TENSES AS ANAPHORA* 
Kurt Eberle Walter Kasper 
Institut ftir Maschinelle Sprachverarbeitung 
Universit£t Stuttgart 
Keplerstr. 17 
West Germany 
Net address: nbaa1423@ds0rus54.bitnet 
Abstract 
A proposal to deal with French tenses in the frame- 
work of Discourse Representation Theory is pre- 
sented, as it has been implemented for a fragment 
at the IMS. It is based on the theory of tenses of 
H. Kamp and Ch. Rohrer. 
Instead of using operators to express the 
meaning of the tenses the Reichenbachian point of 
view is adopted and refined such that the impact of 
the tenses with respect to the meaning of the text 
is understood as contribution to the integration of 
the events of a sentence in the event structure of 
the preceeding text. Thereby a system of relevant 
times provided by the preceeding text and by the 
temporal adverbials of the sentence being processed 
is used. This system consists of one or more ref- 
erence times and ~emporal perspective ~imes, the 
speech time and the location time. The special in- 
terest of our proposal is to establish a plausible 
choice of "anchors" for the new event out of the 
system of relevant times and to update this system 
of temporal coordinates correctly. The problem of 
choice is largely neglected in the literature. In op- 
position to the approach of Kamp and Rohrer the 
exact meaning of the tenses is fixed by the resolu- 
tion component and not in the process of syntactic 
analysis. 
1 Introduction 
On a Reichenbachian analysis tenses are inter- 
preted as relating three kinds of temporal en- 
tities: the time Of the event talked about, the 
speech time (time of utterance) and the refer- 
*The research was done in the context of ACORD (ES- 
PRIT P393) and LILOG, and was also supported by the 
German Science Foundation (Ro245/13-3) 
ence time. In the original version these entities 
are understood as time-points. In the sequel fre- 
quently this system was transformed into interval- 
based approaches to describe the interaction of 
adverbials, tenses and aspect on sentence level 
(cf. v.Eynde(1987), Bras/Sorillo(1988)) or to de- 
scribe the course of events on text level (cf. Hin- 
richs(1986), Partee(1984)l). A detailed criticism of 
the Reichenbachian analysis of tenses can be found 
in B~uerle(1979)). Motivated by text-phenomena 
the Kamp/Rohrer approach (1983,1985) splits the 
Reichenbachian reference time into three contextu- 
ally given parameters by adding temporal perspec- 
tive points and location times. 
Our approach which is based on the 
Kamp/Rohrer analysis differs from treatments of 
the semantics of tenses (and aspects) which charac- 
terize the tenses by some simple operator (usually 
interpreted as a temporal quantifier) in that respect 
that the tenses are described in terms of their con- 
tribution to the problem of how the temporal struc- 
ture of the events talked about can be constructed. 
The problem how to determine the times the con- 
text has to supply and to which the events have 
to be related is largely neglected in theoretical dis- 
cussions of the semantics of tenses. It is the main 
topic of our discussion. Special attention has been 
paid to the interaction of tense, aspect and tempo- 
ral adverbials in determining these relations. The 
approach represents a unified account of tense and 
aspects. Another problem dealt with is the problem 
of tense switch. 
We will restrict the discussion to French 
tenses prevailing in (written) reports about past 
events (imparfait, pass~ simple, passd composd, 
plusqueparfait, conditionnel). The tense system 
x Partee has discussed at length the parallelism between 
nominal and temporal anaphora. The idea of tenses as 
anaphora, as developed there, underlies to a certain extent 
our approach as well. 
- 43 - 
concerning the past, in contrast to that relating 
to the present or future time, is rather elaborated, 
especially in French. A proper theory of tenses has 
to account for this multiplicity. 
2 Representing events in 
DKT 
In the framework of Discourse Representation The- 
ory (DRT) (Kamp(1981)) a Discourse Representa- 
tion Structure (DRS) is a pair < U, K > consisting 
of a set U of discourse referents (DRFs) and a set 
K of conditions. Discourse Referents are assumed 
to be sorted according to the following sort sys- 
tem (for our purposes only the temporal branch is 
relevant): 
all, a 
temporal, t non-temporal, x 
dynamic state, s 
event, e process, p 
We shall use the indicated letters for DRFs of 
the corresponding sort 2. Conditions take the fol- 
lowing forms: 
. 
. 
3. 
P(al,..., an), where P is an n-ary predicate 
symbol and the ai are discourse referents. Con- 
ditions of this form are also called atomic. For 
2-place temporal relations we will also use infix 
notation. 
DRS1 ==~ DRS2 
t: DRS, where t is a temporal DRF 
Thus DRT uses a variant of the Davidsonian 
method of talking about events: they are treated 
as a kind of objects. But DRT deviates from the 
Davidsonian representation in that instead of us- 
ing additional argument places in the predicates 
2For simplicity we will use the term event not just for the 
corresponding subsort of dynamic but also for dynamic or 
temporal objects in general. For the same reasaa we do not 
make a clear distinction between "aspect" and "Aktions- 
arten". The intended meaning should be clear from the 
context. The classification is related to the one given in 
VendieT(1967). 
an event discourse referent is put before a DttS re- 
presenting the nature of the event. This allows for 
a recursive definition of aspects and to account for 
aspectual change. So the process p of z's travelling 
would be written in DRT as 
and the event of x's travelling to Paris would be 
written as 
p x paris 
e': p: I travel(x) 1 
goal(p, paris) 
DRSs containing conditions as these are embedded 
into models by mapping the temporal DRFs onto 
objects in a domain of temporal objects. A proper 
embedding is realized if the value of the event- 
DRF fulfills conditions represented by the sub-DRS 
which the event introduces with respect to an in- 
terpretation function defined among other things 
on predicates such as travel in the example above. 
An advantage of this representation on the one 
hand is that, following the usual definition of acces- 
sibility of DRSs from a DRS used in DRT, restric- 
tions on the accessibility of DRFs as possible an- 
tecedents for anaphoras can be expressed, though 
for our purposes this will play only a subordinate 
role. On the other hand different modes of existence 
can be discriminated for the objects DRFs stand 
for, depending on the position of the sub-DRSs, 
where these DRFs are introduced, in the DRS of 
the whole text. In the case of reported speech for 
instance, the real existence of a reported episode is 
not necessary. Extensions of the original modelthe- 
ory ofDRT (cf. Kamp(1981)) which point in this di- 
rection are given e.g. in (Asher(19S6), Reyle(1985), 
Eberle(1988b)). 
3 Basic Assumptions of the 
Semantics of Tenses and 
Adverbials 
The basic tenses of French narrative texts are im- 
parfait and pass~ simple. The interaction of these 
tense forms is often described by two pairs of op- 
position: On the one hand the pass~ simple is used 
to describe actions of the narrative, "the course of 
events", whereas the imparfait serves to paint the 
-44- 
background of the story. On the other hand the im- 
parfait can be used to describe events in progress 
as viewed from "inside", whereas the pass~ simple 
presents the event as a punctual entity. In order to 
reflect these dichotomies we require that an event 
introduced by pass~ simple serves as new reference 
time which must not start before the old one, and 
that the imparfait introduces a state which includes 
temporally the existing reference time. In this case 
no new reference time Is created. 
In the case of pass~ simple we do not require 
that the new reference time has to follow com- 
pletely the old one in order to deal correctly with 
discourse situations as 'elaboration' or 'contrast' 
and others. To discriminate such textual functions 
an elaborate inference component is needed, which 
at present is not available. In addition there are 
cases where this inference component would need 
information about the proceeding of the whole 
story. This cannot be made available at this stage 
of processing. 
Thus an underspecified relation not-before is 
necessary which can be defined, as other relations 
like subset (needed for the imparfait), out of the re- 
lations overlap and before, the only basic temporal 
relations used in the system s . 
The plusqueparfait can be understood as per- 
fective state giving background information with 
respect to the actual reference time of the story, 
(Jean avait dgj~ mange'), or as introducing or con- 
tinuing a flashback. 
The conditionnel is understood as a counter- 
part to the plusqueparfait describing an anticipa- 
tion with respect to a perspective point in the past. 
We think that pass~ compos~ in (written) 
narrative texts should be treated as analogue to 
the pass4 simple with respect to pure temporal 
relations 4 . 
Temporal adverbials provide a location time 
for events in relation to the temporal structure of 
the preceding text. They can differ from each other 
by their characterization of the location time and 
their anaphoric behavior. Deictic adverbials like de- 
main, ia semaine derni~re for instance create lo- 
cation times of different temporal extension with 
different ordering conditions regarding the evalu- 
ation time (after and before), but they are simi- 
lar in that the evaluation time must be the speech 
3For motivations and definitions cf. Kamp(1979), 
Reyle(l~6), Eberle(1988b)). 
4 For a more elaborate discussion of the French tense sys- 
tem cf. Kamp/Rohrer(1985), Eberle(1988c). 
time, whereas in case of le lendemain, la semaine 
prdcddante the temporal relations and extensions 
are equivalent to the deictic analogues but the time 
of evaluation has to be a past reference time or per- 
spective point. Frame-time adverbials like ce jour- 
la can be distinguished from punctual time adver- 
bials like fi trois heures and from adverbials, like 
puis, which simply state a temporal relation bet- 
ween the event to be introduced and a temporal 
antecedent. Some adverbials, like puis and ensuite, 
do not restrict the nature of the antecedent, it is 
just a reference time. But maintenant e.g. requires 
that the evaluation time is a perspective point of 
the text. The resolution component has to take into 
account such phenomena. 
4 Aspects of Implementation 
4.1 Architecture 
The construction of the semantic representa- 
tion for a discourse proceeds in several stages: 
each sentence is parsed using a Lexical Func- 
tional Grammar (LFG) (Kaplan/Bresnan(1982), 
Eisele/DSrre(19S6)) which analyzes sentences into 
functional structures (f-structures), augmented by 
indices to indicate the linear order of words in 
the input string. The f-structure serves as input 
for the construction of a proto-semantic represen- 
tation (cf. Reyle(1985), Reinhardt/Kasper(1987)). 
The last stage consists in integrating this represen- 
tation into a semantic representation for the dis- 
course, mainly by doing the necessary resolutions 
for anaphoric expressions. Accordingly, the system 
consists of three major modules: 
LFG-Parser: I 
f-structure 
Composer: I 
Proto-DRS 
DRS ~ Knowledge base 
The whole system is implemented in PRO- 
LOG. Here mainly the Composer and the Resolver 
will be discussed with respect to the treatment 
-45- 
of the tenses in these modules. The inference ma- 
chine and knowledge base are at present not imple- 
mented. The proto-semantic representation for a 
sentence built up by the Composer differs from the 
semantic representation proper in that it not just 
contains the semantic information available from 
the sentence but also morpho-syntactic information 
from the f-structure needed to constrain the resolu- 
tion process. Thereby, this information is passed to 
the Resolver which separates semantic and syntac- 
tic information and uses it. What sort of morpho- 
syntactic information is passed will be discussed 
later. 
The concept of resolution here is broader than 
the usual one which comprises mainly determin- 
ing the reference of anaphoric expressions like pro- 
nouns. We use the term as covering all kinds of con- 
text dependency beyond the single sentence level 
where something in the sentence has to be related 
to some entity in the preceding discourse. The term 
temporal resolution will be used to refer to the pro- 
cess of determining the temporal structure of the 
events the discourse is about. 
The Resolver is intended to implement good 
heuristics based on purely linguistic knowledge. 
The evaluation of the readings produced should be 
left to the inference machine which also can access 
non-linguistic world knowledge. 
4.2 Temporal Features in the F- 
Structure 
In the Kamp/Rohrer system the tenses are ana- 
lyzed by means of four features which have tempo- 
ral and aspectual meaning: 
• PERFectivity, 
• PttO Gressivity, 
• TENSE, 
• Temporal Perspective 
Tense forms can have several meanings or func- 
tions in discourse. Plusqueparfait for instance has a 
reading as flashback, and a perfective state reading 
with the temporal perspective "speech-time" or ac- 
tually reached "reference time". Imparfait can have 
the perspective in the past at the reference time or 
at the speech time. Following Kamp/Rohrer(1985), 
the meanings of the main narrative tenses can be 
described in the following way: 
Form TP 
passe s. 
imparfait 
imparfait 
plusquep. 
plusquep. 
plusquep. 
cond. I 
-PAST 
-PAST 
+PAST 
+PAST 
-PAST 
+PAST 
+PAST 
TENSE \[ PROG \[PERF 
past 
past + - 
pres + - 
past 
past + + 
pres + + 
fut +/- - 
Since it is not desirable to represent these am- 
biguities syntactically we use the Kamp/Rohrer 
categories in a slightly different way to get unique 
descriptions of the tenses. It is completely left to 
the resolver to account for these ambiguities. Since 
we exclude the TP-feature we need the additional 
TENSE-value conditionnel. To mark tenses in in- 
direct discourse the transposed-feature is added: 
feature value 
perf +/.- 
prog +/- 
tense past/present/future/conditionnel 
transposed +/- 
Since we do not discuss embedded clauses in this 
paper, in the following the transposed-feature is 
skipped. The tenses are analyzed by these means 
as shown in Table 1. 
4.3 Tenses and Temporal Adjuncts 
in the Composer 
Here we will discuss what sort of information the 
Composer adds to the Proto-DRS when it encoun- 
ters a tense feature or temporal adjunct or sub- 
clause. It consists basically of two kinds: 
1. DRS-conditions, which do not depend on the 
resolution process 
2. Interface structures for the Resolver, called oc- 
currence information and represented as a 6- 
place occ-term. 
The occurrence information is used to transmit 
morpho-syntactic information from the parser to 
the resolver. For the tenses this occurrence infor- 
mation has the form 
oec(DRF, Pos, TF, Tense, tense, Pointer) 
- 46 - 
imparfait: perf-- -, prog - +, tense ---- past 
pass~ simple: perf--- -, prog = -, tense ---- past 
pass~ compos~: perf -- ~-, prog -- -, tense -- present 
plusqueparfait: perf -- -I-, prog ---- _, tense -- past 
conditionnel I: perf -- -, prog = _, tense -- conditionnel 
Table 1: Decomposition of the Tenses into Semantic Markers 
Form DRS Occurrence Information 
present t: DRS 
impaffait t: DRS 
pass~ simple t: DRS 
perf. participle: t: DRS 
conditionneh t: DRS 
occ(t,. ,- ,pres,tense,Pointer) 
occ(t,- ,tf(_ ,÷prog,- ),past,tense, Pointer) 
occ(t,- ,tf(_ ,-prog,_ ),past,tense,Pointer) 
occ(t,_ ,tf(perf,_ ,_ ),_ ,_ ,- ) 
occ(t,_ ,_ ,cond,tense,Pointer) 
Table 2: Temporal Information introduced by the Composer 
with the following slots: 
DttF the temporal discourse referent, which the 
resolution process has to locate, usually intro- 
duced by the verb 
Pos verb position in surface structure, 
TF tf(Perf, Prog, Tr), the temporal features term. 
The Perf-slot marks the analytic tenses, Prog 
serves to distinguish for instance imparfait and 
passd simple, Tr stands for the value of the 
lransposed feature. 
Tense past I present I future I conditionnel, values 
of the tense feature, 
tense marker, indicating that the occ-term stems 
from a tense feature. It is also the trigger for 
temporal resolution. 
Pointer indicates the occurrence of the tense in 
main or embedded clauses. The clauses are in- 
dexed in a unique way (by natural numbers). 
The pointer is a pair consisting of the index 
of the superordinate clause and the index of 
the clause itself. Main clauses point to them- 
selves. Such an indication is necessary for the 
treatment of embedded sentences. The pointer 
encodes a simplified tree structure for the sen- 
tence and allows moving around in the tree. 
Temporal adjuncts and subclauses also provide oc- 
currence information marked in a special way. They 
contribute an occ-term of the following kind: 
occ(DRF, Pos,_ ,Rel, sel- lime, Pointer) 
DRF here represents the time introduced by the 
adverbial 
Pos its position in the surface structure 
Rel the temporal relation introduced. For instance 
trois jours avant introduces before. 
set-time indicates a special resolution mode for 
temporal adjuncts and indicates that this in- 
formation was contributed by a temporal ad- 
junct. In the resolution process the marker will 
cause the DRFs of the tense markers to resolve 
to DRF. 
The information shown in table 2 is introduced by 
the Composer for the tenses (the "" represent ini- 
tially empty slots which get filled in in the process 
of combining the meanings). 
It will be noticed that the tenses do not intro- 
duce new conditions into the DRS since the tem- 
poral relations cannot be determined without re- 
spect to the nature of the temporal 'antecedent' 
and therefore have to be generated in the Resolver. 
4.4 Temporal Resolution 
For temporal resolution the Resolver uses a stack 
of a system of times consisting of quintuples of the 
form: 
- 47 - 
1. reference time (usually the last event) 
2. temporal perspective point 
3. temporal location time (usually identical to 
the reference time) 
4. speech time (at present kept constant for the 
whole discourse) 
5. last resolved tense (with its occurence infor- 
mation) 
Every resolution process generates such a quintu- 
ple which gets stacked. If the temporal perspective 
point is changed (plusqueparfalt and conditionnel), 
a substack is created and used till the original per- 
spective point is restored. 
The resolver removes the occurrence informa- 
tion for the tenses and temporal adjuncts from the 
proto-DRS. The tenses get resolved according to 
the rules discussed below. The presence of tempo- 
ral adjuncts changes the flow of resolution as it re- 
quires that the temporal DRF introduced by the 
adjunct has to be resolved in accordance with the 
DRF introduced by the tense. 
We will illustrate the effect of some resolution 
rules reflecting the heuristics of the system by dis- 
cussing two sample texts. 
sl Ce jour-l~ il pleuvait tr~s fort. 
e2 Jean regarda par la fen~tre. 
s3 Marie n'~tait pas 1£. 
e4 Il mit son impermSable 
e5 et sortit. 
ce jour-ld introduces the location time t for the 
first state, sl, (pleuvoir). The Composer augments 
the DRS by the condition day(t) and the Resolver 
by the condition t C sl. 5 The first times-quintuple 
consists of a variable for the reference time, (no 
event is actually mentioned), the perspective point 
is assumed to be the speech time. The speech time 
is fixed by "now", t is the location time and in the 
last position the occ-term of sl is stored. Since a 
reference time does not yet exist, the integration 
of e2 produces a temporal relation with respect 
to the last location time: e2 C t, i.e. e2 happens 
within t. A second times-quintuple is put onto the 
stack with the reference time e2 and the new occ- 
term. The other time coordinates remain constant. 
SBecause of the definite description ce jour-ld the NP- 
Resolver has to establish an equation between t and a DRF 
of the preceeding text which is a day. If there is no such 
antecedent t has to be accomodated. The latter ease is at 
present not implemented. 
e2 serves now as reference time for s3. The new re- 
lation e2 C s3 is introduced and the information 
which stems from the occ-term of s3 and the old 
tenses is put as a third times quintuple onto the 
stack. The reference time is not changed. It should 
be noticed that for new states the "smallest" avail- 
able location time is used. Normally this is the ref- 
erence time of the previous quintuple if existent. 
For it is not necessary that the explicit given loca- 
tion time, ce jour-lh in the given example, serves 
as location time for subsequently introduced states 
as well. e4 is ordered with respect to e2, e5 with re- 
spect to e4 by the not-before-relation and the times- 
stack is updated in the obvious way. All states and 
events are located obviously before the speech time 
now. We omit tile full DRS of the example here. 
el Le 6 octobre Pierre arriva ~ Paris. 
e2 Le 3 octobre il ~tait parti. 
e3 Le lendemain il avait travers~ l'Espagne. 
sl Et maintenant il ~tait l'a. 
According to the different meanings of the 
plusqueparfait mentioned in section 3 different 
rules are available to deal with the plusqueparfait 
of the second sentence. However, especially in cases 
where a frame time as in the example exists, the 
possibility to introduce a flashback is preferred. A 
stack in the stack is created and the new event 
serves as reference time for subsequent events in 
the flashback. The last reference time of the higher 
level is now regarded as temporal perspective for 
the events occcuring in the flashback. They are lo- 
calized before that perspective, le lendemain in the 
third sentence has to be resolved to an existing ref- 
erence time, i.e. el or e2. Since we are already in 
a flashback, in processing the plusqueparfait of e3 
the continuation of this flashback is preferred. Thus 
a solution with el as antecedent for le lendernain 
would lead to a cyclic structure and should be ruled 
out by the inference component. The correct order- 
ing conditions are given by establishing e2 as an- 
tecedent for the time introduced by le iendemain, 
and t as location time for e3. The perspective time 
is copied from the stack. Thus we get the condi- 
tions: 
e2 < t, e3 C t,e3 < el 
The embedded stack is updated by the new quin- 
tuple. 
The implemented heuristics require that the tense 
switch from a plusqueparfait of a flashback to pass~ 
- 48 - 
simple or pass~ compos~ or the explicit reference 
to the perspective point, for instance by means of 
"maintenant", always lead to a reactivation of the 
initial level, from which the first flashback started, 
that is, all substacks are popped. Without main- 
tenant the imparfait of sl could lead to the con- 
tinuation of the flashback or to the continuation of 
the main story. It is regarded as the state variant 
of both categories. 
The (simpified) DRS of the example above 
thus looks as follows: 
now tl t2 t3 el e2 e3 sl 
6 octobre(tl) 
el: \[arriver(pierre)\[ 
goal(el) = paris 
el c tl 
el < now 
3 octobre(t2) 
e2: I partir(pierre) I 
e2 C t2 
e2 < now 
e2 < el 
day(t3) 
e3: traverser(pierre,espagne) I 
e3 C t3 
e3 < now 
e3 < el 
next-day(t3,t2) 
s3: \[ ~tre-~(pierre,paris) \] 
el C sl 
sl < now 
In our opinion cases as in the example above 
cannot be treated without adding new parame- 
ters to the Reichenbachian system. At least doing 
it facilitates the job. Beyond the imparfait/pass~ 
simple, pass~ compos~-distinction French does not 
make use of an explicit morphological aspect mark- 
ing. Therefore, for instance in the case of condition- 
nel, treated as anticipation of an episode, we use 
the Aktionsart-characterization of the verb stored 
in the temporal sort of the DI~F to specify the value 
of the prog-feature. An episode of states and dy- 
namics then is treated similar to an imparfait-pass~ 
simple-story transposed by the stored (past) per- 
spective time. By this means we get an interaction 
of Aktionsarten and tenses. 
Similar to the case of flashback the tense 
switch from conditionnel to another past tense form 
marks the end of the anticipation and the reactiva- 
tion of the initial level. 
5 Conclusion and Problems 
The implemented system assigns DRSs to natural 
language texts in such a way that the partial event 
structure, substructure of the whole DRS, reflects 
the events and temporal relations of the story. The 
system is incomplete at present in the following re- 
spects: 
The interaction of nominals and aspects is not 
accounted for. 
A principled treatment of tenses in embedded 
clauses is missing. 
The resolution of deictic/anaphoric temporal 
adjuncts is rudimentary. 
Knowledge about event types is at present not 
available to the Resolver. 
Thus the output event structure is generally under- 
specified. The integration of an inference compo- 
nent combined with background knowledge should 
restrict the number of possibilities to order the 
events linearly. In dependance of the Aktionsart the 
events of the DRS can be assigned subevents mark- 
ing the boundaries of the event as suggested by 
Moens and Steedman(1986), (cf. Eberle(1988b)). 
These subevents can be regarded as having no tem- 
poral extension. Thus on the subevent-level before 
and equivalent exhaust the intuitive possibilities of 
temporal relations. By means of this finer gran- 
ulation the (linear) event substructure of a DB.S 
corresponds to a unique (linear) interval struc- 
ture provided the event relations before and over- 
lap are defined in terms of Allen's interval struc- 
tures (cf. Allen(1983)). Thus inferencing systems 
which deal with intervals, as the Allen system, be- 
come available but also systems which deal with 
point-like events as the event calculus of Kowal- 
ski/Sergot(1985). In addition we get the possibility 
to deal with temporal relations on different levels 
of precision. 
In Eberle(1988a) the possibilities of monotonic 
reasoning in partial event structures (in the sense of 
the Kowalski/Sergot-approach) were investigated. 
It is planned to extend the algorithm suggested 
there which adds deduced events to the structure, 
in order to deal with measure statements, knowl- 
edge about hierarchies of event types (e.g. subevent 
- 49 - 
typologies) and temporal frames for event types 
(average duration of an event type). One aim is 
to rule out circular structures as mentioned in sec- 
tion 4.4. For simple cases we are able to do it yet. 
In such conflicting cases one has to backtrack to 
the Resolver, the Composer or the Parser to gen- 
erate other readings. In this sense the suggested 
analysis system is non-monotonic since it generates 
other temporal relations if necessary. When finally 
a consistent reading has been arrived at, the event 
calculus can be used for non-monotonic reasoning, 
e.g.to deal with the problem of the validity of lo- 
cation times with respect to events localized before 
or after events for which an explicit relation of in- 
clusion holds. 
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