Directional PPs and Reference Frames in DRT
Didier Maillat
Oxford University and Fribourg University
English Linguistics Dept.
Miséricorde
CH-1700 Fribourg, Switzerland
didier.maillat@lincoln.ox.ac.uk
Abstract
In this paper, I will argue that the well-
known ambiguity of directional prepo-
sitions between the intrinsic and rela-
tive readings is not lexical, but can be
interpreted as a framework assignment
ambiguity of the type observed in the
temporal domain. A DRT semantics
will then be constructed around a uni-
fied model of framework assignment
which can be applied across the board
to all three universal frame types.
1 Introduction
A great deal of energy has been spent on the
semantic analysis of temporal relations within a
sentence and beyond that, between sentences.
Crucially, it has been a well-established and
intensively studied fact that eventualities (or
events) are interpreted within a temporal frame-
work, i.e. a temporal reference frame with re-
spect to which the temporal relations which
obtain between those events are interpreted (e.g.
Reichenbach 1947, Kamp & Reyle 1993). For
instance, a past tense verb tends, in the absence
of any other external clue, to be temporally in-
terpreted with respect to the time of utterance.
That is to say that a typical semantic contribu-
tion of a simple past is assessed on a two-
dimensional frame centred on the time of utter-
ance, as is the case in sentence (1), where the
past tense is understood as encoding a prece-
dence relation between the event described by
the sentence and the time of utterance n.
(1) Alice took a walk by the river.
Notice that this is not always the case. In a
larger discursive context, a speaker can control
some of these parameters, for instance by em-
bedding a simple past within a larger narrative
sequence. In this configuration, a simple past is
then often interpreted as following the time de-
noted by the preceding simple past. So, that if
we insert (1) within a larger discursive environ-
ment, the past tense of took gets interpreted in a
different way:
(2) Alice slept in that morning. After a hefty
breakfast, she needed some exercise. She
[Alice] took a walk by the river.
Although it is still the case that the action de-
scribed by sentence (1) is located at a time t
which precedes n, the whole context of (2) adds
a further constraint to the interpretation of the
past tense in took as it implies a narrative se-
quence by virtue of which, t has to follow – in
the temporal dimension – t´, the time at which
Alice had her breakfast.
In contrast, similar facts in the spatial domain
have been barely touched upon by the semanti-
cists’ community. In particular, although the
existence of spatial reference frames has been
ascertained again and again, very little has been
done in order to show how, e.g. spatial adverbs
or spatial prepositional expressions interact with
a spatial framework in order to yield a given
meaning. Part of the failure to come up with a
satisfactory account can be explained by the
relative difficulty to tell the semantic from the
pragmatic contributions. It is the goal of this
paper to present a formalism that addresses
these issues and tries to draw a line between
what is encoded and what is contextually de-
rived. In order to do so, I will focus on a set of
framework-sensitive PPs, namely directional (or
projective) PPs: to the left/right of, in front of /
behind, above /below and cardinal directions.
Researchers (see for instance Levinson 1996)
have shown that human languages make use of
only – and up to – three universal reference
frames in order to express spatial relations. All
three of them are instantiated in Indo-European
languages. Thus, a single spatial configuration
can be expressed in three different ways in these
languages, depending on which reference frame
is activated. All three examples below are to be
interpreted as referring to a unique spatial set-
ting.
(3) Le chat est à l’est de la voiture
(4) the cat  is   east   of the car 
ABSOLUTE FRAMEWORK
(5) Le chat est derrière la voiture [with re-
spect to the car]
(6) the cat  is   behind  the car 
INTRINSIC FRAMEWORK
(7) Le chat est à droite de la voiture [with re-
spect to the speaker’s viewpoint]
(8) the cat  is   on the right of the car 
RELATIVE FRAMEWORK
Crucially, while (5) and (7) – in the relevant
readings – are cases of intrinsic and relative
encoding, respectively, it should be clear that
both sentences fit equally well in the other
framework. That is to say that Indo-European
languages are systematically ambiguous be-
tween the intrinsic and relative reference frames,
since directional PPs can be successfully inter-
preted in both frameworks. Notice that this is
also the case in other language families as simi-
lar ambiguities are observed in Japanese, Arabic
and Sesotho. Most importantly, the spatial area
denoted by each of the six directional PPs men-
tioned above (i.e. all but the cardinal directions)
differs depending on the reference frame within
which it is interpreted.
The strong claim that the present study will
make is that – very much in the fashion of how
DRT deals with tenses or temporal adverbials –
spatial expressions, such as directional preposi-
tions, can be ambiguous between several read-
ings, because these expressions are
underdetermined as to how they connect with
the spatial framework. And this choice between
alternative computations triggers the ambiguity
in (5) and (7).
2 Cross-linguistic Variation
Also, despite the striking recurrence of these
facts throughout the Indo-European languages
and other families, it should be pointed out that
some languages display other properties. For
instance, Tzeltal (Mexico) and Guugu Yimithirr
(Australia) rely (almost) exclusively on the ab-
solute framework. Contrary to what can be ob-
served in Indo-European languages where the
absolute frame is relatively rare and used in very
restricted contexts. This type of cross-linguistic
variation makes up a very strong case in favour
of a unified model for all three frame types. If
such a model can be constructed, the upshot
would consist in being able to account for cross-
linguistic variation in terms of parametric set-
up.
Hausa speakers (Africa), on the other hand,
do not apply the Indo-European ‘mirror’ con-
figuration when they construct a relative frame.
In such a language, a tree’s front is not facing
toward the speaker, but away from her/him (i.e.
in a ‘tandem’ configuration). A suitable model
will have to be flexible enough in order to cap-
ture this second type of parametric variation in a
satisfactory way too.
3 What’s in a framework?
A first step towards a full understanding of
the phenomena described above consists in a
thorough analysis of the properties of the three
frameworks. In fact, although superficially they
appear to be fairly different, the three frame-
works are very much alike. In order to see the
similarity between them, let us start by assuming
a set of conceptual primitives that will make up
a spatial framework in the proposed model.
Under the current view, all three frameworks
combine three primitives:
(i) a set of three orthogonal axes (frontal,
lateral and vertical) in a Euclidean
space
(ii) a point of origin
(iii) an orientation vector
Where the orientation vector determines the
orientation of the axial system by fixing the
direction and orientation of the frontal axis (see
below for a detailed explanation of this process).
According to this model any spatial frame-
work is uniquely and sufficiently defined by
means of these three elements, only two of
which are variables, viz. (ii) and (iii) 
1
. In other
words, any spatial reference frame is fully de-
                                                          
1
 Strictly speaking, a third parameter is needed in order to
fully determine such a space, but, for brevity’s sake, I will
gloss over the variability of this third parameter and as-
sume that gravity fixes the vertical axis (see Levelt 1996
for further discussion of the issue).
termined if one knows the value of its origin and
that of its orientation vector. This means that
this is all that is needed in order to evaluate a)
the truth-value of a directional expressions in
that reference frame and b) to select an appro-
priate directional expressions for that frame-
work, given some spatial configuration. Notice
that contrary to what is suggested in the litera-
ture on this topic (e.g. Zwarts & Winter 2000),
the axial structure is not “polarised”, in the
sense that each axis defines a dimension (fron-
tal, lateral and vertical), but the sides are left
unspecified. In other words, the distinction be-
tween, e.g. front and back is not built into the
system and will have to be handled at a later
stage.
To pursue the analogy with the temporal do-
main, the spatial framework will have to be
fixed, by means of assigning a value to the two
parameters origin and orientation. Very much
like the temporal dimension is organised around
its origin, namely the time of utterance n in
DRT. Notice, though, that the addition of one
more dimension in the spatial domain triggers
the need for a second parameter, orientation.
And as we will soon find out, a lot, if not all of
the computation process needed to interpret a
directional preposition goes precisely into the
value assignment for the orientation parameter.
However, what we have seen so far is the
similarity between the various frameworks, but
as we illustrated earlier on, they also differ in a
crucial way, which accounts for the ambiguity
observed in sentences (5) and (7). A close scru-
tiny of the three frame types reveals that they
differ with respect to their orientation vector.
Indeed the computation of the orientation vector
appears to constitute the distinguishing factor
between the three frames as is shown in table 1.
Origin Orientation
Absolute G given
Intrinsic G G
Relative G VPT
Table 1
Table 1 illustrates that while the orientation
parameter of an absolute frame is given as part
of the lexical / encyclopaedic knowledge that
the speaker has of a preposition like east of, the
orientation parameter of an intrinsic frame is
defined by the properties of the ground object
(G in the table). For instance in our example (5),
the functional and geometrical properties of the
car will yield the corresponding intrinsic orien-
tation. For the relative system, the framework is
oriented with respect to a salient viewpoint
(VPT), which often coincides with the location
of the speaker but may also differ at times. Ta-
ble 1 also sums up the facts regarding the other
parameter, namely the origin. We see that in all
cases the origin of the framework is given by the
ground object G (determined by the complement
of P). In other words, we can conclude that in
the present model, the type of reference frame is
fully determined by the sole orientation pa-
rameter, which is formally represented by a
vector whose origin corresponds to the point of
origin of the axial system and whose endpoint is
either lexically encoded, or depends on the in-
trinsic properties of G, or coincides with a sali-
ent VPT (viewpoint). In striking contrast with
the suggestion made by Crow (1989) who
makes use of no less than 4 different variables.
This, however, leaves the question of the
actual denotation of directional prepositions
open. Building on and extending the kind of
vector geometry used above and discussed ex-
tensively by O’Keefe (1996) and Zwarts &
Winter (2000), the proposed analysis assumes
that a directional preposition denotes an angular
range. Crucially, this angular range needs to be
fixed or anchored on a framework in order to be
fully specified. The model poses that the angular
templates are anchored on the reference frame’s
orientation vector. An obvious advantage in
favour of this approach lies in the fact that it
readily accounts for the ambiguity observed in
(5) and (7), in the sense that a single template
can denote two different configurations de-
pending on the value assigned to the orientation
vector; hence, the ambiguity. Figure 1 below
illustrates the kind of templates that are used in
this model (in this case, a bird’s eye view of the
derrière template). One should point out that the
angular range is calculated clockwise from the
orientation vector (represented by the arrow in
the diagram).
180°
F
-90°
90°
Figure 1
A quick word on the actual values of the an-
gular range displayed here is required. Figure 1
instantiates the maximal denotation of the
preposition derrière. That is to say that this
template covers the maximal extensional – and
truth-functionally correct – scope of the prepo-
sition, but it does not reflect the optimal angular
range denoted by the preposition (but see Mail-
lat 2000).
One should also point out the obvious rela-
tion between the proposed denotation for direc-
tional PPs and the denotation suggested by
Kamp & Reyle for temporal adverbials headed
by after and before. Thus, they write:
What the phrase after α does is to divide the
axis of time into two halves and to say of the
described eventuality that it lies in the “upper-
half” ”. [1993:626]
Clearly, the same remark applies to the type
of directional adverbials under consideration in
this paper, as Figure 1 above illustrates. The
main difference between the two types of adver-
bials comes from the dimensionality of the re-
spective domains: while the temporal domain is
strictly one-dimensional, space, on the other
hand, is three-dimensional. As a result, the
purely linear model adopted by Kamp & Reyle
to handle temporal relations is not sufficiently
informative to tackle the spatial domain. How-
ever, the kind of vector geometry chosen here
could of course be successfully applied to the
temporal domain. Crucially, decreasing the di-
mensional arity makes the angular information
about vectors irrelevant: in a one-dimensional
domain all vectors are parallel to one another. In
other words, before and after are the one-
dimensional equivalents of in front of  and be-
hind.
4 Towards a DRT Model
In the last part of this paper, I want to look at
the kind of formalism that could handle the pro-
cesses described in the previous sections. In that
respect, Discourse Representation Theory seems
to be quite appealing as it has been partly devel-
oped in order to come to terms with similar
framework phenomena in connection with the
temporal domain (Kamp & Reyle 1993 and Van
Eijck & Kamp 1997).
CR.Derrière
Triggering configuration:
  α   PP
       P    β
derrière
Choose: Orientation point, Opt, from the
following items:
(i) ST (β)
(ii) ST (salient VPT) & polarity is
 switched to negative
where ST() is an operator which returns the
spatio-temporal ‘slice’ denoted by its argument.
Introduce into U
K
: l, l
1
, l
2
Introduce into Con
K
:
l ⊆ Opt
l
1
 = ST(α)
l
2 
= ST(β)
l
1
 behind l
2
 & orientation vector = l
2
l & rela-
tional vector = l
2
l
1
 & 90°<δ<-90°, where δ ob-
tains between the relational vector and the
orientation vector
Perhaps I should first enter a caveat regard-
ing the underlying ontology of this extended
model. The extension of the DRT model in or-
der to account for the interpretation of direc-
tional prepositional phrases requires that the
ontology include some sort of spatial entity.
Without further explanation, I introduce loca-
tions as a new type of entity. I can only but refer
the reader to the relevant literature (Aurnague
1995 and Asher & Sablayrolles 1994). An alter-
native approach would have consisted in leaving
out the spatial concepts and manipulating only
existing discourse referents with the proviso that
some conceptual module à la Jackendoff and/or
spatial module would have to translate the DR
structures. Here I have opted for a slightly more
explicit representation.
The DRT element under scrutiny is the so-
called construction rule for the preposition der-
rière. This choice emphasises the fact that the
essential constituent which triggers the setting
up of a spatial framework is the spatial adver-
bial. In this sense, this analysis is very much in
line with the treatment of temporal adverbials
offered in DRT. To put it slightly differently, it
is the directional preposition itself which acti-
vates the frame assignment process.
That being said, following the practice of
Kamp & Reyle (1993), I assume that the crucial
orientation parameter is encoded as an Orienta-
tion Point, Opt, reminiscent of the Temporal
Perspective Point and Reference Point (see also
the register r in the dynamic version of DRT
elaborated by Van Eijck & Kamp 1997). This
choice is pragmatic in nature and, hence, the
proposed construction rule is unspecific with
respect to the selection process. Consequently,
the pragmatic orientation assignment process is
appropriately encoded using the Choose opera-
tion. Below is a partial quotation, taken from
Kamp & Reyle (1993:621), presenting a typical
DRS of a temporal adverbial, which relies on a
similar device.
CR.Sunday
[…]
Choose: Origin of Computation t´´ from
the following items:
(a) n
(b) the current TPpt
(c) The current Rpt […]
The proposed representation for derrière also
makes use of the operator ST(), which extracts
the spatio-temporal value of its argument. To be
brief, ST() returns a set of ordered pairs of loca-
tions and times (Aurnague 1995 and Asher &
Sablayrolles 1994). This step is crucial in order
to interpret sentences like
(9) They danced behind the car.
involving an event dance (x,e), and in which the
adverbial describes the location of the whole
activity denoted by the verb, i.e. the dancing.
With respect to the original DRT environment,
one could think of ST() as being modeled on the
dur() operator, which returns the temporal value
of its argument.
Most of the semantic work done by the
preposition is determined by the set of condi-
tions added to the discourse representation
structure. As we can see, the representation pro-
posed here is relatively explicit, and therefore
slightly at odds with standard DRT. But as I
already argued, this does not constitute a serious
obstacle and is meant to illustrate the whole
process more clearly. In fact, this series of con-
ditions simply formalises the information con-
tained in the template illustrated in Figure 1,
once it has been anchored on a reference frame.
It says that the relational vector which obtains
between the Ground object and the Figure object
must be at an angle δ from the orientation vec-
tor, where δ is contained in the angular range
defined above. If one were to prefer the more
implicit kind of representation favoured by
Kamp & Reyle, one should replace the last con-
dition to be introduced in Con
K
 with the fol-
lowing one:
[…] l
1
 behind l
2
 & orientation vector = l
2
l &
relational vector = l
2
l
1
In brief, the semantics of the preposition re-
quires that the figure object be within a deter-
mined angular range. Figure 2 below illustrates
the kind of calculations needed in order to assess
the truth-value of a directional prepositional
expression.
180°
F
-90°
90°
Figure 2
Relational
vector
Orientation
vector
Ground
Figure
Viewpoint
As we can see, the angle between the Orienta-
tion vector and the Relational vector must be
within the given range for the prepositional
phrase to be true.
Although Figure 2 illustrates a case of rela-
tive framework; hence, the ‘Viewpoint’ pa-
rameter. A corresponding figure for an intrinsic
framework would be essentially the same, ex-
cept for the Orientation vector, which would be
fixed by the ground object (not a viewpoint).
In this system, the absolute framework does
not essentially differ from the other two frame
types. The only divergence is that a preposition
like east of is not ambiguous between frames.
Therefore, its construction rule must reflect the
fact that Opt is uniquely determined lexically (at
least in Indo-European languages). Thus,
CR.east_of does not include a Choose opera-
tion, instead a constraint is added to the set of
conditions, such that:
x
magnetic north (x)
Opt:= ST(x)
In addition, because the model is applied
across the board to all three frameworks, ‘abso-
lute’ languages like Tzeltal and Guugu Yimithirr
do not require any special device in order to be
accounted for by the system. Essentially, these
languages express directions like we do, they
just have stronger selection restrictions on Opt.
Also, Hausa can be readily handled by the
theory by means of the polarity index defined in
the Choose operation. This pragmatic index
keeps track of the orientation assignment proc-
ess and switches its default (positive) value in
case Opt is matched with (ii). Hausa-behind –
 and for that matter, Hausa-in front of –, then, sim-
ply makes use of this index to generate the op-
posite vector from the orientation vector. In
Hausa, glossing over the syntactic differences,
CR.Hausa_behind would thus be:
[…] Introduce into Con
K
:
l ⊆ Opt
l
1
 = ST(α)
l
2 
= ST(β)
l
1
 behind l
2
 & orientation vector = <polarity>l
2
l
& relational vector = l
2
l
1
 & 90° <δ< -90°, where
δ obtains between the relational vector and the
orientation vector
The very same formalism elegantly captures
the notorious lateral flip observed in Indo-
European languages. In these languages, the
prototypical denotation of to the left of is at a
90° angle, clockwise, from the front side as
defined in a relative framework, but at a 90°
angle, anticlockwise, from the front side as
defined in an intrinsic framework. And vice
versa for the prototypical denotation of to the
right of. That is to say that the lateral axis is, as
it were, flipped when you pass from one frame-
work to the other. As a result, in a configuration
where the intrinsic and relative denotations of in
front of and behind coincide, the denotations of
intrinsic to the left of will be that of relative to
the right of.
So far, this oddity has been a major obstacle
in every formal approach which has tried to
tackle the problem. For instance, Zwarts &
Winter (2000) – in the latest attempt to date at
formalising the semantics of spatial expressions
– have skipped the lateral flip issue and decided
to treat only the laterally neutral beside
(2000:182). In contrast, the proposed model
offers a single treatment to and generalises over
the Hausa tandem configuration and the Indo-
European lateral flip. So that, the very same
polarity index will also capture the Indo-
European lateral flip by generating the opposite
vector in case option (ii) is selected in the
Choose operation.
Notice that in CR.Derrière, given above, the
polarity index is not used in the set of conditions
which determine the semantic contribution of
the preposition. In other words, derrière is not
sensitive to the type of framework in which it is
interpreted, i.e. its denotation does not vary
across frame types. In the proposed model, this
particular behaviour is a direct consequence of
the built-in symmetry of the spatial templates.
Indeed, as it appears in Figure 1, the defined
angular range is calculated symmetrically on
both sides of the orientation vector.
In fact, there seems to be strong empirical
evidence to support this particular property of
the model. Thus, the literature on language ac-
quisition and psycholinguistics confirm that the
frontal prepositions are acquired faster and are
used with fewer mistakes than the lateral pair. In
that sense, the spatial templates suggested in this
study seem to capture both of these observed
facts.
Finally, as is the case in standard DRT with
temporal reference points (see Kamp & Reyle
1993:603), Opt is reset after the triggering
clause has been fully processed. This last dele-
tion step is needed so that sentences involving
several directional phrases can be treated with-
out imposing the requirement that Opt is held
constant throughout, which would clearly be too
restrictive. So that, in sequence (10) below, we
must be able to assign an intrinsic reading to the
first directional PP, while, at the same, we are
forced to select a relative reading for the second
PP.
(10) I can see three balls. The one in front of
the car [Choose = (ii)] is identical to the
one behind the tree [Choose = (i)].
In other words, the acceptability of (10) un-
der the relevant reading indicates that the orien-
tation assignment process is a “bookkeeping
device” which must be reset after the triggering
component has been processed (Kamp & Reyle
1993:603).
5 Conclusion
To conclude, I hope to have shown that an
elegant DRT model for directional prepositions
can be constructed on a single parameter and
that this model successfully captures some very
puzzling cross-linguistic facts, while, at the
same time, shedding some new light on the cru-
cial interaction between spatial expressions and
spatial reference frames.
Acknowledgements
This research project has been supported by a
generous grant from the Berrow Trust. I would
also like to acknowledge the fruitful feedback
received on an earlier version of this paper pre-
sented at CSP 2000 in Cambridge; as well as
three anonymous reviewers for useful com-
ments.

References
Asher, N. (1993). Reference to Abstract Objects
in Discourse. Kluwer, Dordrecht.
Asher, N. and P. Sablayrolles (1996). A Typol-
ogy and Discourse Semantics for Motion
Verbs and Spatial PPs in French. In: Lexical
Semantics (Pustejovsky, J. and B. Boguraev,
eds), pp. 163-209, Clarendon Press, Oxford.
Aurnague, M. and L. Vieu (1996). A three-level
approach to the semantics of space. In: The
semantics of prepositions from mental proc-
essing to natural language processing (C. Ze-
linsky-Wibbelt, ed.), pp. 393-439. Mouton de
Gruyter, Berlin.
Aurnague, M. (1995). Orientation in French
Spatial Expressions: Formal Representations
and Inferences. Journal of Semantics, 12,
239-267.
Bloom, P., Peterson M. A., Nadel L. and Garrett
M. F. (eds) (1996). Language and Space.
MIT, Cambridge, MA.
Crow, J. (1989). Toward a Semantics for Eng-
lish Spatial Expressions. Ms. PhD Thesis.
University of Texas at Austin.
Herskovits, A. (1986). Language and Spatial
Cognition: An interdisciplinary study of the
prepositions in English. CUP, Cambridge.
Jackendoff, R. (1990). Semantic Structures.
MIT, Cambridge, MA.
Levelt, W. J. M. (1996). Perspective Taking and
Ellipsis in Spatial Descriptions. In: Language
and Space (P. Bloom, M. A. Peterson, L.
Nadel and M. F. Garrett, eds), pp. 77-108.
Levinson, S. C. (1996). Frames of reference and
Molyneux’s question: crosslinguistic evi-
dence. In: Language and Space (P. Bloom, M.
A. Peterson, L. Nadel and M. F. Garrett, eds),
pp. 109-171.
O’Keefe, J. (1996). The spatial prepositions in
English, vector grammar, and the cognitive
map theory. In: Language and Space (P.
Bloom, M. A. Peterson, L. Nadel and M. F.
Garrett, eds), pp. 277-316.
Maillat, D. (1999). On the Interpretation of Di-
rectional Expressions: Empirical and Theo-
retical Considerations. Ms. MPhil Thesis.
University of Oxford.
Maillat, D. (2000). Which Spatial Template for
Directional Prepositions? Ms. University of
Oxford.
Nam, S. (1995). The Semantics of locative
prepositional phrases in English. Ms. PhD
Thesis. University of California, Los Angeles.
Vandeloise, C. (1986). L’espace en français.
Seuil, Paris.
Van Eijck, J. and Kamp, H. (1997). Represent-
ing Discourse in Context. In: Handbook of
Logic and Language (Van Bethem, J. and A.
Ter Meulen, eds), pp. 179-237. Elsevier, Am-
sterdam.
Zwarts, J. and Winter, Y. (2000). Vector Space
Semantics: A Model-theoretic Analysis of
Locative Prepositions. Journal of Logic, Lan-
guage and Information, 9/2, 171-213.
