EFFICIENT PROCESSING OF 
FLEXIBLE CATEGORIAL GRAMMAR 
Gosse Bouma 
Research Institute for Knowledge Systems 
Postbus 463, 6200 AL Maastricht 
The Netherlands 
e-mail(earn): exriksgb@hmarl5 
ABSTRACT* 
From a processing point of view, however, 
flexible categorial systems are problematic, 
since they introduce spurious ambiguity. In 
this paper, we present a flexible categorial 
grammar which makes extensive use of the 
product-operator, first introduced by 
Lambek (1958). The grammar has the prop- 
erty that for every reading of a sentence, a 
strictly left-branching derivation can be 
given. This leads to the definition of a subset 
of the grammar, for which the spurious ambi- 
guity problem does not arise and efficient 
processing is possible. 
1. Flexibility vs. Ambiguity 
Categorial Grammars owe much of their 
popularity to the fact that they allow for 
various degrees of flexibility with respect to 
constituent structure. From a processing 
point of view, however, flexible categorial 
systems are problematic, since they intro- 
duce spurious ambiguity. 
The best known example of a flexible 
categorial grammar is a grammar containing 
the reduction rules application and compo- 
sition, and the category changing rule rais- 
ing 1 • 
* I would like to thank Esther K0nig, 
Erik-Jan van der Linden, Michael Moortgat, 
Adriaan van Paassen and the participants of 
the Edinburgh Categorial Grammar Weekend, 
who made useful comments to earlier 
presentations of this material. All remaining 
errors and misconceptions are of course my 
own. 
1 Throughout this paper we will be us- 
ing the notation of Lambek (1958), in which 
A/B and B\A are a right-directional and a 
(i) application : A/B B ==> A 
B B\A ==> A 
composition: A/B B/C ==> NC 
C\B B~, ==> C~ 
raising : A ==> (B/A)\B 
A ==> B/(A\B) 
With this grammar many alternative con- 
stituent structures for a sentence can be 
generated, even where this does not corre- 
spond to semantic ambiguities. From a lin- 
guistic point of view, this has many advan- 
tages. Various kind of arguments for giving 
up traditional conceptions of constituent 
structure can be given, but the most con- 
vincing and well-documented case in favour 
of flexible constituent structure is coordi- 
nation (see Steedman (1985), Dowty (1988), 
and Zwarts (1986)). 
The standard assumption in generative 
grammar is that coordination always takes 
place between between constituents. Right- 
node raising constructions and other in- 
stances of non-constituent conjunction are 
problematic, because it is not clear what the 
status of the coordinated elements in these 
constructions is. Flexible categorial gram- 
mar presents an elegant solution for such 
cases, since, next to canonical constituent 
structures, it also admits various other con- 
stituent structures. Therefore, the sentences 
in (2) can be considered to be ordinary in- 
stances of coordination (of two categories 
slap and (vp/ap)\vp, respectively). 
(2) a. John sold and Mary bought a book 
s/vp vp/np s/vp vp/np np 
s/np s/np 
left-directional functor respectively, looking 
for an argument of category B. 
- 19- 
b. J. loves Mary madly and Sue wildly 
vp/np np vp\vp np vp\vp 
Mary madly 
np vp\vp 
(vplnp)\vp 
(vp/np)Wp 
A somewhat different type of argument 
for flexible phrase structure is based on the 
way humans process natural language. In 
Ades & Steedman (1982) it is pointed out 
that humans process natural language in a 
left-to-right, incremental, manner. This pro- 
cessing aspect is accounted for in a flexible 
categorial system, where constituents can be 
built for any part of a sentence. Since syn- 
tactic rules operate in parallel with semantic 
interpretation rules, building a syntactic 
structure for an initial part of a sentence, 
implies that a corresponding semantic struc- 
ture can also be constructed. 
These and other arguments suggest that 
there is no such thing as a fixed constituent 
structure, but that the order in which ele- 
ments combine with eachother is rather free. 
From a parsing point of view, however, 
flexibility appears to be a disadvantage. 
Flexible categorial grammars produce large 
numbers of, often semantically equivalent, 
derivations for a given phrase. This spurious 
ambiguity problem (Wittenburg (1986)) 
makes efficient processing of flexible catego- 
rial grammar problematic, since quite often 
there is an exponential growth of the number 
of possible derivations, relative to the length 
of the string to be parsed. 
There have been two proposals for 
eliminating spurious ambiguity from the 
grammar. The first is Wittenburg (1987). In 
this paper, a categorial grammar with compo- 
sition and heavily restricted versions of 
raising (for subject n p's only) is considered. 
Wittenburg proposes to eliminate spurious 
ambiguity by redefining composition. His 
predictive composition rules apply only in 
those cases where they are really needed to 
make a derivation possible. A disadvantage 
of this method, noticed by Wittenburg, is 
that one may have to add special predictive 
composition rules for all general combina- 
- 20 - 
tory rules in the grammar. Some careful 
rewriting of the original grammar has to take 
place, before things work as desired. 
Pareschi & Steedman (1987) propose an 
efficient chart-parsing algorithm for catego- 
rial grammars with spurious ambiguity. In- 
stead of the usual strategy, in which all pos- 
sible subconstituents are added to the chart, 
Pareschi & Steedman restrict themselves to 
adding only those constituents that may lead 
to a difference in semantics. Thus, in (3) 
only the underlined constituents are in the 
chart. The "---" constituent is not. 
(3) John loves Mary madly 
s/vp vp/np np vp\vp 
Combining 'madly' with the rest would be 
impossible or lead to backtracking in the 
normal case. Here, the Pareschi & Steedman 
algorithm starts looking for a constituent 
left adjacent of madly, which contains an el- 
ement X/vp as a leftmost category. If such a 
constituent can be found, it can be concluded 
that the rest of that constituent must 
(implicitly) be a v p, and thus the validity of 
combining vp\vp with this constituent has 
been established. Therefore, Pareschi & 
Steedman are able to work with only a mini- 
mal amount of items in the chart. 
Both Wittenburg and Pareschi & Steed- 
man work with categorial grammars, which 
contain restricted versions of composition 
and raising. Although they can be processed 
efficiently, there is linguistic evidence that 
they are not fully adequate for analysis of 
such phenomena as coordination. Since 
atomic categories can in general not be 
raised in these grammars, sentence (2b) (in 
which the category n p has to be raised) 
cannot be derived. Furthermore, since 
composition is not generalized, as in Ades & 
Steedman (1982), a sentence such as John 
sold but Mary donated a book to the library 
would not be derivable. The possibilities for 
left-to-right, incremental, processing are 
also limited. Therefore, there is reason to 
look for a more flexible system, for which 
efficient parsing is still possible. 
2. Structural Completeness 
In the next section we present a gram- 
matical calculus, which is more flexible than 
the systems considered by Wittenburg 
(1987) and Pareschi & Steedman (1987), and 
therefore is attractive for linguistic pur- 
poses. At the same time, it offers a solution 
to the spurious ambiguity problem. 
Spurious ambiguity causes problems for 
parsing in the systems mentioned above, be- 
cause there is no systematic relationship 
between syntactic structures and semantic 
representations. That is, there is no way to 
identify in advance, for a given sentence S, a 
proper subset of the set of all possible syn- 
tactic structures and associated semantic 
representations, for which it holds that it 
will contain all possible semantic represen- 
tations of S. 
(5) Strong Structural Complete- 
ness 
If a sequence of categories XI .. X n 
reduces to Y, with semantics Y', 
there is a reduction to Y, with se- 
mantics Y', for any bracketing of 
XI..Xn into constituents. 
Grammars with this property, can poten- 
tially circumvent the spurious ambiguity 
problem, since for these grammars, we only 
have to inspect all left-branching syntax 
trees, to find all possible readings. This 
method will only fail if the set of left- 
branching trees itself would contain spuri- 
ous ambiguous derivations. In section 4 we 
will show that these can be eliminated from 
the calculus presented below. 
3. The P-calculus 
Consider now a grammar for which the 
following property holds: 
(4) Structural Completeness 
If a sequence of categories X1 .. X n 
reduces to Y, there is a reduction to 
Y for any bracketing of X1..Xn into 
constituents. (Moortgat, 1987:5) 2 
Structural complete grammars are interest- 
ing linguistically, since they are able to 
handle, for instance, all kinds of non-con- 
stituent conjunction, and also because they 
allow for strict left-to-right processing (see 
Moortgat, 1988). 
The latter observation has consequences 
for parsing as well,, since, if we can parse 
every sentence in a strict left-to-right man- 
ner (that is, we produce only strictly left- 
branching syntax trees), the parsing algo- 
rithm can be greatly simplified. Notice, 
however, that such a parsing strategy is only 
valid, if we also guarantee that all possible 
readings of a sentence can be found in this 
way. Thus, instead of (4), we are looking for 
grammars with the following, slightly 
stronger, property: 
2 Buszkowski (1988) provides a 
slightly different definition in terms of 
functor-argument structures. 
The P(roduct)-calculus is a categorial 
grammar, based on Lambek (1958), which 
has the property of strong structural com- 
pleteness. 
In Lambek (1958), the foundations of 
flexible categorial grammar are formulated 
in the form of a calculus for syntactic cate- 
gories. Well-known categorial rules, such as 
application, composition and category-rais- 
ing, are theorems of this calculus. A largely 
neglected aspect of this calculus, is the use 
of the product-operator. 
The calculus we present below, was 
developed as an alternative for Moortgat's 
(1988) M-system. The M-system is a subset 
of the Lambek-calculus, which uses, next to 
application, only a very general form of 
composition. Since it has no raising, it seems 
to be an attractive candidate for investigat- 
ing the possibilities of left-associative 
parsing for categorial grammar. It is not 
completely satisfactory, however, since 
structural completeness is not fully guaran- 
teed, and also, since it is unknown whether 
the strong structural completeness property 
holds for this system. In our calculus, we 
hope to overcome these problems, by using 
product-introduction and -elimination rules 
instead of composition. 
The kernel of the P-calculus is right- 
and left-application, as usual. Next to these, 
-21 - 
we use a rule for introducing the product- 
operator, and two inference rules for elimi- 
nating products : 
(6) RA : A/B B => A 
LA : B B~, => A 
(product) introduction: 
I : A B => A*B 
inference rules : 
P : AB=>C, DC=>E 
D*A B => E 
P': AB=> C, CD=>E 
A B*D => E 
We can use this calculus to produce left- 
branching syntax trees for any given 
(grammatical) sentence. (7) is a simple ex- 
ample 3 
(7) John loves Mary madly 
s/vp vp/np np vp\vp 
I 
s/vp*vp/np 
(a) 
s/vp*vp 
(b) 
S --> NP VP), or (in CG) with a reduction 
rule (application or composition, for in- 
stance), we now have the freedom to 
concatenate arbitrary categories, completely 
irrespective of their internal structure. 
The P-calculus is structurally complete. 
To prove this, we prove that for any four 
categories A,B,C,D, it holds that : 
(AB)C --> D <==> A(BC) --> D, where --> 
is the derivability relation. From this, 
structural completeness may be concluded, 
since any bracketing (or branching of syntax 
trees) can be obtained by applying this 
equivalence an arbitrary number of times. 
Proof : From (AB)C --> D it follows that 
there exists a category E such that AB --> 
E and EC --> D. BC --> B'C, by I. Now 
A(B*C) --> D, by P', since AB --> E and 
EC --> D. Therefore, by transitivity of -->, 
A(BC) --> D. To prove that A(BC) --> D 
==> (AB)C --> D use P instead of P'. 
Semantics can be added to the grammar, 
by giving a semantic counterpart (in lower 
case) for each of the rules in (6): 
(8) RA : 
I_A: 
A/B:a B:b => A:a(b) 
B:b B~A:a => A:a(b) 
(product) introduction: 
I: A:a B:b => A*B:a*b 
(a) vp/np np => vp,s/vp vp => s/vp*vp 
s/vp*vp/np np ffi> s/vp*vp 
(b) vp vp\vp => vp, s/vp vp => s 
s/vp*vp vp => s 
The first step in the derivation of (7) is the 
application of rule I. The other two reduc- 
tions ((a) and (b)) are instantiations of the 
inference rule P. As the example shows, the 
*-operator (more in particular its use in I) 
does something like concatenation, but 
whereas such operations are normally asso- 
ciated with particular grammatical rules (i.e. 
you may concatenate two elements of category 
N P and V P, respectively, if there is a rule 
3 To improve readability, we assume 
that the operators / and \ take precedence 
over * (X*Y/Z should be read as X*(Y/Z) ). 
inference rules : 
P : A:a B:b => C:c, D:d C:c => E:e 
D*A:d*a B:b => E:e 
P' : A:a B:b => C:c, C:c D:d => E:e 
A:a B*D:b*d => E:e 
We can now include semantics in the 
proof given above, and from this, we may con- 
clude that strong structural completeness 
holds for the P-calculus as well. 
4. Eliminating Spurious 
Ambiguity 
In this section we outline a subset of the 
P-calculus, for which efficient processing is 
possible. As was noted above, in the P-cal- 
culus there is always a strictly left- 
branching derivation for any reading of a 
- 22 - 
sentence S. The restrictions we add in this 
section are needed to eliminate spurious am- 
biguities from these left-branching deriva- 
tions. 
Restricting a parser so that it will only 
accept left-branching derivations will not 
directly lead to an efficient parsing proce- 
dure for the P-calculus. The reason is 
twofold. 
First, nothing in the P-calculus excludes 
spurious ambiguity which occurs within the 
set of left-branching analysis trees. Con- 
sider again example (7). This sentence is 
unambiguous, but nevertheless we can give a 
left-branching derivation for it which dif- 
fers from the one given earlier : 
(9) John loves Mary madly 
s/vp vp/np np vp\vp 
I 
s/vp*vp/np 
I 
(s/vp*vp/np)* np (**) 
s 
The inference step (**) can be proven to be 
valid, if we use P' as well as P. 
An even more serious problem is caused 
by the interaction between I and P,P'. 
(10) AB=>A*B A*B C=>D 
BC=>B*C A B*C=>D 
P 
p, 
A*B C => D 
If we try to prove that A*B and C can be re- 
duced to a category D, we could use P, with I 
in the left premise. To prove the second 
premise, we could use P', also with using I in 
the left premise. But now the right premise 
of P' is identical to our initial problem; and 
thus we have made a useless loop, which 
could even lead to an infinite regress. 
These problems can be eliminated, if we 
restrict the grammar in two ways. First of 
all, we consider only derivations of the form 
C1,...,Cn ==> S, where C1,...,Cn,S do not 
contain the product-operator. This means we 
require that the start symbol of the grammar, 
and the set lexical categories must be prod- 
uct-free. Notice that this restriction can be 
easily made, since most categorial lexicons 
do not contain the product-operator anyway. 
Given this restriction, the inference rule 
P can be restricted: we require that the left 
premise of this rules always is an instance of 
either left- or right-application. Consider 
what would happen if we used I here : 
(11) 
BC=>B*C A B*C=>D 
P 
A*B C => D 
Since the lexicon is product-free, and we are 
interested in strictly left-branching deriva- 
tions only, we know that C must be a prod- 
uct-free category. If we combine B and C 
through I, we are faced with the problem in 
***. At this point we could use I again for in- 
stance, thereby instantiating D as A*(B*C). 
But this will lead to a spurious ambiguity, 
since we know that: 
A*(B*C) E => F iff (A*B)*C E--> F 4. 
A category (A*B)*C can be obtained by ap- 
plying I directly to A*B and C. 
If we apply P' at point ***, we find our- 
selves trying to find a solution for A B => 
E, and then E C => D. But this is nothing 
else than trying to find a left-branching 
derivation for A,B,C => D, and therefore, 
the inference step in (11) has not led to 
anything new. 
In fact, given that the lexicon is product 
free and only application may be used in the 
left premise of P, P' is never needed to derive 
a left-branching tree. 
As a result, we get (12), where we have 
made a distinction between reduction rules 
(right and left-application) and other rules. 
This enables us to restrict the left premise 
of P. The fact that every reduction rule is 
also a general rule of the grammar, is ex- 
pressed by R. P' has been eliminated. 
4 In the P-calculus, this follows from 
the fact that E must be product-free. It is a 
theorem of the Lambek-calculus as well. 
c,-o - 23 - 
(12) RA : A/B B -> A 
LA : B B~A -> A 
(product) introduction: 
I: A B =>A*B 
inference rules : 
P : AB->C, DC=>E 
D*A B => E 
R : AB->C 
sentence like (7) using a shift-reduce pars- 
ing technique, and having only right- and 
left-application as syntax rules. 
(1 4) (remaining) input stack 
s/vp,vp/np,np,vp\vp _ 
S vp/np,np,vp\vp s/vp 
S np,vp\vp s/vp,vp/np 
S vp\vp s/vp,vp/np,np 
R vp\vp s/vp,vp 
S _ s/vp,vp,vp\vp 
R _ s/vp,vp 
R s 
AB=>C 
The system in (12) is a subset of the P- 
calculus, which is able to generate a strictly 
left-branching derivation for every reading 
of a given sentence of the grammar. 
The Prolog fragment in (13) shows how 
the restricted system in (12) can be used to 
define a simple left-associative parsing algo- 
rithm. 
(13) parse(\[C\] ==> C) :- !. 
parse(\[Cl,C21Rest\] ==> S) :- 
rule(\[C1,C2\] ==> C3), 
parse(\[C31Rest\] ==> S). 
%R: 
rule(X --=> Y) :- 
reduction_rule(X ==> Y). 
%P: 
% '+' is used instead of '*' to avoid 
% unnecessary bracketing 
rule(\[X+Y,Z\] ==> W) :- 
reduction_rule(\[Y,Z\] ==> V), 
rule(\[X,V\] ==> W). 
% I: 
rule(\[X,Y\] ==> X+Y). 
% application • 
reduction_rule(\[X/Y,Y\] ==> X). 
reduction_rule(\[Y,Y\X\] ==> X). 
5. Shift-reduce parsing 
It has sometimes been noted that a 
derivation tree in categorial grammar (such 
as (7)) does not really reflect constituent 
structure in the traditional sense, but that it 
reflects a particular parse process. This may 
be true for categorial systems in general, but 
it is particularly clear for the P-calculus. 
Consider for instance how one would parse a 
Shifting an element onto the stack (apart 
form the first one maybe) seems to be equiv- 
alent to combining elements by means of I. 
The stack is after all nothing but a somewhat 
different representation of the product types 
we used earlier. The fact that adding one el- 
ement to the stack (vp\vp) induces two re- 
duction steps, is comparable to the fact that 
the inference rule P may have the effect of 
eliminating more than one slash at time. 
The similarity between shift-reduce 
parsing and the derivations in P brings in 
another interesting aspect. The shift-reduce 
algorithm is a correct parsing strategy, be- 
cause it will produce all (syntactic) ambigu- 
ities for a given input string. This means 
that in the example above, a shift-reduce 
parser would only produce one syntax tree 
(assuming that the grammar has only appli- 
cation). 
If the input was potentially ambiguous, 
as in (15), there are two different deriva- 
tions. 
(15) aJa a a\a 
It is after shifting a on the stack that a 
difference arises. Here, one can either re- 
duce or shift one more step. The first choice 
leads to the left-branching derivation, the 
second to the right-branching one. 
The choice between shifting or reducing 
has a categorial equivalent. In the P-calcu- 
lus, one can either produce a left-branching 
derivation tree for (15) by using application 
only, or as indicated in (16). 
- 24 - 
(16) a/a a a\a 
I 
~a*a 
P 
Note that the P-calculus thus is able to 
find genuine syntactic (or potentially se- 
mantic) ambiguities, without producing a 
different branching phrase structure. The 
correspondence to shift-reduce parsing al- 
ready suggests this of course, since we 
should consider the phrase structure pro- 
duced by a structurally complete grammar 
much more as a record of the parse process 
than as a constituent structure in the tradi- 
tional sense. 
6. Coordination 
The P-calculus is structurally complete, 
and therefore, all the arguments that have 
been presented in favour of a categorial 
analysis of coordination, hold for the P-cal- 
culus as well. Coordination introduces poly- 
morphism in the grammar, however, and this 
leads to some complications for the re- 
stricted P-calculus presented in (12). 
Adding a category X\(X/X) for coordina- 
tors to the P-calculus, enables us to handle 
non-constituent conjunction, as is exempli- 
fied in (17) and (18). 
(17) John loves and Peter adores Sue 
s/vp vp/np X\(X/X) s/vp vp/np np 
I 
s/vp*vp/np s/vp*vp/np 
s/vp*vp/np 
S 
(18) J. loves Mary madly and Sue wildly 
vp/np np vp\vp X~(X/X)np vp\vp 
np*vp\vp np*vp\vp 
np*vp\vp 
vp 
p, 
The restricted-calculus of (12) was de- 
signed to enable efficient left-associative 
parsing. We assumed that lexieal categories 
would always be product-free, but this as- 
sumption no longer holds, if we add X\(X/X) 
to the grammar (since X can be instantiated, 
for instance as s/vp*vp/np). This means 
that left-associative derivations are not 
always possible for coordinated sentences. 
Our solution to this problem, is to add 
rules such as (19) to the grammar, which can 
transform certain product-categories into 
product-free categories. 
(19) A/(B*C) ~> (A/C)/B 
A number of such rules are needed to restore 
left-associativity. 
Next to syntactical additions, some 
modifications to the semantic part of the 
inference rule P had to be made, in order to 
cope with the polymorphic semantics 
proposed for coordination by Partee & Rooth 
(1983). 
7. Concluding remarks. 
The spurious ambiguity problem has 
been solved in this paper in a rather para- 
doxical manner. Whereas Wittenburg (1987) 
tries to do away with ambiguous phrase 
structure as much as possible (it only arises 
where you need it) and Pareschi & Steedman 
(1987) use a chart parsing technique to re- 
cover implicit constituents efficiently, the 
strategy in this paper has been to go for 
complete ambiguity. It is in fact this 
massive ambiguity, which trivializes 
constituent structure to such an extent that 
one might as well ignore it, and choose a con- 
stituent structure that fits ones purposes 
best (left-branching in this case). It seems 
that as far as processing is concerned, the 
half-way flexible systems of Steedman 
(having generalized composition, and heavily 
restricted forms of raising) are in fact the 
hardest case. Simple AB-grammars are in all 
respects similar to CF-grammars, and can di- 
rectly be parsed by any bottom-up 
algorithm. For strong structurally complete 
systems such as P, spurious ambiguity can 
be eliminated by inspecting left-branching 
trees only. For flexible but not structurally 
complete systems, it is much harder to pre- 
dict which derivations are interesting and 
which ones are not, and therefore the only 
solution is often to inspect all possibilities. 
- 25 - 
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