Expressing generalizations 
in unification-based grammar formalisms * 
Marc Moens, Jo Calder 
Ewan Klein, Mike Reape, Henk Zeevat 
Centre for Cognitive Science, University of Edinburgh 
2, Buccleuch Place, Edinburgh EH8 9LW 
Scotland, UK 
Abstract 
This paper shows how higher levels of general- 
ization can be introduced into unification gram- 
mars by exploiting methods for typing grammati- 
cal objects. We discuss the strategy of using global 
declarations to limit possible linguistic structures, 
and sketch a few unusual aspects of our type- 
checking algorithm. We also describe the sort 
system we use in our semantic representation lan- 
guage and illustrate the expressive power gained 
by being able to state global constraints over these 
sorts. Finally, we briefly illustrate the sort system 
by applying it to some agreement phenomena and 
to problems of adjunct resolution. 
1 Introduction 
Since Kay's seminal work (Kay 1979), the util- 
ity of unification as a general tool in computa- 
tional linguistics has gained widespread recogni- 
tion. One major point on which the methodology 
of unification grammars differs radically from that 
assumed by linguistic theories lies in the way they 
deal with generalizations that hold over the do- 
main of description. In unification-based theories, 
such generalizations are typically implicit, or ex- 
tremely limited in their import. The reasons for 
this are easy to pinpoint. First, in such theories 
one has to be explicit about the feature structures 
that the grammar manipulates, and these struc- 
tures have to be described more or less directly. In 
PATR-II for example (Shieber et al 1983) the only 
means of expressing a generalization is via the no- 
tion of template, a structure which merely repre- 
sents recurring information--i.e, information that 
*The work reported here was carried out ae part of ES- 
PRIT project P393 ACORD. A longer version of this paper 
can be found in Calder et a! (1988a). 
recurs in different lexical items, combination rules, 
lexical rules or other templates. A second reason 
why unification-based theories do not lend them- 
selves easily to the expression of general state- 
ments is that there is no explicit quantification in 
unification formalisms. In fact, every statement 
in these formalisms represents a simple existential 
constraint, never a universal generalization. 
The work reported here is an attempt to intro- 
duce higher levels of organization into unification 
grammars. The notions we employ to do this come 
from sorted logics and from strong data typing in 
programming language theory. We will show that 
the typing of grammatical objects offers a way of 
stating structural constraints on, or equivalently 
universal properties of, the objects that constitute 
the grammar. 
The grammatical framework in which these 
ideas have been implemented is Uaificatioa Cat- 
egorial Grammar (UCG) and its semantic repre- 
sentation language InL, both developed as part 
of the ESPRIT-funded project ACORD. Introduc- 
tions to UCG and InL can be found in Calder et al 
(1988b) and Zeevat (1988). For present purposes 
it is sufficient to note that UCG uses a sorted logic 
which requires being able to express complex con- 
straints over clusters of features. While there is no 
real distinction between this technique and that of 
data typing mentioned above, we will nevertheless 
continue to use the term typing only to refer to 
constraints on the global structure of an object 
and reserve the term sort to refer to constraints 
that hold of a variable in InL. 
In the following sections, we will first discuss 
our strategy of using global declarations to limit 
possible linguistic structures. We will briefly de- 
scribe some of the type declarations currently im- 
plemented in UCG and discuss the unusual aspects 
of our type-checking algorithm. We will also infor- 
174 - 
mally describe the InL sort system and will show 
how the ability to express global constraints on 
the sort lattice is both perspicuous and expres- 
sively powerful. Detailed discussion of the under- 
lying formal theory and the implementation can 
be found in Calder et al (1988a) and will not be 
attempted here. 
Next, we will demonstrate the usefulness of the 
sort system by describing ucG's adjunct resolu- 
tion system, the declarative semantics of which de- 
pends crucially on our use of a logic of sorts. This 
treatment allows the grammar writer to write and 
add adjunct resolution conditions using the same 
notation as that used to express sort descriptions 
in the grammar and without having to modify any 
implementation code. 
2 Types in UCG 
Importing the notion of data typing into 
unification-based grammars has several advan- 
tages (cf. also Calder et al 1986, Calder 1987). 
To begin with, the use of data typing allows one 
to show whether a grammar is consistent with a 
set of statements about the possible structures 
allowed within the grammar. This compile-time 
type-checking of the structures designed by the 
grammar writer allows more useful error informa- 
tion to be presented to the grammar writer. We 
have found such information essential in writing 
large grammars for the ACORD project. 
Second, data typing forces the grammar writer 
to make the structure of linguistic objects explicit. 
This higher level of organization makes it easier to 
pinpoint aspects of the grammar which are inele- 
gant or inefficient. 
Finally, the notion of typing represents a fur- 
ther step towards the goal of making local struc- 
tures reflect global restrictions. This move is an 
essential part of the programme of characterizing, 
within a formal computational theory, linguistic 
devices such as GPSG's feature co-occurrence re- 
strictions. 
A standard way of defining categorlal grammars 
is to provide a set of basic categories and one 
or more recursive rules for defining complex cate- 
gories. A very similar definition holds in uCG. Fol- 
lowing Pollard & Sag (1987), we treat every ucG 
object, apart from the rules, as a sign. That is, it 
represents a complex conjunction of phonological, 
syntactic and semantic information. We can fur- 
ther specify a sign by adding constraints on legal 
instantiations of each of the sign's attributes: for 
example, semantics in UCG has a tripartite struc- 
ture, consisting of an index, a predicate and an 
argument list. 
It is obvious that the abstract structure of each 
of these categories must be known in advance to 
the interpreter. The formalism we will use here for 
declaring types is borrowed from Smolka (1988), 
and the following illustrates his matrix notation 
for record structures, where type symbols are writ- 
ten in bold face, and feature symbols are written 
in italics 1: 
sign 
phonology : phonlist LJ basic 
(1) category : complex U beat u basic 
semantics : variable U formula 
The structure as a whole is declared to be of 
type sign, and it is defined for exactly three fea- 
tures, namely phonology , category , and semantics. 
We also show, for each feature, the types of the 
values that it takes; as it happens, these are all 
disjunctive. So, for example, the feature seman- 
tics has a value either of type variable or of type 
formula. 
Obviously, further information has to be given 
about what constitute legal structures of type for- 
mula. As was mentioned above, semantic formu- 
lae in InL are typically tripartite: 
formula 
index : variable 
(2) predicate : basic U list 
arglist : basic U sem_m-gs 
For present purposes, it suffices to know that 
the first element is the index, a privileged variable 
representing the ontological type and the identity 
of the semantic structure. Next, there is the pred- 
icate. This may be basic or a list of atoms. The 
type basic is the only type provided as a primitive 
in the system, and indicates that only instantia- 
tions to an atomic value (in the PROLOG sense of 
atomic) are legal. In the case where the predicate 
is a list, it represents a disjunction over adjunct 
functions, as will be discussed below. 
Further discussion of (1) and (2) is not possible 
within the limited space here. The examples are 
only intended to illustrate how at each level in 
a UCG sign, type specifications can be given that 
indicate restrictions on the value any given feature 
may take on. However, one point deserves further 
XSmolka uses the term 'sort' in place of 'type'; however, 
as already mentioned, we reserve the former for talking 
about InL expressions. 
- 175 - 
comment. It will be recalled that earlier we said 
the structure (1) was ~defined for exactly three 
features". It follows from this that, for example, 
(lt) would not be a legal instantlation of this type: 
sign 
phonology : value_a 
(lt) category : value_b 
8eraantic$ : value_c 
arglist : value_d 
Thus, types in UCG are closed: all features which 
are not explicitly stated as defined in a particular 
type declaration are held to be undefined for that 
type (i.e. they can only be specified as .1_). Con- 
sequently, closed types offer a form of universal 
quantification over features. This device offers a 
way of characterizing the well-formedness of dif- 
ferent dimensions of a sign that is stronger than 
systems based on open types, such as HPSG. 2 
The UCG compiler uses declarations like those in 
(1) and (2) to check variables for consistent typ- 
ing. This involves keeping track of all variables 
introduced by a particular UCG expression as well 
as of the possible types that a variable may be 
assigned. The compiler proves that, for multiple 
occurrences of the same variable, the intersection 
of the sets of possible types induced for each oc- 
currence of the variable is non-empty. If the set is 
empty, the compilation process fails and an error 
is reported. 
This technique has the advantage that one may 
partition the set of variables employed by the sys- 
tem. Thus in ucG, the set of PROLOG variables 
that is used to represent variables in an InL for- 
mula is disjoint from the set used to represent the 
predicate introduced by a sign: the type of vari- 
ables of the first set is stated to be variable, while 
the type of those of the second set is predicate. 
This property is crucial if we wish to check for 
correctness of highly underspecified structures. 
3 The sort system 
The ontological types of InL indices are formalized 
by dividing the set of InL variables into sorts. Tak- 
ing results from work in automated theorem prov- 
ing (Cohn 1984, Walther 1985), the use of sorted 
variables in InL was first presented in Calder et 
al (1986). Similar proposals have also been made 
in the SRI Core Language Engine (Aishawi et al 
2See Uszkoreit (1987) and Bouma et ag C1988) for a sys- 
tem that allows the flexible combination of open and closed 
types. 
1988) and in recent HPSG work on referential pa- 
rameters (Pollard & Sag 1988). 
As a first approximation, InL sorts can be iden- 
tified with bundles of feature-value pairs, such as 
(3) l-Temporal, +Human, +Singu r\] 
However, the standard linguistic notation for 
feature bundles is too restricted, since it only al- 
lows conjunction and negation of atoms. We find 
it useful to use a full propositional language ~ort 
for expressing sortal information, where each fea- 
ture specification of the form -/-F is translated into 
~oort as an atomic proposition F, and each spec- 
ification -F is translated as a negated atom -~F. 
Thus, in place of (3) we write the following: 
(4) -.Temporal ^ Human A Singular 
This is construed as a partial description of el- 
ements in the semantic domain used to interpret 
InL. In order to calculate the unification of two 
sorted variables, we conjoin the associated sort for- 
mulae and check for consistency. 
The design of the sort structure as a theory of 
propositional logic also allows the incorporation of 
background constraints or axioms with which ev- 
ery possible description in the structure is consis- 
tent. Let's call the theory Tsort. A few examples 
of these background axioms in Teort are given in 
(5) to (9): 
(5) Temporal --* Neuter V Plural 
(6) Neuter --* Singular A --Human 
(7) Singular --* Objectual 
(8) Measure --* 
Objectual A (Tmeasure V Lmeasure) 
(9) Stative --, Eventual 
From (5), (6) and (7) it follows that the unifi- 
cation of an index of sort Temporal and an index 
of sort Neuter should give us an index of sort 
(10) Objectual ^ Singular ^ --Human 
And from (8) it follows that anything that is 
Tmeasure is also Objectual. This implicit deduc- 
tive capacity is useful in specifying concisely and 
accurately the sort of an index. 
A few examples will help clarify these distinc- 
tions. Below are listed the lexical definitions for 
some of the nouns in the current lexicon. In these 
definitions, the items preceded by the symbol ~Q" 
are templates, in the sense of PATR-II. Templates 
whose names are the unabbreviated form of sort 
names instantiate the indez of the aemantiee of 
a sign to the corresponding sort. For example, 
UQExtended" specifies the sort of the InL vari- 
ables as Eztended, ~QNeuter ~ as Neuter, etc. 
tomato: \[QNoun, QNeuter, QExtended, 
:pred -- tomato\]. 
176 - 
o4 
Tempor   ! 
MassPlur Singular 
uter 
PlUral~ Mass Male Female 
actual 
~asure /\ 
Tmeasure 
J 
~vventual 
Stative Nonstative 
Lmeasure/ ~ 
Process Event / i 
Figure 1: Sort lattice (overview) 
inquiry: \[~Noun, ~Temporal, QNeuter, 
:pred = inquiry\]. 
organlsation: \[QNoun, ~Neuter, QAbstract, 
:pred = organisation\]. 
miles: \[~Noun, QLmeasure, ~Plural, 
:pred = mile\]. 
night: \[~Noun, QNeuter, QTmeasure, 
:pred = night\]. 
A tomato is obviously an object with spatial ex- 
tent. It is also Neuter, which implies---given the 
axiom in (6) above--that it is also 8iagular, and 
not Humaa. An inquiry is also Neuter, but it has 
a temporal dimension; a time span can be predi- 
cated of it. An organisation is an abstract entity; 
it is, moreover, Neuter (implying it is a singular 
object). Finally, miles has the index Lmeasure 
since it can be used in measure phrases to express 
the length of something; and night is Tmeasure 
which means it can be used to express the tempo- 
ral duration of something. 
The standard consequence relation over these 
partial descriptions (i.e. the formulae of ~,ort) in- 
duces a lattice (cf. Mellish 1988). Moreover, the 
sets of models associated with these partial de- 
scriptions (i.e. the truth assignments to the formu- 
lae) also form a lattice, ordered by the set inclusion 
relation. This lattice is isomorphic to the lattice of 
descriptions. The model sets can be encoded as bi- 
nary bit strings where a zero bit indicates that the 
corresponding model is not a member of the model 
set and a one bit indicates the opposite. Model 
set intersection is equivalent to bitwise conjunc- 
tion and model set union to bitwise disjunction. 
Testing for the satisfiability of the conjunction of 
two descriptions can consequently be performed 
in two machine instructions, viz. taking the bit- 
wise conjunction of two model" set encodings and 
testing for zero (el. Proudian & Pollard 1985). 
Such a model set encoding is obviously linear in 
the number of models it generates; in the worst 
case, the number of models is exponential in the 
number of propositional constants mentioned in 
T,o,t, but typically it is much less. This means 
that the exponential complexity involved in test- 
ing for satisfiability can be compiled away offline; 
the resulting model set encoding can be used with 
equal computational efficiency. 
As illustrated above, the statements that de- 
fine the lattice of sorts can be arbitrary state- 
ments in classical propositional logic. This is in 
distinction to systems discussed by Mellish (1988) 
and Alshawi et al (1988), in which the set of logi- 
cal connectives is restricted to those for which an 
encoding exists using PROLOG terms without re- 
peated variables and for which PROLOG unification 
provides an immediate test of the compatibility 
of two descriptions. The resulting sort definition 
language is therefore more expressive. The major 
drawback of such an approach is that the encoding 
- 177- 
Objectual 
Measur~ ~ " Temporal Extended Smgular "~-~stract 
~ ~ ~ Neuter / 
\ f 
fjf_¾J- \/ \ / 
mile inquiry pig butter love 
Figure 2: Sort lattice plus examples (detail) 
in terms of sets of satisfying models prevents the 
statement of reentrant dependencies between fea- 
tures in the sort system and features in the rest 
of the grammar. A more general, but computa- 
tionally less efficient approach would use general 
disjunction and negation over feature structures, 
as discussed by Smolka (1988), and so give a uni- 
form encoding of sortal and general grammatical 
information. 
Figure 1 depicts part of our current lattice of 
sorts. It is not complete in that not all the sorts 
we currently use are represented in Figure 1, nor 
are all the meets of the sorts in Figure 1 repre- 
sented. Figure 2 gives an enlarged fragment of 
Figure 1, showing a more complete picture of the 
sorts related to Neuter, as well as some instantia- 
tions of these sorts in English. 
The fact that the lattice soon becomes rather 
complicated isn't particularly worrisome: the 
grammar writer need only write simple back- 
ground axioms in Taort, like the ones in (5) to 
(9), to extend or otherwise change the sort lattice. 
To check for plausibility, the grammar writer can 
also ask for the models or truth assignments to the 
properties of the sort system. 
In UCG, sortal restrictions have been used to 
capture certain agreement phenomena. Collective 
nouns like committee, for example, are lexlcally 
marked as being either Neuter or Plural (for which, 
of course, the term Collective can be introduced). 
In British English, this allows anaphoric reference 
by means of a singular as well as a plural pronoun: 
(11) The committee met yesterday. It/They re- 
jected the proposal. 
Proper binding of the pronoun in (11) requires 
the index associated with it or they to be identical 
with that introduced by committee. Since com- 
mittee is marked as either Neuter or Plural, both 
bindings are possible. 
However, once the choice has been made (as in 
(12a) and (b)) the referential index for committee 
has become specified more fully (as being either 
singular or plural) and further pronominal refer- 
ence in the discourse is restricted (as illustrated 
in (c) and (d)) (cf. Klein & Sag 1982, and more 
recently Pollard & Sag 1988 on this issue): 
(12a) The committee has rejected its own pro- 
posal 
(12b) The committee have rejected their own 
proposal. 
(12c) *The committee has rejected their own 
proposal. 
(12d) *The committee have rejected its own pro- 
posal 
Note that sorts like Plural or Neuter are not syn- 
tactic features, but are part of the internal struc- 
ture of referential indices introduced through the 
usage of certain expressions. These indices are ab- 
stract objects whose function in a discourse repre- 
sentation it is, amongst other things, to keep track 
of the entities talked about in the discourse. 
Of course, sorts like Plural or Human also have 
a semantic import in that they permit real-world 
non-linguistic objects to be distinguished from 
one another (cf. Hoeksema (1983) and Chierchia 
(1988) on a similar use of indices in theories of 
agreement and binding). Nevertheless, the aim of 
the sort system is not to reflect the characteris- 
tics of real world objects and events referred to by 
linguistic expressions, but rather to systematize 
the ontological structure evidenced by linguistic 
expressions. 
The usefulness of being able to express global 
constraints over the sort lattice can best be illus- 
trated by considering the treatment of adjunct res- 
olution in UCG. It is to a brief account of this that 
we turn next. 
- 178 - 
4 Adjunct resolution 
Ambiguity in the attachment of prepositional 
phrases is a longstanding problem in the area of 
natural language processing. We suggest that this 
ambiguity has two basic causes. First, there is 
structural ambiguity in that prepositional phrases 
may modify at least nouns and verb phrases. This 
structural ambiguity is a cause of inefficiency in 
processing. Second, prepositions may have sev- 
eral distinct, if related, meanings. (This problem 
becomes even more acute in a multilingual set- 
ting with a common semantic representation lan- 
guage). Such ambiguity then represents an in- 
determinacy for theorem provers and knowledge 
bases that deal with the output of a natural lan- 
guage component. 
The mechanisms we have introduced above al- 
low us to address both these problems simulta- 
neously. We use the term adjunct resolution to 
describe the situation in which the possible mean- 
ings of a preposition, perhaps drawn from a uni- 
versal set of possible prepositional meanings, and 
the possible attachments of a prepositional phrase 
are mutually constraining. 
To consider the problem from the multilingual 
point of view, the way in which a particular lan- 
guage uses its prepositions to decompose the set of 
spatial and temporal relations that obtain between 
objects and events may well be inconsistent with 
the decomposition shown in othdr languages. For 
example, the French preposition dana can express 
spatial location (il eat dans la ehambre - he is in 
the room), spatial inclusion (dans un rayon de 15 
kilomdtres - within a radius of 10 m//es), spatial 
path (il passerait dans le feu pour ells - he'd 9o 
through fire for her sake), spatial source (copier 
quelque chose dans un liars - copy somethin9 from 
a book), and several other relations. 
In the semantic representation language InL, 
the meaning of a preposition is a relation between 
two InL indices. Thus the translation of a sentence 
like 
(14) John walked to the store 
would be 
(15) \[e\]\[walk(e,john) & store(x) 
& direction(e,x)\] 
where "direction(e,x) ~ represents a relation be- 
tween the going event and the store. However, as 
noted above, a preposition will typically introduce 
a disjunction over relations. The French preposi- 
tion dana, for example, will have as its translation 
a disjunction of spatial location, spatial inclusion, 
spatial source and spatial path. Some of these it 
will share with the English preposition in; others 
will be shared with within, through and the other 
prepositions mentioned above. 
Let us look at an English example in some more 
detail An adjunct phrase introduced by with can 
express (without aiming to be exhaustive) an ac- 
companiment relation (as in 18a), the manner in 
which an act was carried out (18b), the instrument 
with which it was carried out (illustrated in 18c), 
or something which is part of something or owned 
by someone (as in 18d). 
Sortal restrictions on the arguments of these re- 
lations are expressed by means of the three-place 
predicate sort_restriction: 
(16) sort_restriction(RELATION, 
HEAD.INDEX, 
MODIFIER_INDEX). 
In (16), RELATION is a possible adjunct rela- 
tion (or a list of adjunct relations, interpreted 
disjunctively), HEAD_INDEX represents the condi- 
tions on the index of the expression modified by 
the adjunct, and MODIFIER_INDEX likewise states 
restrictions on the index of the object that is part 
of the modifier phrase. 
An instance of this schema is (17): 
(17) sort_restriction(instrument, 
-"Stative A Eventual, 
Extended A --Human) 
The declaration in (17) restricts instruments to 
be non-human, extended objects. They can, more- 
over, only be combined with nonstative or event 
expressions. This rules out an instrumental read- 
ing for the wit~phrases in (lSa) and (b) (since 
teacher will be marked in the lexicon as Human, 
and effort is Abstract), and for (18d) (since the 
man is not EventuaO, but allows it for (c): 
(18a) Lisa went to Rome with her teacher. 
(18b) He ran with great effort. 
(18c) He broke the window with a hammer. 
(18d) There's the man with the funny nose. 
The restrictions on accompaniment, manner and 
possession are given as follows: 
(19) sort_restriction(accompaniment, 
Eventual, 
Extended) 
(20) sort_restriction(manner, 
--Stative A Eventual, 
Abstract) 
(21) sort.restriction(possession, 
Objectual, 
Extended A "-Human) 
It is easy to verify that (19) rules out an ac- 
companiment reading for (18b) (since effort is not 
g, tende  and for (18d) (since man is not Even- 
- 179 - 
tual). (20) renders a manner reading impossible 
for (18a), (c) and (d), since neither teacher, ham- 
mer or nose are Abstract. Finally, (21) rules out a 
possession relation for (18a) and (b). 
In some cases the sortal restrictions will reduce 
the disjunction of possible readings to a single one, 
although this is obviously not a goal that is al- 
ways obtainable or even necessary for the seman- 
tics component of a natural language system. 
As the discussion of the with-clauses shows, in 
some cases PP attachment ambiguity may be re- 
duced by restrictions associated with particular 
adjunct prepositions. A standard example of such 
an ambiguity is 
(22) John saw the man with a telescope. 
There are two readings to this sentence, repre- 
sented by these two bracketings: 
(23a) \[vpsaw \[Npthe man \[ppwith a telescope\]\]\] 
(23b) \[vP \[vpsaw the man\]\[ppwith a telescope\]\] 
Due to the restrictions given above, only the pos- 
session relation may hold between man and tele- 
scope in (23a), while in (b) only the relations ac- 
companiment or instrument may hold between the 
telescope and the event of seeing. 
In some cases, the sortal restrictions may actu- 
ally remove prepositional attachment ambiguities 
altogether. Examples (24) are predicted by most 
theories to be ambiguous: 
(24a) John will eat the tomato in two hours. 
(24b) John will eat the tomato in his ofllce. 
The ambiguity arises because the prepositional 
phrase may attach low, to the noun phrase, or 
high, modifying the verb phrase. In the system 
described here, the first sentence is not ambigu- 
ous. The preposition in introduces a disjunction 
between (amongst other things) spatial location 
and duration. The former can relate an object 
with any other object or event. The latter rela- 
tion can only hold of expressions involving some 
temporality; as was illustrated above, tomato has 
no temporal extent, therefore does not allow this 
kind of temporal time-span to be predicated of it. 
As a result, the prepositional phrase in (24a) can 
only get high attachment. 
Although the discussion has been limited to the 
use of sortal information in adjunct resolution and 
the treatment of certain agreement phenomena, it 
should be clear that exactly the same mechanism 
may be used to indicate sortal restrictions asso- 
ciated with any other predicates of the system. 
Thus we have one way of expressing the linguis- 
tic concept of selectional restrictions. We realize 
that care has to be taken here, since there is no 
well-defined point at which statements about nor- 
tal correctness become clearly inappropriate. For 
instance, we might be tempted to treat the ambi- 
guity associated with the verb bank as in Ronnie 
banked the cheque and Maggie banked the MIG by 
invoking a feature monetary for the first example 
and a feature manoeuvrable for the second. If we 
had a clear picture of precisely those properties 
that might be invoked for lexical disambiguation, 
this approach might be tenable. It seems more 
likely to be the case that the features and axioms 
about those features used in a particular case are 
ad hoc and domain-specific, as their creation and 
definition would be governed by just those lexi- 
cal items one wanted to distinguish. Also they 
are language-specific, as patterns of homography 
presumably do not hold cross-linguistically. It is, 
nevertheless, plausible (following Kaplan 1987) to 
assume that the techniques we have introduced 
could be employed in the automatic projection of 
non-lexical knowledge into the lexicon. 
The notation we have presented above for the 
definition of sorts and the relations between sorts 
that prepositions represent may appear somewhat 
removed from the notation introduced in section 2 
in our discussion of typed grammatical objects. It 
is however worth noting that the use of ~order- 
sorted algebras" (Meseguer et al 1987) as the 
mathematical basis of feature structures allows 
not only the statement of such restrictions on the 
structure of grammatical and semantic objects, 
but also the definition of relations, like our prepo- 
sitional relations above, whose interpretation is 
dependent on the interpretation of the structures 
they relate. Such formalisms may well provide 
a useful foundation for a more general theory of 
prepositional meaning and its relation to syntac- 
tic structure. 
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