particular, we
show that multi-component TAG does not
necessarily retain the well-nestedness con-
straint, while this constraint is inherent to
Coupled Context-Free Grammar (Hotz and
Pitsch, 1996).
1 Introduction
The ability to assign ‘limited cross-serial depen-
dencies’ to the words in a sentence is a hallmark
of mildly context-sensitive grammar formalisms
(Joshi, 1985). In the case of TAG, an exact def-
inition of this ability can be given in terms of
two graph-theoretic properties of the dependency
structures induced by TAG derivations: the gap de-
gree restriction and the well-nestedness constraint
(Bodirsky et al., 2005).
Gap degree and well-nestedness can be seen as
the formal correspondents of what Joshi (1985)
refers to as ‘a limited amount of cross-serial depen-
dencies’ and ‘the nesting properties as in the case
of context-free grammars.’ More specifically, the
gap degree of a dependency structure counts the
number of discontinuities in a dependency subtree,
while well-nestedness constrains the positions of
disjoint subtrees relative to one another. The depen-
dency structures that correspond to the derivations
in a lexicalized TAG are well-nested, and their gap
degree is at most1.
In the present paper, we compare formalisms
from two strands of extensions to TAG in the con-
text of the question, what classes of dependency
structures they are able to induce.
We are particularly interested in formalisms that
induce only well-nested dependency structures.
This interest is motivated by two observations:
First, well-nestedness is interesting as a generaliza-
tion of projectivity (Marcus, 1967)—while more
than 23% of the 73088 dependency structures in
the Prague Dependency Treebank of Czech (Ha-
jiˇc et al., 2001) are non-projective, only 0.11%
are not well-nested (Kuhlmann and Nivre, 2006).
Second, well-nestedness is interesting for process-
ing. Specifically, parsers for well-nested grammar
formalisms are not confronted with the ‘crossing
configurations’ that make the universal recogni-
tion problem of Linear Context-Free Rewriting Sys-
tems NP-complete (Satta, 1992). In summary, it
appears that well-nestedness can strike a successful
balance between empirical coverage and computa-
tional tractability. If this is true, then a formalism
that has the well-nestedness constraint hardwired
is preferable over one that has not.
The results of this paper can be summarized
as follows: Derivations in lexicalized multi-com-
ponent TAGs (Weir, 1988; Kallmeyer, 2005), in
which a single adjunction adds a set of elemen-
tary trees, either induce exactly the same depen-
dency structures as TAG, or induce all structures
of bounded gap degree, even non-well-nested ones.
This depends on the decision whether one takes
‘lexicalized’ to mean ‘one lexical anchor per tree’,
or ‘one lexical anchor per tree set’. In contrast,
multi-foot extensions of TAG (Abe, 1988; Hotz
and Pitsch, 1996), where a single elementary tree
may have more than one foot node, only induce
well-nested dependency structures of bounded gap
degree. Thus, from the dependency point of view,
they constitute the structurally more conservative
extension of TAG.
121
2 Dependency structures for TAG
We start with a presentation of the dependency
view on TAG that constitutes the basis for our work,
and introduce the relevant terminology. The main
objective of this section is to provide intuitions; for
the formal details, see Bodirsky et al. (2005).
2.1 The dependency view on TAG
Let s D w1 SOHSOHSOHwn be a sentence (a sequence of
tokens). By a dependency structure fors, we mean
a tuple.W;!;RS/, whereW D fw1;:::;wng, and
! D f.wi;wj/2W STXW jwj depends onwi g
RS D f.wi;wj/2W STXW ji <j g
To interpret a grammar formalism as a specifica-
tion for a set of dependency structures, we need to
assign meaning to the relation ‘depends’ in terms
of this formalism. For TAG, this can be done based
on the Fundamental Hypothesis that ‘every syntac-
tic dependency is expressed locally within a single
elementary tree’ (Frank, 2002). More specifically,
a derivation in a (strongly) lexicalized TAG can
be viewed as a dependency structure as follows:
The setW contains the (occurences of) lexical an-
chors involved in the derivation. For two anchors
wi;wj 2 W, wi ! wj if the elementary tree an-
chored atwj was substituted or adjoined into the
tree anchored atwi. We then havewi RSwj ifwi
precedes wj in the yield of the derived tree cor-
responding to the derivation. Notice that the rela-
tion ! in such a dependency structure is almost
exactly the derivation tree of the underlying TAG
derivation; the only difference is that elementary
trees have been replaced by their lexical anchors.
Figure 1 shows a TAG grammar together with a
dependency structure induced by a derivation of
this grammar. Tokens in the derived string are rep-
resented by labelled nodes; the solid arcs between
the nodes represent the dependencies.
2.2 Gap degree and well-nestedness
An interesting feature of the dependency structure
shown in Figure 1 is that it violates a standard
constraint on dependency structures known as pro-
jectivity (Marcus, 1967). We introduce some termi-
nology for non-projective dependency structures:
A set T DC2 W is convex, if for no two tokens
w1;w2 2 T, there exists a token w from W NULT
such that w1 RS w RS w2. The cover of T, C.T/,
is the smallest convex set that contains T. For
w 2 W, we write #w for the set of tokens in the
S;
a T D
B C
T;
a T D
B ? C
B;
b
C;
c
D;
d
.
a1 a2 b2 b1 c1 c2 d2 d1
Figure 1: TAG grammar for anbncndn, and a de-
pendency structure induced by this grammar
subtree rooted atw (includingw itself). A gap in
#wis a largest convex set inC.#w/NUL#w. The gap
degree ofw, gd.w/, is the number of gaps in #w.
The gaps in #wpartition #winto gd.w/NUL1largest
convex blocks; we write #iw to refer to the i-th
of these blocks, counted from left to right (with
respect to RS). The gap degree of a dependency
structureisthemaximumoverthegapdegreesofits
subtrees; we writeDg for the set of all dependency
structures with a gap degree of at mostg.
The gap degree provides a quantitative measure
for the non-projectivity of dependency structures.
Well-nestedness is a qualitative property: it con-
strains the relative positions of disjoint subtrees.
Let w1;w2 2 W such that #w1 and #w2 are dis-
joint. Four tokensw11;w21 2 #w1,w12;w22 2 #w2
interleave, if w11 RS w12 RS w21 RS w22. A depen-
dency structure is well-nested, if it does not contain
interleaving tokens. We write Dwn for the set of all
well-nested dependency structures.
For illustration, consider again the dependency
structure shown in Figure 1. It has gap degree 1:
a2 is the only tokenw for which #w is not convex;
the set fb1;c1g forms a gap in #a2. The structure
is also well-nested. In contrast, the structure shown
in the right half of Figure 2 is not well-nested; the
tokensb;c;d;e interleave. Bodirsky et al. (2005)
show that TAG induces precisely the set Dwn \D1.
3 Multi-component extensions
Multi-component TAG (MCTAG) extends TAG with
the ability to adjoin a whole set of elementary trees
(components) simultaneously. To answer the ques-
tion, whether this extension also leads to an ex-
tended class of dependency structures, we first need
to decide how we want to transfer the Fundamental
Hypothesis (Frank, 2002) to MCTAGs.
122
 
A;
a B1 C1 B2 C2
8 
<
 :
B;1
b
B;2
D
9>
=
>;
8 
<
 :
C;1
c
C;2
E
9>
=
>;
D;
d
E;
e
Figure 2: An MCTAG and a not well-nested dependency structure derived by it.
3.1 One anchor per component
If we commit to the view that each component of
a tree set introduces a separate lexical anchor and
its syntactic dependencies, the dependency struc-
tures induced by MCTAG are exactly the structures
induced by TAG. In particular, each node in the
derivation tree, and therefore each token in the
dependency tree, corresponds to a single elemen-
tary tree. As Kallmeyer (2005) puts it, one can
then consider an MCTAG as a TAG G ‘where cer-
tain derivation trees inG are disallowed since they
do not satisfy certain constraints.’ The ability of
MCTAG to perform multiple adjunctions simultane-
ously allows one to induce more complex sets of
dependency structures—each individual structure
is limited as in the case of standard TAG.
3.2 One anchor per tree set
If, on the other hand, we take a complete tree set
as the level on which syntactic dependencies are
specified, MCTAGs can induce a larger class of de-
pendency structures. Under this perspective, tokens
in the dependency structure correspond not to in-
dividual components, but to tree sets (Weir, 1988).
For each tokenw, #w then contains the lexical an-
chors of all the subderivationsstartinginthetreeset
corresponding tow. As there can be a gap between
each two of these subderivations, the gap degree
of the induced dependency structures is bounded
only by the maximal number of components per
tree set. At the same time, even non-well-nested
structures can be induced; an example is shown in
Figure 2. Here, #b is distributed over the compo-
nents rooted at B1 and B2, and #c is distributed
overC1 andC2. The elementary tree rooted atA
arranges the substitution sites such thatb;c;d;ein-
terleave. Note that the MCTAG used in this example
is heavily restricted: it is tree-local and does not
even use adjunction. This restricted form suffices
to induce non-well-nested dependency structures.
4 Multi-foot extensions
A second way to extend TAG, orthogonal to the
multi-component approach, is to allow a single el-
ementary tree to have more than one foot node.
For this kind of extension, the Fundamental Hy-
pothesis does not need to be re-interpreted. Prob-
ably the most prominent multi-foot extension of
TAG is Ranked Node Rewriting Grammar (RNRG)
(Abe, 1988); however, the properties that we are
interested in here can be easier investigated in a
notational variant of RNRG, Coupled Context-Free
Grammar (Hotz and Pitsch, 1996).
Terminology Multi-foot formalisms require a
means to specify which foot node gets what ma-
terial in an adjunction. To do so, they use ranked
symbols. A ranked alphabet is a pair˘ D.˙;SUB/,
where˙ is an alphabet, andSUB2˙ !N is a total
function that assigns every symbolESC 2˙ a (pos-
itive) rank. Define˘Œrc141 WD fESC 2 ˙ j SUB.ESC/ D rg.
The components of ESC, comp.ESC/, are the elements
of the set f.ESC;i/j1DC4i DC4SUB.ESC/g. We writeESCi in-
stead of.ESC;i/. Let comp.˘/WDSESC2˘ comp.ESC/.
4.1 Coupled Context-Free Grammar
Coupled Context-Free Grammar (CCFG) is a gener-
alization of context-free grammar in which non-ter-
minals come from a ranked alphabet, and compo-
nents of a non-terminal can only be substituted si-
multaneously. The ‘TAG-ness’ of CCFG is reflected
in the requirement, that the RHS of productions
must be words from a bracket-like language, and
thus have the same hierarchical structure as ele-
mentary trees in a TAG. As an example, the second
elementary tree from Figure 1 can be linearized as
hT1aT1B1;C1T2D1T2i;
where each pair.T1;T2/of matching components
corresponds to an inner node in the tree, and the
boundary between the first and the second part of
the tuple marks the position of the foot node. The
required structure of the RHS can be formalized as
follows:
Definition 1 Let ˘ be a ranked alphabet, and
let ˙ be an unranked alphabet. The extended
semi-Dyck set over ˘ and ˙, ESD.˘;˙/, is the
smallest set that satisfies the following properties:
123
(a) ˙ETX DC2 ESD.˘;˙/; (b) ˘Œ1c141 DC2 ESD.˘;˙/;
(c) if s1;:::;sk 2 ESD.˘;˙/ and EM 2 ˘ŒkC1c141,
then EM1s1EM2 SOHSOHSOHEMkskEMkC1 2 ESD.˘;˙/; (d) if
s1;s2 2 ESD.˘;˙/, thens1s2 2 ESD.˘;˙/.
Definition 2 Let N be a ranked alphabet of non-
terminals, and let T be an (unranked) alphabet
of terminals. A ranked rewriting system over
ESD.N;T/is a finite, non-empty set of productions
of the form X ! h˛1;:::;˛ri, where X 2 NŒrc141,
and˛ WD˛1 SOHSOHSOH˛r 2 ESD.N;T/.
We writeSUB.p/to refer to the rank of the non-termi-
nal on the LHS of a productionp.
RNRG and CCFG are notational variants because
each RNRG elementary tree withr NUL1foot nodes
can be linearized into the RHS of a production
X ! h˛1;:::;˛ri in a ranked rewriting system,
as indicated by the example above.
Definition 3 A coupled context-free grammar is a
tuple G D .N;T;P;S/ where: N is a ranked al-
phabet of non-terminal symbols;T is an unranked
alphabet of terminal symbols;P is a ranked rewrit-
ing system over ESD.N;T/; S 2 NŒ1c141 is a start
symbol.
We say that a CCFGG is anr-CCFG, if the maximal
rank among all non-terminals inG isr.
Definition 4 PutV WD comp.N/[T, and let
RS 2VETX D u1X1u2 SOHSOHSOHurXrurC1
 2VETX D u1˛1u2 SOHSOHSOHur˛rurC1
such thatu2;:::;ur 2 ESD.N;T/, andX 2 NŒrc141.
We say that can be derived fromRS in one step,
and write RS )G  , if G contains a production
X ! h˛1;:::;˛ri. The string language of G is
the setL.G/WD fs 2TETX jS )ETXG sg.
Based on this definition, the notions of derivation
tree and derived tree are defined in the usual way.
In particular, the nodes of the derivation tree are
labelled with productions, while the nodes of the
corresponding derived tree are labelled with com-
ponents from comp.˘/(inner nodes) and terminal
symbols (leaves). We write .T];T[/ to refer to a
derivation in CCFG: T] stands for the derivation
tree,T[ for the corresponding derived tree.
4.2 The dependency view on CCFG
A CCFG G is strongly lexicalized, if each produc-
tionp contains exactly one terminal symbol, writ-
ten as anchor.p/. Just as in the case of TAG, a
strongly lexicalized CCFG G can be interpreted as
a dependency grammar: Let.T];T[/be a deriva-
tion in G. Since G is strongly lexicalized, there
is a one-to-one mapping between the nodes of the
derivation treeT] (labelled with productions) and
the leaves of the derived treeT[ (labelled with ter-
minals); we refer to this mapping by the namefL.
Definition 5 A dependency structureD is induced
by a derivation.T];T[/, written.T];T[/`D, if
(a) anchor.p1/ ! anchor.p2/ in D if and only
if p1 ! p2 in T]; (b) anchor.p1/ RS anchor.p2/
inD if and only iffL.p1/RSfL.p2/inT[.
We write D.G/for the set of all dependency struc-
tures induced by derivations inG. Figure 3 shows
a sample CCFG G, a derivation in G, and the de-
pendency structure induced by this derivation.
4.3 Projections
To reason about the structural properties of the
dependency languages induced by CCFGs, we need
some additional definitions. In the following, we
use the notation .uWESC/ to refer to a node u with
labelESC in some given labelled tree.
LetD 2 D.G/be a dependency structure such
that.T];T[/`D, and let.uWp/2T] be a node.
Somewhere in the course of the derivation repre-
sented byT], theSUB.p/components of the non-ter-
minal on the LHS of the productionp are simulta-
neously rewritten. LetfI.u/be theSUB.p/-tuple of
nodes inT[ that correspond to these components.
Note that, whilefL maps nodes in the derivation
tree T] to leaves in the derived tree T[, fI takes
nodes inT] to tuples of inner nodes inT[. Define
down.u/D fv ju!ETX v inT] g;
proj.u;i/D fv jfI.u/i !ETX fL.v/inT[ g:
The set down.u/contains the lexical anchors in the
sub-derivation starting atu. The set proj.u;i/iden-
tifies that part of this sub-derivation that is derived
from thei-th component of the non-terminal at the
LHS of the production corresponding tou. For the
derivation shown in Figure 3, we have
fI.p2/D hB1;B2;B3i; proj.p2;1/D fp2g:
Lemma 6 For all nodesu2T],
down.u/DU1DC4iDC4SUB.p/ proj.u;i/:
4.4 Results
In this section, we prove the main technical re-
sults of this paper: that all dependency structures
124
GrammarGETX: p1W A! hai; p2W B ! hb;D1;D1i; p3W C ! hA1B1cA1B2A1B3i; p4W D ! hdi
p3
p1 p2 p1 p1
p4 p4
(a) Derivation tree (b) Derived tree
a b c a d a d
(c) Induced dependency structure
Figure 3: A CCFG derivation and the dependency structure induced by it
induced by an r-CCFG have a gap degree that is
bounded by r; that they are all well-nested; and
that each well-nested structure with a gap degree
bounded byr can be induced by anr-CCFG. In the
following, letG be anr-CCFG, and writeGr for the
set of allr-CCFGs.
Lemma 7 D.G/DC2 DrNUL1
Proof Let.T];T[/`D, and let.uWp/2T]. By
definition of proj, for each 1 DC4 i DC4 SUB.p/, the set
proj.u;i/ forms a contiguous region of the sen-
tence derived by T]. Using Lemma 6, we then
see that down.u/ is distributed over at most SUB.u/
contiguous regions of that sentence. This means
that the dependency subtree rooted at anchor.p/
has at mostSUB.p/NUL1gaps.
Lemma 8 D.G/DC2 Dwn
Proof Choose aD 2 D.G/, and assume thatD is
not well-nested. Then there is a governor u 2 D
with two distinct dependents v;w such that #v
contains tokens v1;v2, and #w contains tokens
w1;w2 such that v1 RS w1 RS v2 RS w2. For the
derivation.T];T[/that inducesD, this means that
there is a node .uWp/ with children .vWpv/ and
.wWpw/inT] such that
9.v1;v2 2 down.v//W 9.w1;w2 2 down.w//W
fL.v1/RSfL.w1/RSfL.v2/RSfL.w2/ inT[:
Sincedown.v/anddown.w/aredisjoint;v1 andv2
must come from distinct convex blocks in down.v/,
and w1 and w2 must come from distinct convex
blocks in down.w/. Therefore,
v1 2 proj.v;i1/; v2 2 proj.v;i2/; i1 <i2 and
w1 2 proj.w;j1/; w2 2 proj.w;j2/; j1 <j2:
By definition, proj.x;k/(x 2 fv;wg) is the projec-
tion of a nodefI.x/k inT[; the label of this node
is LHS.px/k. Assume now that the non-terminal
on the LHS of pv is V, and that the non-terminal
on the LHS ofpw isW. Given thatpv andpw are
used to rewrite p, RHS.p/ contains the substring
Vi1 SOHSOHSOHWj1 SOHSOHSOHVi2 SOHSOHSOHWj2. This contradicts the fact
that RHS.p/2 ESD.N;T/.
Lemma 9 Dwn \DrNUL1 DC2SG2Gr D.G/
Proof LetD D.W;!;RS/be a dependency struc-
ture from Dwn \DrNUL1. We construct anr-CCFG
G D .N;T;P;S/that inducesD. For the ranked
alphabetN of non-terminals, put
N D fNw jw 2W g; SUB.Nw/D gd.w/C1:
The setS of start symbols is fN>g, where > is the
root ofD. For the terminal alphabet, putT DW.
The setP consists of jWj productions of the form
Nw ! E˛, where w 2 W, and E˛ is a tuple with
arity gd.w/C1 that contains the terminal w and
non-terminal components for all children ofw as
follows. Consider the following family of sets:
Cw D ffwgg[f#iv jw !v;1DC4i DC4 gd.v/C1g:
All sets in Cw are disjoint, and their union equals
the set #w. We define a functionŒSOHc141that interprets
the elements of Cw as elements from N [T as
follows: Œfwgc141 WD w, and Œ#ivc141 WD Nvi . Now the
RHS of a rule Nw ! E˛ is fully specified by the
following equivalences, whereC 2 Cw:
ŒCc141occurs in˛i iff C DC2 #iw
ŒC1c141precedesŒC2c141in E˛ iff C1 STXC2 DC2 RS
Applied to the dependency structure of Figure 3c,
this constructs the given grammarGETX. Note that,
due to the well-nestedness ofD, the RHS of each
rule forms a valid extended semi-Dyck word.
125
5 Summary
Starting from the fact that TAG is able to derive
well-nested dependency structures with a gap de-
gree of at most1, we have investigated how multi-
component and multi-foot extensions of TAG alter
this expressivity. Our results are as follows:
SI For multi-component TAG, the notion of ‘in-
duced dependency structures’ depends on the
assumed notion of lexicalization. Therefore,
either the same structures as in TAG, or arbi-
trary gap-bounded dependency structures are
derivable. In the former case, MCTAG has the
same structural limits as standard TAG; in the
latter case, even non-well-nested dependency
structures are induced.
SI The multi-foot extension CCFG (and its equiv-
alent RNRG) is restricted to well-nested de-
pendency structures, but in contrast to TAG, it
can induce structures with any bounded gap
degree. The rank of a grammar is an upper
bound on the gap degree of the dependency
structures it induces.
Since the extensions inherent to MCTAG and
CCFG are orthogonal, it is possible to combine
them: Multi-Component Multi-Foot TAG (MMTAG)
as described by Chiang (2001) allows to simulta-
neously adjoin sets of trees, where each tree may
have multiple foot nodes. The structural limita-
tions of the dependency structures inducible by
MCTAG and CCFG generalize to MMTAG as one
would expect. As in the case of MCTAG, there
are two different understandings of how a depen-
dency structure is induced by an MMTAG. Under
the ‘one anchor per component’ perspective, MM-
TAG, just like CCFG, derives well-nested structures
of bounded gap-degree. Under the ‘one anchor
per tree set’ perspective, just like MCTAG, it also
derives non-well-nested gap-bounded structures.
Acknowledgements We thank Jan Schwingham-
mer, Guido Tack, and Stefan Thater for fruitful dis-
cussions during the preparation of this paper, and
three anonymous reviewers for their detailed com-
ments on an earlier version. The work of Marco
Kuhlmann is funded by the Collaborative Research
Centre ‘Resource-Adaptive Cognitive Processes’
of the Deutsche Forschungsgemeinschaft.

References
Naoki Abe. 1988. Feasible learnability of formal
grammars and the theory of natural language acqui-
sition. In 12th International Conference on Compu-
tational Linguistics, pages 1–6, Budapest, Hungary.
Manuel Bodirsky, Marco Kuhlmann, and Mathias
Möhl. 2005. Well-nested drawings as models of
syntactic structure. In Tenth Conference on For-
mal Grammar and Ninth Meeting on Mathematics
of Language (FG-MoL), Edinburgh, UK.
David Chiang. 2001. Constraints on strong gener-
ative power. In 39th Annual Meeting and Tenth
Conference of the European Chapter of the Associa-
tion for Computational Linguistics, pages 124–131,
Toulouse, France.
Robert Frank. 2002. Phrase Structure Composition
and Syntactic Dependencies. MIT Press.
Jan Hajiˇc, Barbora Vidova Hladka, Jarmila Panevová,
Eva Hajiˇcová, Petr Sgall, and Petr Pajas. 2001.
Prague Dependency Treebank 1.0. LDC, 2001T10.
GüntherHotzandGiselaPitsch. 1996. Onparsingcou-
pled-context-free languages. Theoretical Computer
Science, 161:205–233.
Aravind K. Joshi. 1985. Tree adjoining grammars:
How much context-sensitivity is required to provide
reasonable structural descriptions? In David R.
Dowty, Lauri Karttunen, and Arnold M. Zwicky,
editors, Natural Language Parsing, pages 206–250.
Cambridge University Press, Cambridge, UK.
Laura Kallmeyer. 2005. A descriptive charac-
terization of multicomponent tree adjoining gram-
mars. In Traitement Automatique des Langues Na-
turelles (TALN),volume1, pages457–462, Dourdan,
France.
Marco Kuhlmann and Joakim Nivre. 2006. Mildly
non-projective dependency structures. In 22nd In-
ternational Conference on Computational Linguis-
tics and 43rd Annual Meeting of the Association for
Computational Linguistics (COLING-ACL), Com-
panion Volume, Sydney, Australia.
Solomon Marcus. 1967. Algebraic Linguistics: An-
alytical Models, volume 29 of Mathematics in Sci-
ence and Engineering. Academic Press, New York.
Giorgio Satta. 1992. Recognition of linear context-
free rewriting systems. In 30th Meeting of the Asso-
ciation for Computational Linguistics (ACL), pages
89–95, Newark, Delaware, USA.
David J. Weir. 1988. Characterizing Mildly Context-
Sensitive Grammar Formalisms. Ph.D. thesis, Uni-
versity of Pennsylvania, Philadelphia, USA.
