A LOGICAL FORMALISM FOR THE REPRESENTATION OF DETERMINERS f 
Barbara Di Eugenio~ Leonardo Lesmo, Paolo Pogliano, 
Pietro Torasso, Francesco Urbano 
Dipartimento di Informatica - Universita' di Torino 
Via Valperga Caluso 37 ~ 10125 Torino ~ Italy 
ABSTRACT 
Determiners play an important role in conveying the meaning of 
an utterance, but they have often been disregarded, perhaps 
because it seemed more i~portant to devise methods to grasp the 
global meaning of a sentence, even if not in a precise way. 
Another problem with determiners is their inherent a~bigulty. 
In this paper we propose a logical formaliem, which, among 
other things, is suitable fc~ representing determiners without 
forcing a particular interpretation when their meaning is still 
not clear. 
INTRODUCTION 
Ambiguity of determiners is one of the most striking phenomena 
of natural language; what is strange is the case with ~hich humans 
use them: it seems that the molteplicity of interpretations of a 
ncun phrase including a determiner is not explicitly perceived by 
human users of rmtural language \[Hobbs 1983\]. The approach we 
chose tries to model this behavier: each determiner has a charac ~ 
teristic semantic interpretation, which is different from that of 
ot\]qer determiners and which can be furtherly specified an the 
basis of the information contents gathered from the overall ean ~ 
text and frfxn the remsining part of the sentence. If such an 
information contents is rot sufficient, then the meaning of the 
determiner remains ambiguous. What is of paramount i~rtance is 
that any determiner has a "single" meaning, that can be furtherly 
specified by the context. 
Of course, we need to express the se~sntics of determiners by 
means of a suitable representation. The one that we propose seems 
to be intuitively acceptable, formally precise and suitable for a 
cm~positional analysis of natural language that, even if questien - 
able in some particular cases, is still one of the approaches that 
guarantee the most reasonable degree of generality in semantic 
interpretation. 
It is obvious that the representation of a sentence in such a 
fon~sliem may contain ambiguities; therefore a further step is 
needed in order to obtain an unambigtmus deep specification of its 
meaning. Contrarily to the intermediate logical formalism we are 
going to discuss, this firml specification will rot be given in 
declarative form, but in terms of operations on an underlying 
knowledge base. 
REPRESENTATION FORMALISM 
Ct~ main goals in designing the representation formalism that 
will be used in the following sections were: 
I ) To maintain a close relationship between the pieces of informs- 
tion that are intuitively present in the input sentence and the 
predicates appearing in its interpretation. 
2) To make explicit the distinction between surface objects and 
semantic entities: words on one side and concepts, individuals, 
classes etc. on the other. 
3) To maintain a co~itional analysis of language, where the 
starting point is provided by the dependency tree built by the 
rule~bassd syntactic cogent of the FIDO system \[Le~no, 
Torasso 84; Les~o, Torasso 85a; Lesmo, Toraseo 85b\]. 
+ Partially supported by MPI Project "Architetture Software per 
Sistemi Intelligenti" 
Supported by a fellowship of CSI Pienmnte 
344 
4) To devise a set of predicates allowing an easy translation 
between the obtained representation and the corresponding 
operations cn a Knowledge Base. 
A first example concerns a very simple sentence: 
I) Bob loves Lucy. 
The representation is (lower case strings refer to variables; 
upper case ones to constants or predicates): 
Ir) REF(S,x,BOB) & REF(S,y,LOVE) & RFF(S,z,LUCY) & AG~F(y,x) & 
OBJECT(y,z) 
This can be read as: there are three internal entities (x,y,z); 
the speaken (S) is referring to the first of them by using the 
word BOB, to the second with TO LOVE, to the third with LUCY; the 
agent of y is x, its obiect is z. Fig.1 depicts, in terms of nodes 
and arcs, the proposed representation. REF predicates are meant to 
indicate the mapping between words and internal nodes. Consider 
now ex.2: 
2) The boy loves a girl 
The representation reported below disregards the information con ~ 
tents gathered from the determiners: 
2r) REF(S,x,BOY) & REF(S,y,LOVE) & R\[F(S,y,GIRL) & A(~r(y,x) & 
OBJECT(y, z) 
The representation is analogous to the previous one. On the other 
hand, some problems arise in this case; they concern the com~uni~ 
eative impact of ex.2, and which were not evident in the previous 
exan~)le. If we say "Bob loves Lucy", we assume that whoever hears 
this sentence knows both Bob and Lucy, so that he is able to 
reconstruct the right semantic interpretation, and to identify the 
specific individuals to whom the speaker is referring. But how 
can the hearer convey such kind of informmtion when explicit names 
are not available? And, on the opposite side, how can the speaker 
tell the hearer that he is not referring to any specific indivi- 
dual, but he wants to mention a general property of the class? We 
believe that the discriminating information is carried by deter- 
miners. If we take them into account, we should state that ex.2 
expresses something as: "BOY (this word should suffice for yca to 
identify whom I'm talking about) LOVES GIRL (this word is not 
specific enough to allow you to identify the correct referent)" 
or, if we think of a knowledge base represented as a semantic net ~ 
work: "Dear hearer, you should find a node satisfying the 'BOY' 
description (and if ysu consider the context, this can be done 
unambiguously), then yca should create a new node of type 'GIRL' 
and connect them via a nede which is an 'ACT~OF 'LOVE' " 
CONUEP~ NET 
~ "THING" 
SS - i ~ ~ OBJ 
--~, \ 
IS-A ~ \ IS-A \\ ACT-OF 
BOB 
LUCY 
TO LOVE 
SURFACE WORDS 
INSTANCE NET 
fig.l 
We 'can give now the complete representation of ex.2: 
2r' ) RFF(S,x,BOY) & IDENTIFIABLE(S,S,x) & ENABLESAF~REF(S,H,BOY) 
& REF(S,y,LOVE) & REF(S,z,GIRL) & IDENTIFIABLE(S,S,z) and 
not ENABII~AMF~,'(S,H,GIRL) & AGF/~\]'(y,x) & OBJECT(y,z) 
that is: "The .~)eaker is referring to entity x by ~eans of the 
~rd BOY, he assumes that x is identifiable to himself and that 
the description used (BOY) enables the hearer to refer to the same 
entity; there Ls also an act of loving (y) and another entity (z) 
what\] he is referring to by means of the word GIRL; z is identifi- 
able to himself, but the word GIRL will not enable the hearer to 
refer to the same entity he is thinking about. Finally, x is the 
agent of y and z is the object of y". 
Actually, 2r' does not correspond exactly to sentence 2. In 
fact Ex.2 is mrbiguous whilst 2r' is not. The source of ambiguity 
is the NP "a girl". In the previous discussions we. assumed that 
the speaker knows the girl loved by "the boy", but this is not 
necessarily true. Tne "specific" reading is given in 2r' by the 
presence of the predicate IDENTIFIABLE(S,S,z). Now, how can we 
account for the inherent ambiguity of the indefinite determiner? 
Simply droppi~ from its sei&nntics the "IDENTIFIABLE" prcdicate: 
it will be added in case the context provides m~fficient clues to 
infer the "specific" interpretation, or its negation ("not IDEN = 
TIFIABLE") will be added in case some evidence about a "generic" 
interpretetion is available. No predicate is added (and the sen ~- 
tence remains a~bi\[~ous, as it actually is) if no disambign~ating 
criterium is provided by the context. 
the approach exemplified above will be de~.~ribed in the next 
section, covering the definite and Jndef\]nite determiners. The 
predicates used are listed below, together with an explanation of 
their intuitive meaning. 
REF(x,y,z): Individual x is able to refer to eatity y by by means 
of expression z. 
ENABLESAMERI.Im(x,y,z): Ir~\]ividual x assures that individual y is 
able to identify, by i~eans of expression z, the s~me entity 
which he refers to. 
IDENTIFIABLE(x,y,z): Individual x assumes that individual y is 
able to identify (or that y knows) entity z. 
Sl~2(x): Entity x is a set composed of at least t~c elements. 
ARBITRARY(x,z) : Any member of the class x identified by be expres- 
sion z necessarily satisfies the property expressed by the pro ~ 
position in which z occurs. 
REPRFSFN£ATION CF DEI~ERMINERS 
We wi\] 1 de~.ribe ~ne representations we have adopted for deter- 
miners, fell.owing the classification introduced in \[Croft 85\], 
which we. report here (note, however, that the ARBITRARY predicate 
introduced above does not correspond to 'arbitrary' in Croft's 
c\]assification, only to its 'not defeasible' subclass): 
Pereeptual\]y available (this,that) 
- Not percept~ally available: 
Identifiable (the, anaphoric pronouns) 
Not identifiable: 
'-- Spnoific (specific, epistemic a) 
Arbitrary: 
.... Defeasible (generic / intensional a) 
~'~ Not defeasible (any) 
Table I lists the various representations we have adopted, b)t us 
consider first the definite determiners (we are not going to dis- 
cuss what Enoft refers to as 'perceptually available'-referent 
determiners, i.e. de~mstratives like 'this' and 'that'). 
Tne representation for 'the' reported in table I can be para" 
phrased as: "there exists an entity that the speaker is able to 
refer to by ma~s of the expression following the determiner; the 
speaker asam~s that that expression will enable the hearer to 
refer to the ~me entity; the speaker is ~le to identiDI the 
referred entity". An example is provided by 
3) I1 ragazzo mangia (The boy is eating) 
3r) REF(S,x,BOY) & ENABLESAF~REF(S,H,BOY) & IDENTIFIABLE(S,S,x) 
& REF(S,y,TO EAT) & A .C6lJT(y,x) 
It ~st be noted that it is not written anywhere that the entity x 
has to be m~ individual. In principle, it could be a generic 
entity (i.e. an 'intensional' node of a semantic not), thus ful- 
filling the role of 'prototype individual' \[Grosz, Jo~qni, ~in~ 
stein 83\]. 
A few ~rds now to discuss plurals. For example: 
4) I ragazzi mangiano (The boys are eating) 
4r) REF(S,x,BOY) & ~NABLESAF~_,'.REF(SjI,BOY) & IDEN£IFIABLE (S,S,x) 
& R~(S,y,TO RAT) & AGENT(y,x) & SEIT(X) 
The enly difference is the presence of the predicate SET2(x), 
~hich states that the entity x is a set. We use the r~ SEF2 to 
evidentiate that it refers to the pretheoretical notion of set as 
'a group' ca~posed by mere than ore element. 
As regards indefinite determiners, the representations given in 
Table I can be paraphrased as: "There is an entity that the 
speaker is able to refer to by means of the expression following 
the determiner; the speaker cannot assume that that expression 
will enable the hearer to refer to the seme entity". Let us con- 
sider first the 'specific' meaning of the determiner 'a': 
5) Un uorr~ entre' adagio nel\]a stanza (A man quietly entered the 
r(x\]~) 
5r) HEF(S,x,MAN) & not ~NAB\[%SA~REF(S,I-I,MAN) & REF(S,y,DNTER) & 
REF(S,z,ROOM) & ENABLF/IA~.REF(S,H,ROOM) & AG,~\]'(y,x) & IrE 
(y,z) & MDD(y,w) & REF(S,w,QUIETLY) & not ARBITRARY(x,MAN) 
(note that the speaker assumes that the use of the lexeme 'room' 
enables the he~'er to identify the specific room he is thinking 
about). This interpretation is the simplest one, since it 
directly encodes the basic m~ning of' the indefinite determiner, 
i.e. the reference to an unspecified entity. 
A first problem is how to get the 'generic' rreaning f~ii this 
representation (epistemic and intensio~l interpretations will be 
analyzcwl afterw~lrds, since they do not appear as subjnots of sen- 
tences). In: 
6) Un onso va in letargo in inverno (A bear hibernates in 
winter) 
you could probably perceive a ~r~@tning such as: "If yea randar6y 
pick an iedividtml bear, then yea will see that it hibernates in 
winter; of course, the bear y~l will select will probably rmt be 
the same bear I am thinking of, but it still hibernates in 
winter". Notice that this paraphrase (as we aassme it is) does 
not i\[~oly the existence of a 'prototypieal' bear to khich the gen- 
eral property of 'hibernating in winter' <'~ihould apply: we are 
referring to an arbitrary element of the class we are talking 
about, although ~.~ are not saying that no exceptions exist. It is 
this non~identlfiability of the element for which the property is 
predicated that allows the he;men to obtain the state general 
result. 
But now, what is the difference between ex.5 and ex.6? In the 
first ease (specific interpretation), the speaker is referring to 
a particular individt~al, ~hereas in the second one he is not. We. 
can state that in the specific interpretation IDE~\['IFIABLE(S,S,x), 
whereas in the generic interpretation 'not ID\[~I'IFIARLE(S,S,x)'. 
Of course, in both eases the presence of ' not 
ENABLESA~REF(S,H,EXP) ' should allow to infer that 'not 
IDE~'IFIABLE(S,H,x) ' , that is, to the speaker's knowledge, x is 
not identifiable by the hearer by ire~mns of the expression EXP 
used. Note that this does not mean that the hearer will not be 
able to identify x, but only t~at the speaker is not willing to 
asssme so (s~m examples will be provided afterwards). The 
representation we get for ex.6 is: 
~ R~(s,×,ExP) ~ ~ABm~___~MZREF(S,H,EXP) ~ ID~IF~IZ(S\[S,x) I 
REF(S,x, EXP) & not ENABLESAFEREF ( S, H, EXP ) & not 
ARBITRARY(x ) 
~(S,x, ,~) ~ ~(x) \] 
Table I: Sem~tie representation of the r~_aning of determiners. 
Note that the representation includes the R~,' predicate, which 
will be actually built up on the basis of the expression following 
the determiner,. This has been done in order to provide a means of 
unifying the variable x occurring in the other predicates with the 
one appearing in the representation of the relmining NP. 
345 
6r) REF(S,x,BEAR) & not ENABLESA~RBE(S,H,BEAR) & not 
IDENTIFIABLE(S,S,x) & not .~gBITRARY(x,BEAR) & REF(S,y,TO 
HIBERNATE) & REF(S,z,WINTER) & AC~(y,x) & TI~(y,z). 
It could be argued that there is no reason why in the analysis of 
definite determiners we allowed the 'expression' following the 
determiner to refer to an intensional object, whereas in the inde~ 
finite case we do not. However, lar~uage works just because we 
assume (sometimes incorrectly) that a given lexeme refers to the 
same concept for the whole comr~nity of language users. This means 
that ~e cannot accept a reading where 'not ENABLNSAFEREF(S,H,EXP)' 
occurs and where EXP is intended to refer to a generic concept. 
In order to discuss the other two interpretations of indefinite 
determiners, we need to refer to their use in eases different from 
the subject of the sentence, or, more precisely, in sententlal 
contexts where there is another partecipant, different f~em the 
speaker, who has an 'active' role. In these cases, the representa ~ 
tion m~st account for the existence of a referentiality predicate 
attributed to someone different from the speaker and the hearer. 
The first well known example is provided by a 'desire' verb, that 
is 'to want': 
7) John wants to nmrry a Norwegian 
Some different meanings can be characterized by the hearer's dif- 
ferent replies : 
7a) No, Ingrid isn't a Norwegian. 
7b) Who is she? 
7c) How does he think he can find one? 
In the first case, the speaker is using the word 'Nor~ngian' to 
refer to John's future wife, bat the speaker does net agree on 
that word (*). In the second case, the hearer assumes that the 
speaker is referring to a specific girl whom he does net knew. In 
the third case, he assumes the speaker is not referring to any 
par ticu\]ar Norwegian. 
In all cases there is a common core in the representation of 
the initial sentence; it is: 
7r) REF(S,x,JOI~) & REF(S,y,TO WANT) & REF(S,z,TO MARRY) & 
REF(S,w,NORWEGIAN) & not ENABLESAMEREF(S,H,NORWEOIAN) & not 
ARBITARY ( w, NDRWEGIAN ) & AG~(y,x) & OBJECT(y, z) & 
AGENT(z,x) & OBJECT(z,w) 
To this basic interpretation, scme different predicates are added 
for each different case: 
7ar) IDE~FfIFIABLE(S,S,w) for the standard "specific" interpreta- 
tion of the indefinite determine~; IDENTIFIABLE(H,H,w) & not 
REF(H,w,NDRWEGIAN) to state the hearer's disagreement. 
7br) IDENTIFIABLE(S,S,w) & not IDENTIFIABLE(H,N,w) 
7cr) not ID~TIFIABLE(S,S,w) 
But now ~e have the possibility to characterize two subcases of c: 
in the first one (ci) S does not know the Norwegian that John 
wants to m~rry, but John does know her; in the second ease (c2) 
the identification is generic fer both of them: 
7cir) IDenTIFIABLE(x, x,w) 
7e2r) not IDENTIFIABLE(x,x,w) 
The last determiner (in Croft's analysis) is "any". Its represen ~ 
tation is reported in table I, but lack of space prevents us from 
discussing it (mereover, not all students agree on its status of 
determiner ~ vs. quantifier ~ and no Italian lexeme has a meaning 
exactly equivalent to "any"). 
We list below the rules m~re strictly concerned with the opera- 
tional inter1~retation of the predicates associated with deter ~ 
miners: 
RI (Definite) : 
if REF(S,x,exp) & ~ABLESA~RBE(S,H,x) & IDENTIFIABLE(S,S,x) 
then leeatenode( exp, x) 
R2 (Specific indefinite): 
if REF(S,x,exp) & not ENABLESA~REF(S,H,x) 
IDENTIFIABLE(S, S, x) 
then createnode(exp,x), n~rk(x, 'INDIVIDUAL' ) 
(*) Note that neither the speaker nor the bearer are necessari ~ 
ly right. Fer instance, the speaker could reply "She was born in 
O~Io", and the hearer "~t last year she got the U.S. citizen ~ 
ship". 
346 
R3 (Plural definite) : 
if REF(S,x,exp) & ~ABLESA~R~F(S,H,x) & SET2(x) 
then lecateset(exp,x) 
R4 (Generic indefinite): 
if REF(S,x, exp) & not ENABBE~AMEBEF(S,H,x) & not 
IDEN£IFIABLE(S,S,x) & not ARBITRARY(x,exp) 
then ereatenode(exp,x), r~rk(x, 'GENERIC~BEFgASIBLE ' ) 
A few words on the functions used in the acticn part of the 
rules: 
locatenode looks first for individual referents; if none is 
available it considers generic nodes. 
createnede builds a new instance of the most specific available 
concept identified by exp. 
iooateset works exactly as looatenode, but the node that it 
looks for nust represent a set. 
Tnese rules are not complete, as they do not take into account 
~istemic and Intensional Indefinite: in fact, both the represen ~ 
tations of these interpretations nust include the hypothetical 
knowledge of another individual and, as we said before, we did not 
treat belief contexts. 
CONCLUDING REMARKS 
Interpretation of de~terminers and quantifiers is usually over- 
simplified in many natural language interfaces. We think the for~ 
melism discussed in this paper constitutes a significant step in 
representing the meaning of the sentence at a more abstract level 
than many interfaces do; at the same time we can directly exploit 
the features of this representation to build the actual update 
cor~nand or query. 
Other approaches use a direct translation of the sentence from 
its surface form (or from a purely syntactic tree) into a 
representation language which is actually a KB ~ement or a DB 
query language. The formalism discussed in this pages does not 
make any assumption on the language used for actually accessing 
the KB (and for this reason the formalism does represent the mean- 
ing of a sentence in a natural or at least 'neutral' way \[Hobbs 
1985, Schubert & Pelletier 1982\]). 
On the other hand, the formalism is not too far from the way 
the domain kncwledge is (or could be) represented inside a KB or 
DB, so that it is easy to develop translation rules stating ~nat 
operations on the KB or DB should be done. 
The constraints on the available space prevented us ~ dis- 
cussing the problem of using the context to disambiguate among the 
different meanings of a given determiner (e.g. specific vs. 
unspecific "a"). Some efforts were made and the results are 
encouraging, though in irony cases it is only very high-level 
information (e.g. ~utual knowledge and beliefs) can provide the 
basis for selecting the right interpretation. 

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