Categorial Unification Grammars* 
Hans Uszkoreit 
Artificial Intelligence Center 
SRI International and 
Center for the Study of Language 
and Information at Stanford University 
Abstract 
Categorial unification grammars (CUGs) embody 
the essential properties of both unification and categorial 
grammar formalisms. Their efficient and uniform way of 
encoding linguistic knowledge in well-understood and 
widely used representations makes them attractive for 
computational applications and for linguistic research. 
In this paper, the basic concepts of CUGs and simple 
examples of their application will be presented. It will be 
argued that the strategies and potentials of CUGs justify 
their further exploration in the wider context of research 
on unification grammars. Approaches to selected 
linguistic phenomena such as long-distance 
dependencies, adjuncts, word order, and extraposition 
are discussed. 
0. Introduction 
The work on merging strategies from unification 
grammars and categorial grammars has its origins in 
several research efforst that have been pursued in 
parallel. One of them is the grammar development on 
the PATR system (Shieber et al., 1983; Shieber, 1984) at 
SRI. For quite a while now I have been using the 
excellent facilities of PATR for the design and testing of 
experimental\[ CUGs. Such grammars currently run on 
two PATR implementations: Stuart Shieber's Zetalisp 
version on the Symbolics 3600 and Lauri Karttunen's 
Interlisp-D w:rsion on the XEROX 1109. The work on 
CUGs has influenced our efforts to develop a larger 
PATR grammar, and will do so even more in the future. 
On the theoretical side, this work is part of ongoing 
research on such topics as word order variation, 
modification, and German syntax within projects at SRI 
and CSLI (Stanford University). 
The structure of the paper reflects the diverse nature 
of the enterprise. In the first section, I will introduce the 
basic notions of CUGs and demonstrate them through 
examples in PATR notation. The second section 
discusses the motivation for this work and some of its 
theoretical implications. The third section sketches a 
linguistically motivated CUG framework with a strong 
lexical syntax that accomodates word order variation. 
The paper concludes with a brief discussion of possible 
CUG approaches to long-distance dependencies. 
1. Basic Notions of Categorial Unification 
Grammars 
1.2. Unification Grammars and Categorial 
Grammars 
Both terms, unification grammar (UG) and 
categorial grammar (CG), stand for whole families of 
related grammar formalisms whose basic notions are 
widely known.l Yet, for the characterization of the class 
of formalisms I want to discuss, it will be useful to review 
the most central concepts of both UG and CG. 
Unification grammar formalisms employ complex 
feature structures as their syntactic representations. 
These structures encode partial information about 
constituents. Either term or graph unification is utilized 
as the main operation for checking, propagating, and 
merging of the information in these complex 
representations. Most unification grammars also use the 
complex feature structures for the linking of syntactic 
and semantic information. 
In traditional categorial grammars, all information 
about possible syntactic combinations of constituents is 
encoded in their categories. Those grammars allow only 
binary combinations. One of the two combined 
constituents, the functor, encodes the combination 
funtion, the other constituent serves as the argument to 
this function. Instead ot7 phrase structure rules, the 
grammar contains one or, in some formalisms, two 
combination rules that combine a functor and an 
argument by applying the function encoded in the 
functor to the argument constituent. Most categorial 
grammars only combine constituents whose terminal 
strings concatenate in the input string, but this need not 
be so. In most categorial grammar formalisms, it is 
assumed that the syntactic functor-argument structure 
in the corresponding compositional semantics. 
187 
There are usually two types of grammatical 
categories in a categorial grammar, basic and derived 
ones. Basic categories are just category symbols, derived 
categories are functions from one (derived or basic) 
category to another. A derived category that encodes a 
function from category A to category B might be written 
B/A if the functor combines with an argument to its right 
or B~, if it expects the argument to its left. Thus, if we 
assume just two basic categories, N and S, then N/S, S/N, 
N\S, S\N, (S\N)/N, (N/S\(S\(N/N)), etc. are also categories. 
Not all of these categories will ever occur in the 
derivation of sentences. The set of actually occurring 
categories depends on the lexical categories of the 
language. 
Assume the following simple sample grammar: 
(2) Basic categories: N, S 
lexical categories: N (Paul, Peter) 
(S\N)fN (likes) 
The grammar is used for the sample derivation in (3): 
(3) Peter likes Paul 
N (S\N)fin N 
SkN 
S 
It should be clear from my brief description that the 
defining characteristics of unification grammar have 
nothing to do with the ones of categorial grammar. We 
will see that the properties of both grammar types 
actually complement each other quite wetl. 
1.2. A Sample CUG in PATR Notation 
Since the first categorial unification grammars were 
written in the PATR formalism and tested on the PATR 
systems implemented at SRI, and since PATR is 
especially well suited for the emulation of other 
grammar formalisms, I will use its notation. 
The representations in PATR are directed acyclic 
graphs (DAGs) 2 . Rules have two parts, a head and a 
body. The head is a context-free rewrite rule and the 
body is a DAG. Here is an example, a simple rule that 
forms a sentence by combining a noun phrase with a verb 
phrase. 
188 
(4) head XO -~ X1, X2 
body in unification notation 
<X0 cat> = S 
<X1 cat> = NP 
<X2cat> = VP 
<X1 agr> = <X2agr> 
body in graph notation 
xo 
r 
S NP 
The rule states that two constituents X1 and X2 can 
combine to form a constituent X0 if the terminal string 
covered by X1 immediately precedes the terminal string 
of X2 and if the DAGs of X0, X1, and X2 unify with the 
X0, X1, and X2 subgraphs of the rule body, respectively. 
I will now show the most straight-forward encoding 
of a categorial grammar in this notation. There are two 
types of constituent graphs. Constituent graphs for basic 
categories are of the following form: 
(5) 
N S 
Of course, there might be more features associated with 
the constituent: 
(6) 
/oe 7 
N S Finite 3 Sg 
Derived constituents have graphs of the following form: 
(7) 
arg 
(t0b) Backward Functional Application (BFA) 
value -~ functor argument 
< value > = < functor val> 
<argument> = <functor arg> 
<flmctor dir> :--: Left. 
This is the graph associated with the VP likes Paul: 
in graph notation: 
(8) 
,. /~ Left /~ agr 
ca~//pers / form cat/pers~nu m 
S Finite N 3 Sg 
It corresponds to the derived-category symboh 
(9) 
S \ N 
form : Finite pers : 3 num: Sg 
(10a) and (10b) are the rules that combine constituents. 
As in traditional categorial grammars, two such rules 
sufice. 
(10a) Forward Functional Application (FFA) 
value -~ functor argument 
<value> = <functorval> 
<argument> = <functorarg> 
<functor dir> = Right. 
in graph notation: 
val u e~J-~~'~. 
/funct°rl . ~rgu 
Right 
ment 
val u e ~--J J-~-~rg u ment 
/ 
Left 
If Backward Functional Application is used to 
combine the constituents Peter and likes Paul, the result 
is a finite sentence. 
However, if the same rule is applied to the identical 
constituents likes Paul and likes Paul, again a finite 
sentence is obtained. '\]\['his is so because the graph for 
likes Paul actually unifies with the value of arg in the 
same graph. This can be easily remedied by modifying 
the graph for the VP slightly. By stipulating that the 
argument must not have an unfilled argument position, 
one can rule out derivcd categories as subject arguments 
tbr the VP: 
(II) 
/0o-i /°e?Tum 
S Finite N 3 Sg 
1.3. Extensions to the Basic Formalism 
In this subsection \[ want to discuss very briefly a few 
extensions of' the basic model that make it more suitable 
for the encoding of natural-language grammars. The first 
one is the sorting of fimctors according to their own 
syntactic category. This move might be described 
alternatively as defining the type of a constituent as 
being defined by both a set of syntactic (and semantic) 
189 
attributes and a function from categories to categories. 
This function is also expressed as the value of an 
attribute. For a basic category the value of the function 
attribute is NIL. The following graph is a simplified 
example of a functor category (prenominal adjective in a 
language with case and number agreement within the 
NP). 
~ ~/ ~ ~'~unction 
ca;~/ ~s::m --~gr 
The combination rules need 
accordingly. This is the modified 
functional application. 
to be changed 
rule of forward 
value -~ functor argument 
<value> = <functor function val> 
< argument > = < functor function arg > 
<functor function dir> = Right. 
In a traditional categorial grammar, a derived 
category is exhaustively described by the argument and 
value categories. But often, syntacticians want to make 
more fine grained distinctions. An example is VP 
modification. In a traditional categorial grammar, two 
different VP modifiers, lets say an adverb and an 
adverbial clause, would receive the same translation. 
(12) Peter called him angrily 
N (S\N)fN N (S\N)/(S~q) 
(13) Peter called him at work 
N (S\N)/N N (S\N)/(S~aN) 190 
But what should be the category for very? If it receives 
the category ((S\N)\(S\N))/((S\N)\(S~N)) to allow the 
derivation of (14), the ungrammatical sentence (15) is 
also permitted. 
(14) Peter called him very angrily 
N (S\N)/N N ((S\N)\(SLN))/ (S\N)/(S~X\[) 
((S\N)\(S~N')) 
(15) *Peter called him very 
N (S\N)/N N ((S\N)\(S~))/ 
((S\N)\(S\N)) 
at work 
(S\N)/(S~) 
If functor categories are permitted to carry features 
of their own that are not necessarily bound to to any 
features of their argument and value categories, this 
problem disappears. Adverbs and adverbial clauses could 
receive different features even if their categories encode 
the same combination function. 
Another solution to the problem involves the 
encoding of the difference in the value part of the functor. 
Yet this solution is not only unintuitive but also 
contradicts a linguistic generalization. It is unintuitive 
because there is no difference in the distribution of the 
resulting VPs. The only difference holds between the 
modifiers themselves. The gene~:alization that is violated 
by the encoding of the difference in the value subgraphs 
is the endocentricity of the VP. The modified VP shares 
all syntactic features with its head, the lower VP. Yet 
the feature that indicates the difference between adverbs 
and adverbial phrases could not be in both the argument 
and the value parts of the functor, otherwise iterations of 
the two types of modifiers as they occur in the following 
pair of sentences would be ruled out. 
(16a) Peter called him very angrily at work. 
(16b) Peter called him at work very angrily. 
Another augmentation is based on the PATR 
strategy for linking syntax and semantics. Most 
grammars written in PATR use the constituent graphs 
also for encoding semantic information. Every 
constituent has an attribute called trans or semantics. 
The value of this attribute contains minimally the 
internal semantic fnnction-argument structure of the 
constituent, but may also encode additional semantic 
information. The separate encoding of the semantics 
allows for a compositional semantics even in construction 
in which syntactic and semantic structure divert as in 
certain raising constructions. The following graph for a 
ficticious prenominal adjective that was introduced 
earlier contains translation attributes for the functor, 
the argument and the value. The meaning of the 
adjective is indicated by the atom Red. 
cat ~ / functi% ~rans 
Adj 
Acc ing ~._~g 
Red 
At first glance, the lexical graphs--even the ones 
that are used in the highly simplified examples--seem to 
exhibit an excessive degree of complexity and 
redundancy. However, the lexical approach to syntax is 
built on the assumption that the lexicon is structured. To 
create a lexicon that is structured according to linguistic 
generalizations, we introduced lexical templates early on 
in the development of PATR. 
Templates are graphs that contain structure shared 
by a class of lexical entries. Lexical graphs can be 
partially or fully defined in terms of templates, which 
themselves can be defined in terms of templates. If a 
template name appeam in the definition of some graph, 
the graph is simply unified with the graph denoted by the 
template. 
The next augmentation is already built into the 
formalism. Categorial grammarians have recognized the 
limitations of fimctional application as the sole mode of 
combining constituents for a long time. One of the 
obvious extensions to classical categorial grammar was 
the utilization of functional composition as a further 
combination mode. A good example of a categorial 
grammar that employs both functional application and 
functional composition is Steedman (1985). Forward 
functional composition permits the following 
combination of categories: 
(21) A/B + B/C = A/C 
The resulting category inherits the argument place for C 
from the argument B/C. 
Neither Steedman's nor any other CG I am aware of 
permits functional composition in its full generality. In 
order to prevent overgeneration, functional composition 
as well as other combination modes that are discussed by 
Steedman are restricted to apply to certain categories 
only. This somehow violates the spirit of a categorial 
grammar. Steedman's combination rules, for instance, 
are net universal. 
In CUG, functional composition is subsumed under 
functional application. It is the functor category that 
determines whether simple functional application, or 
functional composition, or either one may take place. 
Conjunction is a good case for demonstrating the 
versatility. 
Consider the following sentences: 3 
(22a) Peter andPaul like bananas. 
(22b) Peter likes bananas and Paul likes oranges. 
(22c) Peter likes and buys bananas. 
The conjunction and may combine two simple 
argument categories (22a), two functors with one unfilled 
argument position (22b), or two functors with more than 
one unfilled argument position (22c). If the conjuncts 
have unfilled argument positions, the conjoined phrase 
needs to inherit them through functional composition. 
The simplified lexical graph for and is given under (23). 
In order to avoid a thicket of crossing edges, I have 
expressed some of the relevant bindings by indices. 
191 
(23) c~ 
.... r ( 
The most appealing feature of this way of utilizing 
functional composition is that no additional combinators 
are required. No restriction on such a rule need to be 
formulated. It is only the lexical entries for functors that 
either demand, permit, or forbid functional composition. 
Extensions to the formalism that I have 
experimented with that cannot be discussed in the frame 
of this paper are the use of multiple stacks for leftward 
and rightward arguments and the DCG-like encoding of 
the ordering positions in the graphs. In Sections 3. and 
4., I will discuss further extensions of the formalism and 
specific linguistic analyses. The following section 
contains a summary of the motivations for working on 
and with CUG and the main objectives of this work. 
2. Motivation and Theoretical implications 
Both terms, unification grammar and categorial 
grammar are used for classes of grammar formalisms, for 
individual grammar formalisms, and finally for 
grammars that are written in these formalisms. In 
addition, they might also be used by linguists to denote 
linguistic theories that are built around or on top of such 
a formalism. This is the type of terminological 
overloading that linguists have learned to live with--or 
at least gotten accustomed to. 
As I indicated in the previous section, I consider 
CUG to stand for a family of grammar formalisms that 
might be described as the intersection of categorial and 
192 
unification grammar formalisms. What has been 
proposed so far is therefore not a new grammar 
formalism and even less a linguistic framework. 
The proposal is simply to further explore the 
usefulness and formal properties of subclasses of CUG. 
This proposal can be supported by a number of reasons. 
Both types of formalisms have clear advantages. 
Categorial grammars have been hailed for their 
conceptual clarity and their potentials for linking syntax 
and semantics. The fact that they have been around for a 
long time and that they are currently enjoying a 
renaissance in the works of Steedman, Bach, Dowty, and 
many others demonstrates their virtues. Unification 
grammars are spreading last and lend themselves to 
powerfifl but efficient computer implementations. 
Traditionally, categorial grammars have been 
lacking syntactic sophistication. In a functor category 
such as A/B, only domain and range of the function are 
specified but nothing is said about bow they are related; 
how, for instance, the features of the argument influence 
the features of the value. The graph notation expresses 
the relation between argument and value categories 
quite well; it is expressed in a set of bindings between 
subgraphs of the two categories. 
In the context of this discussion, some remarks are in 
order on the specific role PATR has played for the 
experiments with CUGs. The philosophy behind the 
development of PATR has been to provide a tool for 
writing, testing, and comparing grammars of very 
different types in a powerful formalism with 
well-understood formal properties and a well-defined 
semantics (Shieber 1984). 
Thus PATR could be useful for writing grammars, 
designing grammar formalisms, and for exploring classes 
of such formalisms. The work on exploring categorial 
unification formalisms has not only benefitted from the 
features of PATR but it has in a way also influenced the 
development of the PATR formalism. It was, for 
instance, essential for the writing of categorial 
grammars to allow category variables in the context-free 
phrase structure part of the rules. How else could one 
formulate the rules of functional application. The 
implementation of this facility through Stuart Shieber, 
however, raised interesting problems in connection with 
the prediction aspect of the Earley-parser. Original 
Earley prediction works on category symbols. An answer 
to these problems was presented by Shieber (1985) who 
proposed to do Earley prediction on the basis of some 
finite quotient of all constituent DAGs which can be 
specified by the grammar writer. 
Another example for the influence of the CUG efforts 
on the development of PATR is a new template notation 
introduced by Lauri Karttunen in his Interlisp-D version 
of PATR. Since categorial grammars exhibit an 
extensive embedding of categories within other 
categories, it is useful to unify templates not only with 
the whole lexical DAG but also with its categorial 
subgraphs. The @-notation permits this use of templates 
(Karttunen, 1986)3 
3. A CUG Grammar Model that Aecomodates Word 
Order Variation 
Worder order variation has always been one of the 
hardest problems for categorial grammars. Functional 
composition together with type-raising can be used to 
obtain all permutations of the sentences that are 
generated by a traditional categorial grammar. Totally 
free word order does therefore not pose an 
unsurmountable problem to the categorial approach. As 
with other types of grammar formalisms, it is semi-free 
word order that is difficult to accommedate. 
GPSG, LFG, and FUG all have mechanisms for 
encoding ordering regularities. Such a device does not 
exist in the categorial grammars that i am aware of. 
However, Uszkoreit (1985a,b) argues (on the basis of 
data fl'om German) for an application of l/near 
precedence rules to the valency list of syntactic functors. 
This approach presupposes that the valency list contains 
adjuncts as well as complements as the flmetor's 
syntactic arguments) 
The model can be summarized as follows. The 
lexicon lists uninstantiated entries. For functors, these 
entries contain a set of thematic roles. The 
uninstantiated lexical entry may also state whether 
thematic roles have to be filled, whether they may be 
filled more than once, and whether idiosyncratic 
properties of the fnnetor predetermine the syntactic 
features of certain syntactic arguments. 
There are three types of rules that instantiate lexical 
entries: feature instantiation rules, valency 
instantiation rules, and order instantiation rules. 
An instantiated functor has an ordered valency list 
containing syntactic specifications of complements and 
adjuncts together with the appropriate semantic 
bindings. The model can account for the interspersing of 
complements and adjuncts as they occur in many 
languages including English. The model can also 
account for right-extraposition phenomena. 
t Therefore, the valency list may constain adjuncts 
that do not fill a thematic role of the functor but combine 
semantically with some constituent inside a linearily 
preceding member of the same valency listfi 
In the proposed model, the dependency between the 
extraposed phrase and its antecendent is neither 
established by functional application/composition nor by 
feature passing. It is assumed that there is a different 
matching process that combines the noncontiguous 
phrases. A process of this kind is independently needed 
for the matching of adjuncts with thematic roles that are 
embedded in the meaning of the functor: 
(26a) Tellme about French history. 
(26b) Start in 1700. 
The year 1700 is obviously not the start time for the 
telling. 
(27a) His call was very urgent. 
(27b) lie tried desperately from every phone booth on 
campus. 
It is not try that supplies here the source role but the 
implicit theme of try. If the theme role is filled, everybody 
would analyze the from PP as semantically belonging to 
the theme of try: 
(28) He tried to call her desperately from every phone 
booth on campus. 
I want to conclude this discussion with a remark on 
the parsing problem connected with the proposed model. 
In older PATR Phrase-Structure grammars as well as in 
the categorial PATR grammars, all graphs that may be 
connected with a word in the input string are either 
retrieved from the lexicon or from a cache of ah'eady built 
lexical graphs, or they are constructed on the spot fi'om 
the \[exical entries through the morphology and through 
lexical rules. 
For obvious reasons, this approach cannot be used in 
conjunction with the categorial model just proposed. If 
all adjuncts are included in the valency list, and if 
moreover all acceptable linearizations are performed in 
the extended lexicon, there is no upper bound on the 
number of acceptable lexieal graphs for functors. This 
means that lexical entries cannot be fully instantiated 
when the word is recognized. \]'hey need to be 
instantiated incrementally as potential arguments are 
encountered. 
In Uszkoreit (1985b) it is argued that the ordered 
valency lists of a functor admitted by the lexical 
\[nstantiation rules form a regular language. \[f further 
research confirms this hypothesis, the incremental 
{nstantiation of valency lists could be performed through 
sets of finite state machines. 
193 
4. A Note on Long-distance Dependencies in CUGs 
In Steedman's (1985) categorial grammars, 
long-distance dependencies are endcoded in the 
function-argument structure of categories, The 
categories that form the path between filler and gap in a 
derivation tree all carry a valency slot for the filler. This 
uniform encoding of both subeategorization and 
long-distance dependencies in the argument structure of 
categories seems at first glance superior to the HPSG or 
PATR approaches to long-distance dependencies, in 
which the two types of information are marked in 
different feature sets. However, it turns out that the 
Steedman grammars have to mark the long-distance 
valency slots in order to distinguish them from other 
valency slots. 
There could still be a justification for encoding the 
two types of dependencies in the same argument stack. 
One might loose important nesting information by 
separating the two types of slots. However, I have not yet 
seen a convincing example of nesting constraints among 
subcategorization and long-distance dependencies. 
Therefore, I consider the question of the appropriate 
place for encoding long-distance dependencies still open. 
A last remark on long-distance dependencies. In a 
unification based system like PATR it is not trivial to 
ensure that gap information is passed up from one 
daughter constituent only when a rule is applied. There 
are two ways to enforce this constraint. The first one 
involves a multiplication of rules. For a binary rule A 
--> B C, for instance, one could introduce three new 
rules, one of which does not do any gaP passing, another 
one the passing of a gap from B to A, and the third the 
passing of a gap from C to A. 
PATR uses a little more elegant method which has 
been first suggested by Fernando Pereira. Two features 
are threaded through every tree, one of which carries a 
gap up a tree, passing through all the constituents to the 
left of the gap, and a second one that is set to NIL if a gap 
has been found and that is then sent through all the 
constituents to the right of the gap, unifying it on the 
way with potential gaps. It requires that information 
about the two special features be added to every rule. In 
PATR a preprocessor of rules adds this information for 
all rules in which the grammar writer did not include 
any gap threading information herself, e.g., for encoding 
island constraints. 
In a CUG that only contains two (or at least very 
• few) rules, the first method of duplicating rules appears 
preferrable over the gap threading approach. Rules that 
propagate gap information might also include rules that 
permit parasitic gaps along the lines of Steedman's rules 
of functional substitution. 
194 
Notes 
*The research for this paper was made possible through a gift by the 
System l)evelopment Foundation. 
tFor an introduction to the family of unification grammar models 
refer to Shieber (forthcmning). A good introduction to the basic 
notions ofcategorial grammar is Bar \[tiliel (1964). 
2The PATti implementations that arc currently used at SR\[ actually 
permit cyc l ic graphs. 
:IRight-Node-Raising (RNR) which leads to sentences as: Peter likes 
and Paul buys bananas will be neglected here (although RNR is an 
attractive lopic for catcgorial grammarians and one of my grammars 
~ctnally handles many cases of RNR.) 
IAn even lilt)re general notation can })e used that does not 
distinguish between root templates and subgraph templates. As long 
as template names are marked by some typographic convention. 
could be freely used wherever a graph is described. 
~The version of t!le linear precedence rule component proposed b~ 
Uszkoreit {1982. 1986) is {\tlI> compatible with this approach. The., 
proposal permit> the formalization of partially free word order as \[t 
results fl'om the interaction of potentially conflicting ordering 
principles and as it probably occurs to some degree in all natural 
languages 
6Sag (1985) proposes a mechanism for IIPSG that allows the 
syntactic binding of an extraposcd phrase to a complement or 
adjunct slot of a complement or adjunct. However, this approach is 
too restricted. Although there is a strong tendency to only extrapose 
complements and adiunets of top-level complements and adjuncts, 
there is certainly no such constraint in languages like English or 
German. The following sentence could not be handled since the 
extraposed relative clause modifies an adjunct of the subject. 
Petitions from those people were considered who had not filed a 
complaint before. 
7Mark Johnson (1986) has worked out a quasi-categorial solution of 
this phenomenon in the framework of HPSG. 

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