COMBINATORY CATEGORIAL GRAMMARS: GENERATIVE POWER AND 
RELATIONSHIP TO LINEAR CONTEXT-FREE REWRITING SYSTEMS" 
David J. Weir Aravind K. Joshi 
Department of Computer and Information Science 
University of Pennsylvania 
Philadelphia, PA 19104-6389 
Abstract 
Recent results have established that there is a family of 
languages that is exactly the class of languages generated 
by three independently developed grammar formalisms: 
Tree Adjoining Grammm~, Head Grammars, and Linear 
Indexed Grammars. In this paper we show that Combina- 
tory Categorial Grammars also generates the same class 
of languages. We discuss the slruclm'al descriptions pro- 
duced by Combinawry Categorial Grammars and com- 
pare them to those of grammar formalisms in the class of 
Linear Context-Free Rewriting Systems. We also discuss 
certain extensions of CombinaWry Categorial Grammars 
and their effect on the weak generative capacity. 
1 Introduction 
There have been a number of results concerning the rela- 
tionship between the weak generative capacity (family of 
string languages) associated with different grammar for- 
malisms; for example, the thecxem of Oaifman, et al. \[3\] 
that Classical Categorial Grammars are weakly equivalent 
to Context-Free Grammars (CFG's). Mote recently it has 
been found that there is a class of languages slightly larger 
than the class of Context-Free languages that is generated 
by several different formalisms. In pardodar, Tree Ad- 
joining Grammars (TAG's) and Head Grammars (HG's) 
have been shown to be weakly equivalent \[15\], and these 
formalism are also equivalent to a reslriction of Indexed 
Grammars considered by Gazdar \[6\] called Linear In- 
dexed Grammars (LIG's) \[13\]. 
In this paper, we examine Combinatory Categorial 
Grammars (CCG's), an extension of Classical Catego- 
rial Grammars developed by Steedman and his collab- 
orators \[1,12,9,10,11\]. The main result in this paper is 
*This work was partially mpported by NSF gnmts MCS-82-19116- 
CER. MCS-82-07294, DCR-84-10413, ARO grant DAA29-84-9-0027. 
and DARPA gnmt N0014-85-K0018. We are very grateful to Mark 
Steedmm, \]C Vijay-Shanker and Remo Pare~:hi for helpful disctmiem. 
that CCG's are weakly equivalent to TAG's, HG's, and 
LIG's. We prove this by showing in Section 3 that Com- 
binatory Categorlal Languages (CCL's) are included in 
Linear Indexed Languages (LIL's), and that Tree Adjoin- 
ing Languages (TAL's) are included in CCL's. 
After considering their weak generative capacity, we 
investigate the relationship between the struclzwal descrip- 
tions produced by CCG's and those of other grammar for- 
malisms. In \[14\] a number of grammar formalisms were 
compared and it was suggested that an important aspect 
of their descriptive capacity was reflected by the deriva- 
tion structures that they produced. Several formalisms 
that had previously been descn2~d as mildly context- 
sensitive were found to share a number of properties. In 
particular, the derivations of a grammar could be repre- 
senled with trees that always formed the tree set of a 
context-free grammar. Formalisms that share these prop- 
erties were called Linear Context-Free Rewriting Systems 
('LCFRS's) \[14\]. 
On the basis of their weak generative capacity, it ap- 
pears that CCG's should be classified as mildly context- 
sensitive. In Section 4 we consider whether CCG's should 
be included in the class of LCFRS's. The derivation tree 
sets traditionally associated with CCG's have Context-free 
path sets, and are similar to those of LIG's, and therefore 
differ from those of LCFRS's. This does not, however, 
nile out the possibility that there may be alternative ways 
of representing the derivation of CCG's that will allow 
for their classification as LCP'RS's. 
Extensions to CCG's have been considered that enable 
them to compare two unbounded sU'uctures (for example, 
in \[12\]). It has been argued that this may be needed in 
the analysis of certain coordination phenomena in Dutch. 
In Section 5 we discuss how these additional features 
increase the power of the formalism. In so doing, we 
also give an example demonstrating that the Parenthesis- 
free Categorial Grammar formalism \[5,4\] is moze pow- 
erful that CCG's as defined here. Extensions to TAG's 
(Multicomponent TAG) have been considered for similar 
278 
reasons. However, in this paper, we will not investigate 
the relationship between the extension of CCG's and Mul- 
ticomponent TAG. 
2 Description of Formalisms 
In this section we describe Combinatory Categorial Gram- 
mars, Tree Adjoining Grammars, and Linear Indexed 
Grammars. 
2.1 Combinatory Categoriai Grammars 
Combinatory Categorial Grammar (CCG), as defined here, 
is the most recent version of a system that has evolved in 
a number of papers \[1,12,9,10,11\]. 
A CCG, G, is denoted by (VT, VN, S, f, R) where 
VT is a finite set of terminals (lexical items), 
VN is a finite set of nonterminals (atomic categories), 
S is a distinguished member of VN, 
f is a function that maps elements of VT U {e} to 
finite subsets of C(VN), the set of categories*, where 
V N g C(VN) and 
if CI, C 2 e C(VN) then 
(el/c2) E C(VN) and (c1\c2) E C(VN). 
R is a finite set of combinatory rules, described below. 
We now give the combinatory rules, where z, y, z are 
variables over categories, and each Ii denotes either \ or 
/. 
1. forward application: 
2. backward application: 
u (z\u) -. z 
3. generaliT~d forward composition for some n _> 1: 
(... I.z.) -. 
4. generalized backward composition for some n E 1: 
(...(yll~x)12... I-=-) (~\~) --' 
(--. (~11=x)12... I~z.) 
z Note that f can assign categoric8 to the empty suing, ~, though, 
to our knowledge, this feature has not been employed in the linguistic 
applications ¢~ C'CG. 
Restrictions can be associated with the use of the com- 
binatory rule in R. These restrictions take the form of 
conswaints on the instantiations of variables in the rules. 
These can be constrained in two ways. 
1. The initial nonterminal of the category to which z is 
instantiated can be restricted. 
2. The entire category to which y is instantiated can be 
resuicted. 
Derivations in a CCG involve the use of the combi- 
natory rules in R. Let the derives relation be defined as 
follows. 
~c~ F ~clc2~ 
if R contains a combinawry rule that has czc2 --* c as 
an instance, and a and ~ are (possibly empty) strings of 
categories. The string languages, L(G), generated by a 
CCG, G', is defined as follows. 
{al... 
c, ~ f(aO, a, ~ VT U {~}, 1 _< i _< .} 
Although there is no type-raising rule, its effect can be 
achieved to a limited extent since f can assign type-raised 
categories to lexical items, which is the scheme employed 
in Steedman's recent work. 
2.2 Linear Indexed Grammars 
Linear Indexed Grammars (LIG's) were introduced by 
Gazdar \[6\], and are a restriction of Indexed Grammars 
introduced by Aho \[2\]. LIG's can be seen as an exten- 
sion of CFG's in which each nonterrninal is associated 
with a stack. 
An LIG, G, is denoted by G = ( Vjv , VT , Vs , S, P) where 
VN iS a finite set of nontenninals, 
VT is a finite set of terminals, 
Vs is a finite set of stack symbols, 
S E VN is the start symbol, and 
P is a finite set of productions, having the form 
A\[\] - 
A\[..1\] -* AI\[\]...Ai\["\]...A.\[\] 
A\[..\]--a~\[\]...Ad..t\]...A.\[\] 
where At .... A. E VN, l E Vs, and a E VT O {~}. 
The notation for stacks uses \[. •/\] to denote an arbi- 
Wary stack whose top symbol is I. This system is called 
L/near Indexed Grammars because it can be viewed as a 
279 
restriction of Indexed Grammars in which only one of the 
non-terminals on the right-hand-side of a production can 
inherit the stack from the left-hand-side. 
The derives relation is defined as follows. 
~A\[Z,, ... ht\]~ ~ ~A,\[\] ... A,\[Z,,... t~\].., a,\[\]~ 
if A\[.. l\] -. ~,\[\]...A,\[..\]...A,\[\] ~ P 
otA\[lm.., ll\]~ o =~ aAl\[\]... Ai\[lm... ill\]... An\[\]/~ 
if A\[..\] --. A,\[\]...A,\[-. Z\]...A,,\[\] ~ P 
: c,a\[\]a ~ ,ma 
if A\[\]--.a~P 
The language, L(G), generated by G is 
2.3 Tree Adjoining Grammars 
A TAG \[8,7\] is denoted G = (VN, VT, S, I, A) where 
VN is a finite set of nontennlnals, 
VT is a finite set of terminals, 
S is a distinguished nonterminal, 
I is a finite set of initial trees and 
A is a finite set of auxiliary trees. 
Initial trees are rooted in S with w E V~ on their fron- 
tier. Each internal node is labeled by a member of 
VN. 
Auxiliary trees have tOlAW2 E V'~VNV~ oll their fron- 
tier. The node on the frontier labeled A is called 
the foot node, and the root is also labeled A. Each 
internal node is labeled by a member of VN. 
Trees are composed by tree adjunction. When a tree 
7' is adjoined at a node ~/in a tree .y the tree that results, 
7,', is obtained by excising the subtree under t/from 
and inserting 7' in its place. The excised subtree is then 
substituted for the foot node of 3 / . This operation is 
illustrated in the following figure. 
~': $ 
r'." x 
Y": s 
Each node in an auxiliary tree labeled by a nonterminal 
is associated with adjoining constraints. These constraints 
specify a set of auxiliary trees that can be adjoined at 
that node, and may specify that the node has obligatory 
adjunction (OA). When no tree can be adjoined at a node 
that node has a null adjoining (NA) constraint. 
The siring language L(G) generated by a TAG, G, is 
the set of all strings lYing on the frontier of some tree that 
can be derived from an initial trees with a finite number 
of adjunctions, where that tree has no OA constraints. 
3 Weak Generative Capacity 
In this section we show that CCO's are weakly equivalent 
to TAG's, HG's, and LIO's. We do this by showing the 
Inclusion of CCL's in L1L's, and the inclusion of TAL's in 
CCL's. It is know that TAG and LIG are equivalent \[13\], 
and that TAG and HG are equivalent \[15\]. Thus, the two 
inclusions shown here imply the weak equivalence of all 
four systems. We have not included complete details of 
the proofs which can be found in \[16\]. 
3.1 CCL's C LIL's 
We describe how to construct a LIG, G', from an arbi- 
trary CCG, G such that G and G' are equivalent. Let 
us assume that categories m-e written without parentheses, 
tmless they are needed to override the left associativity of 
the slashes. 
A category c is minimally parenthesized if and 
only if one of the following holds. 
c= A for A E VN 
c = (*oll*xl2... I,,c,,), for, >_ 1, 
where Co E VN and each c~ is mini- 
mally parenthesize~ 
It will be useful to be able to refer to the components of 
a category, c. We first define the immediate components 
of c. 
280 
when c = A the immediate component is A, 
when c = (col:xh...I.c.) the immediate 
components are co, cl,. • •, e.,,. 
The components of a category c are its immediate com- 
ponents, as well as the components of its immediate com- 
ponents. 
Although in CCG's there is no bound on the number 
of categories that are derivable during a derivation (cate- 
gories resulting from the use of a combinatory rule), there 
is a bound on the number of components that derivable 
categories may have. This would no longer hold if unre- 
stricted type-raising were allowed during a derivation. 
Let the set Dc(G) he defined as follows. 
c E De(G) if c is a component of d where 
c' E f(a) for some a E VT U {e}. 
Clearly for any CCG, G, Dc(G) is a finite set. Dc(G) 
contains the set of all derivable components because for 
every category e that can appear in a sentential form of 
a derivation in some CCG, G, each component of c is in 
Dc(G). This can be shown, since, for each combinatory 
rule, ff it holds of the categories on the left of the rule 
then it will hold of the category on the right. 
Each of the combinatory rules in a CCG can be viewed 
as a statement about how a pair of categories can be com- 
bined. For the sake of this discussion, let us name the 
members of the pair according to their role in the rule. 
The first of the pair in forward rules and the second of 
the pair in backward rules will be named the primary cate- 
gory. The second of the parr in forward rules and the first 
of the pair in backward rules will be named the secondary 
category. 
As a resuit of the form that combinatory rules can take 
in a CCG, they have the following property. When a 
combinatory rule is used, there is a bound on the number 
of immediate components that the secondary categories of 
that rule may have. Thus, because immediate constituents 
must belong to De(G) (a finite set), there is a bound on 
the number of categories that can fill the role of secondary 
categories in the use of a combinatory rule. Thus, theae is 
a bound on the number of instantiations of the variables y 
and zi in the combinatory rules in Section 2.1. The only 
variable that can be instantiated to an unbounded number 
of categories is z. Thus, by enumerating each of the finite 
number of variable bindings for y and each z~, the number 
of combinatory rules in R can be increased in such a way 
that only x is needed. Notice that z will appears only 
once on each side of the rules (Le, they are linear). 
We are now in a position m describe how to represent 
each of the combinatory rules by a production in the LIG, 
G'. In the combinatory rules, categories can be viewed 
as stacks since symbols need only be added and removed 
from the right. The secondary category of each rule will 
be a ground category: either A, or (AIlcl\[2... \[ncn), for 
some n >__ I. These can be represented in a LIG as A\[\] 
or A\[hCl\[2... InCh\], respectively. The primary category 
in a combinatory rule will be unspecified except for the 
identity of its left and rightmost immediate components. 
Its leftmost component is a nonterminal, A, and its right- 
most component is a member of De(G), c. This can be 
represented in a LIG by A\[.. el. 
In addition to mapping combinatory rules onto produc- 
tions we must include productions in G' for the mappings 
from lexical items. 
If c E f ( a ) where a E VT U {e} then 
if e = A then A\[\] ...* a E P 
if c-'(ahcll2...I,c,) then 
A\[llC112.." \]nOn \] .-o, a e P 
We are assuming an extension of the notation for produc- 
tions that is given in Section 2.2. Rather than adding or 
removing a single symbol from the stack, a fixed number 
of symbols can be removed and added in one produc- 
tion. Furthermore, any of the nonterminals on the right 
of productions can be given stacks of some fixed size. 
3.2 TAL's C CCL's 
We briefly describe the construction of a CCG, G' from 
a TAG, G, such that G and G' are equivalent. 
For each nonterminal, A of G there will be two nonter- 
minals A ° and A c in G'. The nonterminal of G' will also 
include a nonterminal Ai for each terminal ai of the TAG. 
The terminal alphabets will be the same. The combinatory 
rules of G' are as follows. 
Forward and backward application are restricted to 
cases where the secondary category is some X ~, and 
the left immediate component of the primary cate- 
gory is some Y°. 
Forward and backward composition are restricted to 
cases where the secondary category has the form 
((XChcl)\[2c2), and the left immediate component 
of the primary category is some Y% 
An effect of the restrictions on the use of combinatory 
rules is that only categories that can fill the secondary role 
during composition are categories assigned to terminals by 
f. Notice that the combinatory rules of G' depend only 
281 
on the terminal and nonterminal alphabet of the TAG, and 
are independent of the elementary trees. 
f is defined on the basis of the auxiliary trees in G. 
Without loss of generality we assume that the TAG, G, 
has trees of the following form. 
I contains one initial tree: 
$ OA 
I 
Thus, in considering the language derived by G, we 
need only be concerned with trees derived from auxiliary 
trees whose root and foot are labeled by S. 
There are 5 kinds of auxiliary trees in A. 
1. For each tree of the following form include 
A"/Ca/B ~ ~ f(e) and A°/C*/B + ~ f(O 
A NA 
B OA C OA 
I I 
AI~ e 
2. For each tree of the fonowing form include 
Aa\Ba/C ¢ E f(e) and A¢\Ba/C ¢ E f(e) 
A NA 
BOA C OA 
I I 
A NA 
3. For each tree of the following form 
Aa/B¢/C e.E f(e) and Ae/Be/C ¢ E f(e) 
ANA 
I 
B OA 
I 
COA 
I 
A NA 
include 
4. For each tree of the following form include A°\AI E 
f(e), A*\AI E f(e) and A, E f(a,) 
ANA 
al A NA 
5. For each tree of the following form include A °/Ai E 
f(e), AC/Ai E f(e) and Ai E f(al) 
ANA 
A NA a i 
The CCG, G', in deriving a string, can be understood as 
mimicking a derivation in G of that suing in which trees 
are adjoined in a particular order, that we now describe. 
We define this order by describing the set, 2~(G), of all 
trees produced in i or fewer steps, for i >_ 0. 
To(G) is the set of auxiliary trees of G. 
TI(G) is the union of T~_x(G) with the set of all trees 7 
produced in one of the following two ways. 
1. 
2. 
Let 3 / and 7" be trees in T~-I(G) such that 
there is a unique lowest OA node, I?, in 7' that 
does not dominate the foot node, and 3/' has no 
OA nodes. 7 is produced by adjoining 7" at 
in 7'. 
Let 7' be trees in T~-I(G) such that there is 
OA node, 7, in 7' that dominates the foot node 
and has no lower OA nodes. 7 is pmduceA by 
adjoining an auxiliary tree ~ at 17 in 7'- 
Each tree 7 E 2~(G) with frontier wiAw2 has tbe prop- 
erty that it has a single spine from the root to a node that 
dominates the entire string wlAw2. All of the OA nodes 
remaining in the tree fall on this spine, or hang immedi- 
ately to its right or left. For each such tree 7 there will 
be a derivation tree in a', whose root is labeled by a 
ca~gory c and with frontier to 1W2, wher~ c encodes the 
remaining obligatory adjunctions on this spine in 7. 
Each OA nodes on the spine is encoded in c by a slash 
and nonterminal symbol in the appropriate position. Sup- 
pose the OA node is labeled by some A. When the OA 
node falls on the spine c will contain /.4 ¢ (in this case 
the direction of the slash was arbiwarfly chosen to be for- 
ward). When the OA node faUs to the left of the spine c 
will contain \A% and when the OA node fall~ to the right 
of the spine c will contain/A °. For example, the follow- 
ing tree is encoded by the category A\A~/AI/A~\A ~ 
282 
A 
i 
A I OA A2OA 
/\ 
Wl w2 
We now give an example of a TAG for the language 
{ a"bn I n >_ 0} with crossing dependencies. We then 
give the CCG that would be produced according to this 
construction. 
S NA 
S 10A S2OA 
I I 
£ SNA 
S2NA 
I 
S OA 
I 
$30A 
I 
$2 NA 
S I NA $3 NA 
a SINA S3NA b 
NA 
£ SNA 
s'\s~/s~ ~ f(O s'\sf/s~ ~ f(O 
S~\A ~ f(O S~\A ~ f(O 
A e f(~) B ~ f(b) 
Sa\S, 6 f(¢) S¢\S, 6 f(¢) 
S, E f(6) 
4 Derivations Trees 
Vijay-Shanker, Weir and Joshi \[14\] described several 
properties that were common to various conswained 
grammatical systems, and defined a class of such 
systems called Linear Context-Free Rewriting Systems 
(LCFRS's). LCFRS's are constrained to have linear non- 
erasing composition operations and derivation trees that 
are structurally identical to those of context-free gram- 
mars. The intuition behind the latter restriction is that 
the rewriting (whether it be of strings, trees or graphs) 
be performed in a context-free way; i.e., choices about 
how to rewrite a structure should not be dependent on 
an unbounded amount of the previous or future context 
of the derivation. Several wen-known formalisms fall 
into this class including Context-Free Grammars, Gener- 
alized Phrase Structure Grammars (GPSG), Head Gram- 
mars, Tree Adjoining Grammars, and Multicomponent 
Tree Adjoining Grammars. In \[14\] it is shown that each 
formalism in the class generates scmilinear languages that 
can be recognized in polynomial time. 
In this section, we examine derivation trees of CCG's 
and compare them with respect to those of formalisms that 
are known to be LCFRS's. In order to compare CCG's 
with other systems we must choose a suitable method for 
the representation of derivations in a CCG. In the case of 
CFG, TAG, HG, for example, it is fairly clear what the 
elementary structures and composition operations should 
be, and as a result, in the case of these formalisms, it is 
apparent how to represent derivations. 
The traditional way in which derivations of a CCG 
have been represented has involved a binary tree whose 
nodes are labeled by categories with annotations indicat- 
ing which combinatory rule was used at each stage. These 
derivation trees are different from those systems in the 
class of LCFRS's in two ways. They have context-free 
path sets, and the set of categories labeling nodes may 
be infinite. A property that they share with LCFRS's is 
that there is no dependence between unbounded paths. In 
fact, the derivation trees sets produced by CCG's have 
the same properties as those produced by LIG's (this is 
apparent from the construction in Section 3A). 
Although the derivation trees that are traditionally as- 
sociated with CCG's differ from those of LCFRS's, this 
does not preclude the possibility that there may be an al- 
ternative way of representing derivations. What appears 
to be needed is some characterization of CCG's that iden- 
tities a finite set of elementary structures and a finite set 
of composition operations. 
The equivalence of TAG's and CCG's suggests one way 
of doing this. The construction that we gave from TAG's 
to CCG's produced CCG's having a specific form which 
can be thought of as a normal form for CCG's. We can 
represent the derivations of grammars in this form with 
the same tree sets as the derivation tree sets of the TAG 
from which they were constructed. Hence CCG's in this 
normal form can be classified as LCFRS's. 
283 
TAG derivation trees encode the adjanction of specified 
elementary trees at specified nodes of other elementary 
trees. Thus, the nodes of the derivation trees are labeled 
by the names of elementary trees and tree addresses. In 
the construction used in Section 3.2, each auxiliary tree 
produces assignments of elementary categories to lexicai 
items. CCG derivations can be represented .with trees 
whose nodes identify elementary categories and specify 
which combinatory rule was used to combine it. 
For grammars in this normal form, a unique derivation 
can be recovered from these trees, but this is not true 
of arbitrary CCG's where different orders of combination 
of the elementary categories can result in derivations that 
must be distinguished. In this normal form, the combina- 
tory rules are so restrictive that there is only one order in 
which elementary categories can be combined. Without 
such restrictions, this style of derivation tree must encode 
the order of derivation. 
5 Additions to CCG's 
CCG's have not always been defined in the same way. 
Although TAG's, HG's, and CCG's, can produce the 
crossing dependencies appearing in Dutch, two additions 
to CCG's have been considered by Steedman in \[12\] 
to describe certain coordination phenomena occurring in 
Dutch. For each addition, we discuss its effect on the 
power of the system. 
5.1 Unbounded Dependent Structures 
A characteristic feature of LCFRS's is that they are un- 
able to produce two structures exhibiting an unbounded 
dependence. It has been suggested that this capability 
may be needed in the analysis of coordination in Dutch, 
and an extension of CCG's has been proposed by Steed- 
man \[12\] in which this is possible. The following schema 
is included. 
X* COnj x ~ x 
where, in the analysis given of Dutch, z is allowed to 
match categories of arbitrary size. Two arbitrarily large 
structures can be encoded with two arbitrarily large cat- 
egories. This schema has the effect of checking that the 
encodings are identical The addition of rules such as 
this increases the generative power of CCG's, e.g., the 
following language can be generated. 
{(wc)" I w e {a,b} °} 
In giving analysis of coordination in languages other than 
Dutch, only a finite number of instances of this schema 
are required since only bounded categories are involved. 
This form of coordination does not cause problems for 
LCFRS's. 
5.2 Generalized Composition 
Steedman \[12\] considers a CCG in which there are an 
inf~te number of composition rules for each n _> 1 of 
the form 
(~lv) (...(vhz~)l~...I.z.) - 
(-.. (~l:dln- .. I,z,) 
(...(VllZl)l,...I,z,) (~\y) -" 
(... (~1:012... I,z,) 
This form of composition is permitted in Parenthesis-free 
Categorial Grammars which have been studied in \[5,4\], 
and the results of this section als0 apply to this system. 
With this addition, the generative power of CCG's in- 
creases. We show this by giving a grammar for a language 
that is known not to be a Tree Adjoining language. Con- 
sider the following CCG. We allow um~stricted use of 
arbitrarily many combinatory rules for forward or back- 
wards generalized composition and application. 
f(e) = {s} 
/(al) = {At} 
.~(a2) = {A2} 
f(Cl) = {S\AI/D1/S\BI} 
f(c2) -- {S\A21D21S\B2} 
f(bx) = {Bx} 
f(b2)'-{B2} 
f(dl) = {DI} 
f(d2)= {D2} 
When the language, L, generated by this grammar is in- 
tersected with the regular language 
we get the following language. 
nl ~3 ~1 ftl ft2 ft 3 2 1 {a I G 2 b I C 1 b 2 C 2 d~2 d~l I nl,n 2   0} 
The pumping lemma for Tree Adjoining Grammars \[13\] 
can be used to show that this is not a Tree Adjoining 
Language. Since Tree Adjoining Languages are closed 
under intersection with Regular Languages, L can not be 
a Tree Adjoining Language either. 
6 Conclusions 
In this paper we have considered the string languages 
and derivation trees produced by CCL's. We have shown 
that CCG's generate the same class of string languages 
284 
as TAG's, HG's, and LIG's. The derivation tree sets nor- 
mally associated with CCG's are found to be the same 
as those of LIG's. They have context-free path sets, and 
nodes labeled by an unbounded alphaboL A consequence 
of the proof of equivalence with TAG is the existence of 
a normal form for CCG's having the property that deriva- 
tion trees can be given for grammars in this normal form 
that are structurally the same as the derivation trees of 
CFG's. The question of whether there is a method of 
representing the derivations of arbitrary CCG's with tree 
sets similar to those of CFG's remains open. Thus, it is 
unclear, whether, despite their restricted weak generative 
power, CCG's can be classified as LCFRS's. 
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