Splitting the Reference Time: Temporal Anaphora and 
Quantification in DRT 
Rani Nelken 
Tel-Aviv University 
Tel-Aviv 69978, Israel 
nelken@math, tau. ac. il 
Nissim Francez 
Computer Science Department 
The Technion 
Haifa 32000, Israel 
francez~cs, technion, ac. il 
Abstract 
This paper presents an analysis of tem- 
poral anaphora in sentences which con- 
tain quantification over events, within 
the framework of Discourse Representa- 
tion Theory. The analysis in (Partee, 
1984) of quantified sentences, introduced 
by a temporal connective, gives the 
wrong truth-conditions when the tempo- 
ral connective in the subordinate clause 
is before or after. This problem has been 
previously analyzed in (de Swart, 1991) 
as an instance of the proportion problem, 
and given a solution from a Generalized 
Quantifier approach. By using a care- 
ful distinction between the different no- 
tions of reference time, based on (Kamp 
and Reyle, 1993), we propose a solu- 
tion to this problem, within the frame- 
work of Dt~T. We show some applica- 
tions of this solution to additional tem- 
poral anaphora phenomena in quantified 
sentences. 
1 Introduction 
The analysis of temporal expressions in natural 
language discourse provides a challenge for con- 
temporary semantic theories. (Partee, 1973) in- 
troduced the notion of temporal anaphora, to ac- 
count for ways in which temporal expressions de- 
pend on surrounding elements in the discourse for 
their semantic contribution to the discourse. In 
this paper, we discuss the interaction of tempo- 
ral anaphora and quantification over eventualities. 
Such interaction, while interesting in its own right, 
is also a good test-bed for theories of the semantic 
interpretation of temporal expressions. We dis- 
cuss cases such as: 
(1) Before John makes a phone call, he always 
lights up a cigarette. (Partee, 1984) 
(2) Often, when Anne came home late, Paul had 
already prepared dinner. (de Swart, 1991) 
(3) When he came home, he always switched on 
the tv. He took a beer and sat down in his 
armchair to forget the day. (de Swart, 1991) 
(4) When John is at the beach, he always squints 
when the sun is shining. (de Swart, 1991) 
The analysis of sentences such as (1) in (Partee, 
1984), within the framework of Discourse Repre- 
sentation Theory (DRT) (Kamp, 1981) gives the 
wrong truth-conditions, when the temporal con- 
nective in the sentence is before or after. In DRT, 
such sentences trigger box-splitting with the even- 
tuality of the subordinate clause and an updated 
reference time in the antecedent box, and the 
eventuality of the main clause in the consequent 
box, causing undesirable universal quantification 
over the reference time. 
This problem is analyzed in (de Swart, 1991) as 
an instance of the proportion problem and given a 
solution from a Generalized Quantifier approach. 
We were led to seek a solution for this problem 
within DRT, because of DRT's advantages as a 
general theory of discourse, and its choice as the 
underlying formalism in another research project 
of ours, which deals with sentences such as 1-4, 
in the context of natural language specifications 
of computerized systems. In this paper, we pro- 
pose such a solution, based on a careful distinction 
between different roles of Reichenbach's reference 
time (Reichenbach, 1947), adapted from (Kamp 
and Reyle, 1993). Figure 1 shows a 'minimal pair' 
of DRS's for sentence 1, one according to Par- 
tee's(1984) analysis and one according to ours. 
2 Background 
An analysis of the mechanism of temporal 
anaphoric reference hinges upon an understand- 
ing of the ontological and logical foundations of 
temporal reference. Different concepts have been 
used in the literature as primitives. These range 
from temporal instants in Tense logic (Prior, 67), 
through intervals of time (Bennet and Partee, 
1978(1972)) as in the analysis of temporal con- 
nectives in (Hein~mKki, 1978), to event structures 
(Kamp, 1979) as in Hinrichs' (1986) analysis of 
temporal anaphora. 
261 
el rl 
el Cr0 rl < el el:l  phono I 
I~,Z r 0 
John(z) 
e2 
e2 ~ rl 
z light up I 8t 
nz8 
John(z) n C s 
et 
t = lot(e) 
e: Iz phone I 
et ~t 
=:~ t ~ < t e I __ t ~ 
e': \[z light up I 
Figure 1: a:Partee's analysis b:Our analysis 
An important factor in the interpretation of 
temporal expressions is the classification of situ- 
ations ihto different aspectual classes (or Aktion- 
sarten), which is based on distributional and se- 
mantic properties. In this paper, we only consider 
events and states, together termed eventualities 
in (Bach, 1981). In narrative sequences, event 
clauses seem to advance the narrative time, while 
states block its progression. The mechanism used 
to account for this phenomena in (Hinrichs, 1986) 
and (Partee, 1984), is based on the notion of ref- 
erence time, originally proposed by Reichenbach 
(1947). 
Reichenbach's well - known account of the inter- 
pretation of the different tense forms uses the tem- 
poral relations between three temporal indices: 
the utterance time, event time and reference time. 
The reference time according to (Reichenbach, 
1947) is determined either by context, or by tem- 
poral adverbials. 
2.1 St unified analysis of temporal 
anaphora 
Hinrichs' and Partee's use of a notion of refer- 
ence time, provides for a unified treatment of tem- 
poral anaphoric relations in discourse, which in- 
clude narrative progression especially in sequences 
of simple past tense sentences, temporal adverbs 
and temporal adverbial clauses, introduced by a 
temporal connective. This concept of reference 
time is no longer an instant of time, but rather, 
an interval. This approach can be summarized 
as follows: in the processing of a discourse, the 
discourse-initial sentence is argued to require some 
contextually determined reference time. Further 
event clauses in the discourse introduce a new 
event, which is included within the then-current 
reference time. Each such event also causes the 
reference time to be updated to a time 'just after' 
(Partee, 1984) this event. State clauses introduce 
new states, which include the current reference 
time, and do not update it. 
As an example of such an analysis consider the 
following narrative discourse (Partee, 1984): 
(5) John got up, went to the window, and raised 
the blind. It was light out. He pulled the blind 
down and went back to bed. He wasn't ready 
to face the day. He was too depressed. 
Figure 2 shows a DRS for the first two sen- 
tences of this discourse, according to Hinrichs' and 
Partee's analysis. The 'n' in the top DRS is a 
mnemonic for 'now'- the utterance time. The first 
event in the discourse, el - John's getting up - is 
interpreted relative to a contextually understood 
reference time, r0. The event et is included in the 
current reference time, r0. A new reference time 
marker, rl is then introduced, rl lies immediately 
after r0 (recorded as r0 _~ rl). rl serves as the cur- 
rent reference time for the following event e2. We 
continue in this fashion, updating the reference 
time, until the second sentence in the discourse 
is processed. This sentence denotes a state, sl, 
which includes the then-current reference time. 
ro el rl e2 r2 e3 81 
ro < n el C_ ford ~_ rl 
rl < he2 C rlrl ~r2 
r2 < ne 3 C r2r2 C Sl 
el: I John get up J 
e2: . . . 
Figure 2: 
Adverbial phrases, whether phrasal (e.g. 'On 
Sunday') or clausal (e.g. 'When Bill left'), are pro- 
cessed before the main clause. They introduce a 
reference time, which overrides the current refer- 
ence time, and provides an anaphoric antecedent 
for the tense in the main clause. This mechanism 
is used to explain how tense and temporal adver- 
bials can combine to temporally locate the occur- 
rence, without running into problems of relative 
scope (ttinrichs, 1988). The tense morpheme of 
the main clause locates the event time with re- 
spect to the reference time, whereas temporal ad- 
verbials are used to locate the reference time. 
When-clauses, for example, introduce a new ref- 
erence time, which is ordered after the events de- 
scribed in the preceding discourse. The eventual- 
ity in the when-clause is related to this reference 
time as discussed earlier with respect to narrative 
progression: a state includes its reference time, 
while an event is included in it. The eventuality 
in the main clause is interpreted with respect to 
this reference time. If the main clause is an event- 
262 
ro @ 
<~ rl < esee 
r2 r3 
<r4 
Figure 3: 
clause, this event introduces a new reference time, 
just after the event time of the main clause. As an 
example, consider the following discourse (Partee, 
1984): 
(6) Mary turned the corner. When John saw her, 
she crossed the street. She hurried into a 
store. 
Following Partee (1984), we will not construct 
a full DRS for this discourse, but illustrate it with 
a diagram in Figure 3, with circles denoting inclu- 
sion. 
2.2 Quantification over events 
(Partee, 1984) extends Hinrichs' treatment of tem- 
poral anaphora to the analysis of sentences, which 
contain a temporal adverbial and quantificatiort 
over eventualities. According to her analysis, 
these trigger box-splitting as do if or every clauses 
in DRT (Kamp, 1981). Consider the following ex- 
ample from (Partee, 1984): 
(7) Whenever Mary telephoned, Sam was asleep. 
nxyro I Mary(y) 
Sam(=) 
l 
el rl 
el C r0 el < n 
el "~ rl rl < n 
el: \[y telephone\] 
=:~ 
81: \[ 
81 
rl c_ sl 
z sleep \] 
Figure 4: 
The subordinate clause cannot be interpreted 
relative to a single reference time, since Mary's 
telephoning is not specified to occur at some spe- 
cific time. Still, the sentence needs to be inter- 
preted relative to a reference time. This reference 
time can be a large interval, and should contain 
each of the relevant occurrences of Mary's tele- 
phoning during which Bill was asleep. This refer- 
ence time is represented as r0 in the top sub-DRS. 
The 'whenever' triggers box-splitting. The 
event marker - ez is introduced in the antecedent 
box, with the condition that it be temporally in- 
cluded in the current reference time, r0 and be 
prior to n. The 'whenever' also causes the intro- 
duction of rl, a new reference time marker, rl lies 
'just after' el. The stative clause causes the intro- 
duction of Sl, which includes the reference time 
rl. 
The embedding conditions for the whole con- 
struction are just like those for a regular 'if' or 'every' 
clause, i.e. the sentence is true, if every 
proper embedding of the antecedent box can be 
extended to a proper embedding of the combina- 
tion of the antecedent and the consequent boxes. 
This means, as desired, that for each choice of 
an event el of Mary's telephoning, and reference 
time rl 'just after' it, there is a state of Sam's 
being asleep, that surrounds rl. 
A sentence such as (Ta) which is the same as 
sentence 7, except the 'whenever' is replaced by 'when', 
and 'always'is added in the main clause, 
would get the same DtLS. 
(Ta) When Mary telephoned, Sam was always 
asleep. 
2.3 Extending the analysis 
As noted in (Partee, 1984), this analysis does not 
extend in a straightforward manner to cases in 
which the operator when is replaced by (an un- 
restricted) before or after, in such quantified con- 
texts. Constructing a similar DRS for such sen- 
tences gives the wrong truth conditions. For ex- 
ample, Figure la shows a DRS for sentence 1, 
according to the principles above, rl - the ref- 
erence time, used for the interpretation of the 
main clause is placed in the universe of the an- 
tecedent box. Because the temporal connective 
is 'before', rl is restricted to lie before el. The 
embedding conditions determine, that this refer- 
ence time be universally quantified over, causing 
an erroneous reading in which for each event, el, 
of John's calling, for each earlier time rl, he lights 
up a cigarette. Paraphrasing this, we could say 
that John lights up cigarettes at all times preced- 
ing each phone call, not just once preceding each 
phone call. We did not encounter this problem 
in the DRS in Figure 4, since although the refer- 
ence time rl, is universally quantified over in that 
DRS as well, it is also restricted, to immediately 
follow el. It is similarly restricted if 'before' is re- 
placed with 'just before' or 'ten minutes before'. 
But, (unrestricted) 'before' is analyzed as 'some 
time before', and thus the problem arises. We 
will henceforth informally refer to this problem as 
Partee's quantification problem. 
Partee (1984) suggests that in these cases we 
somehow have to insure that the reference time, 
rz, appears in the universe of the consequent DRS, 
causing it to be existentially quantified over, giv- 
ing the desired interpretation. De Swart (1991) 
notes that simply moving rl to the right-hand box 
does not agree with Hinrichs' assumption, that 
temporal clauses are processed before the main 
clause, since they update the reference time, with 
respect to which the main clause will be inter- 
263 
preted. In our proposed solution, the 'reference 
time' is indeed moved to the right box, but it is a 
different notion of reference time, and (as will be 
shown) exempt from this criticism. 
3 The proportion problem 
De Swart (1991) sees Partee's quantification prob- 
lem as a temporal manifestation of the proportion problem, 
which arises in cases such as (Kadmon, 
1990): 
(8) Most women who own a cat are happy. 
The sentence is false in the case where out of 
ten women, one owns 50 cats and is happy, while 
the other nine women own only one cat each, and 
are miserable. This will not be predicted by the 
unselective binding of quantifiers in DRT, which 
quantify over all the free variables in their scope, 
in this case women-cat pairs. According to (de 
Swart, 1991) Partee's quantification problem is 
similar- the universal quantifier in sentences such 
as (1) binds pairs of events and updated reference 
times, where the desired quant.ificational scheme 
is universal quantification for the event and exis- 
tential for the reference time. 
De Swart (1991) offers a solution from a Gener- 
alized Quantifier approach, based on the analysis 
of quantified NPs in transitive sentences. In this 
analysis, the reference time is an implicit variable, 
which is needed in the interpretation of the tempo- 
ral relation, but is not part of the quantificational 
structure. 
Temporal connectives are viewed as relations, TC, 
between two sets of events: 
(9) {< el, e2 > I < el, e2 >E TC} 
The quantificational structure of such sentences 
can be analyzed either by an iteration of monadic 
quantifiers, or as a single dyadic quantifier of type 
< 1, 1, 2 >. In the first approach, adverbs of quan- 
tification (Q-adverbs) are assigned the structure: 
(10) Q(S,, {e,13(S,,,TC,,)}) 
In 10, S~ and Sm denote, respectively, the sets 
of events described by the subordinate and the 
main clause, TCe, denotes the image set of el un- 
der the temporal connective TC, i.e. the set of 
events e2 which are related to el via the relation TC, 
(presented in 9). In the second approach, the 
structure is: 
(11) \[Q,3\](S,,Sm,TC) 
De Swart's solution does overcome Partee's 
quantification problem, although not within DRT. 
As such, the existential quantification in 11 has to 
be stipulated, whereas our analysis acquires this 
existential quantification 'for free'. 
4 Splitting the role of reference 
time 
Our analysis of Partee's quantification problem 
uses a different notion of reference time than that 
used by the accounts in the exposition above. Fol- 
lowing (Kamp and Reyle, 1993), we split the role 
of the reference time, used to account for a large 
array of phenomena, into several independent 
mechanisms. This separation allows for an anal- 
ysis in DRT of temporal subordinate clauses in 
quantified sentences, which avoids Partee's prob- 
lem altogether. The mechanisms we discuss are: 
the location time, Rpt and perf 1. DRSs will 
contain temporal markers corresponding to loca- 
tion times and Rpts. 
The location time is an interval, used to tem- 
porally locate eventualities, in accordance with 
their aspectual classification. Events are included 
in their location time (recorded in the DRS as 
e C t on the respective markers), while states 
temporally overlap their location time (recorded 
as s O t). The verb tense determines the relation 
between the location time and the utterance time 
e.g. if the tense is simple past, the location time 
lies anteriorly to the utterance time. When it is 
simple present, the location time coincides with 
the utterance time 2. Temporal adverbials restrict 
the location time: temporal adverbs introduce a 
DRS-condition on the location time, while tem- 
poral subordinate clauses introduce a relation be- 
tween the event time s of the subordinate clause 
and the location time of the main clause. The ex- 
act temporal relation denoted by a temporal con- 
nective depends on the aspectual classes of the 
eventualities related by it 4. For example, in the 
following sentence 12, the event triggers the intro- 
duction of an event marker e, and location time 
marker t, into the DRS with the DRS-condition 
e c_ t. The past tense of the verb adds the con- 
dition t < n. In sentence 13, the location time of 
the event in the main clause is restricted to fall 
(just) after the event time of the event of the sub- 
ordinate clause. 
(12) Mary wrote the letter. 
(13) Mary wrote the letter when Bill left. 
1An additional mechanism is the TPpt, which for 
simplicity's sake will not be discussed in this paper. 
~Since the utterance time, n is a point in (Kamp 
and Reyle, 1993), the overlap relation between a state 
that holds in the present and n reduces to inclusion. 
3The event time t of an eventuality e is the smallest 
interval which includes e (recorded as t = loc(e)). 
4For the sake of the current presentation, we as- 
sume the following relations for When: if both the when-clause 
and the main clause denote states, then 
their respective time indices overlap. If both are 
events then the times are temporally close, with the 
exact relation undetermined. When one is a state and 
one an event, then the time index of the state includes 
that of the event cf. (Hinrichs, 1986). 
264 
Narrative progression is dealt with by using the 
feature Rpt (or reference point). The Rpt can 
be either an event or a time discourse marker, al- 
ready present in the DRS (recorded as assignment 
Rpt := e). Eventualities are interpreted with re- 
spect to the Rpt - events are taken to follow the 
current Rpt, while states include it. The Rpt is 
reset during the processing of the discourse. Note 
that in a 'terminal' DRS (ready for an embedding 
test), all the auxiliary Rpts 'disappear' (do not 
participate in the embedding). 
The perfect is analyzed by using the notion of a 
nucleus (Moens and Steedman, 1988) to account 
for the inner structure of an eventuality. A nucleus 
is defined as a structure containing a preparatory 
process, culmination and consequent state. The 
categorization of verb phrases into different as- 
pectual classes can be phrased in terms of which 
part of the nucleus they refer to. The perfect is 
seen in (Kamp and Reyle, 1993) as an aspectual 
operator. The eventualities described by the per- 
fect of a verb refer to the consequent state of its 
nucleus. For example, the following sentence 14 
denotes the state, s, holding at the present, that 
Mary has met the president. This state is a result 
of the event e, in which Mary met the president." 
Temporally, the state s starts just when e ends, 
or as it is put in (Kamp and Reyle, 1993):e and s 
abut, (represented as e DCs). 
(14) Mary has met the president. 
5 An alternative solution 
By extending the analysis of temporal subordinate 
clauses in (Kamp and Reyle, 1993), to sentences 
which include quantification over eventualities, we 
can propose an alternative DRT solution to Par- 
tee's quantification problem. As in (Partee, 1984), 
such sentences trigger box-splitting. But now, the 
location time of the eventuality in the subordi- 
nate clause serves as the antecedent for the loca- 
tion time of the eventuality in the main clause. 
In this approach, each of the relevant temporal 
markers resides in its appropriate box, yielding 
the correct quantificational structure. This quan- 
tification structure does not need to be stipulated 
as part of the Q-adverb's meaning, but arises di- 
rectly from the temporal system. We illustrate 
this analysis by constructing a DRS in Figure lb 
for sentence 1. 
In this DRS, n denotes the utterance time. The 
subordinate clause triggers the introduction of an 
event marker, e, with its event time marker t. 
The main clause triggers the introduction of an 
event marker e ~, and its location time marker t ~, 
with the DRS-condition e' C t ~. The assymetry in 
using the event time for e and the location time 
for e ~ arises from the interpretation rules of tem- 
poral connectives (for both quantified and non- 
quantified sentences). Since the temporal connec- 
tire in this sentence is before, the relation between 
these two markers is one of precedence. 
We adopt a suggestion by Chierchia in (Partee, 
1984), that the whole implication be rendered as 
a state. This state is no longer an atomic eventu- 
ality. It is a complex state denoting John's habit. 
This state holds during the present, and so its lo- 
cation time is n. 
This solution is not prone to de Swart's (1991) 
criticism against the naive solution of moving the 
reference time to the right DRS. The temporal 
clause may be processed before the main clause, 
since t', the location time of e', which 'replaces' 
rl, the reference time of Partee's analysis, as the 
temporal index of the eventuality in the the main 
clause, arises from processing the main clause (not 
updating the reference time of the subordinate 
clause). 
6 Additional phenomena 
In this section we present some applications of our 
analysis to related constructions. First, we con- 
sider the past perfect, as in sentence 2. De Swart 
(1991) gives this example to illustrate the inability 
to interpret temporal connectives without the use 
of the reference times. According to (de Swart, 
1991), the subordinate clause determines the ref- 
erence time of the verb, which lies anteriorly to the 
event time. Trying to use the event times would 
give the wrong analysis. This would seem to be 
troublesome for our approach, which uses the loca- 
tion time of the event in the main clause, and not 
its reference time. However, this is not a problem, 
since our analysis of the perfect by the use of the 
operator perf, analyses the eventuality referred to 
by the main clause, as the result state of a previ- 
ous event. The temporal relation in the sentence 
is inclusion between the event time of Anne's com- 
ing home, and the location time of the result state 
of Paul's already having prepared dinner. 
Next, we consider narrative progression in 
quantified contexts, as in sentence 3. The basic 
construction is just the same as in the paradigm 
structure, but now we have narrative progression 
in the consequent box. This narrative progres- 
sion is handled as ordinary narrative progression 
in (Kamp and Reyle, 1993), i.e. by resetting the 
Rpt. The DRS in Figure 5 describes the complex 
state sl, that after each event of John's coming 
home, there is a sequence of subsequent events 
according to his activities. 
Finally, we deal with sentences such as (4), 
which contain an iteration of an implicit generic 
quantifier and always. The situation described by 
John's always squinting when the sun is shining is 
analyzed as a complex state s3. This state holds 
whenever John is at the beach, recorded by the 
condition that the location time t~ of sa overlaps 
the event time, tl of John's being at the beach, s2 
in Figure 6. 
265 
Sl: 
S2 tl 
loc(s~) = tl 
s2:\[ z is at the beach \[ 
aohn(z) 
=¢, 
83 
nzysl 
the sun(y) n C sl 
szt2 
sa Ot~t~ Or1 
t3 84 
loc(s4)=t3 
s4: y is shining 
elt4 
el _C t4 
t4 C t3 
el:\[ z squint 
Figure 6: 
nXSltl 
slOt1 Q <n 
$1 
el t2 
el:l x c.n. I 
e2 e3 e4 t3 t4 t5 
e2 Ctzt2 <t3 
e2:\[z sw. on t~ 
Rpt := e~ 
ea C t4 e2 < ea 
ca: Ix take beer I 
Rpt := e3 
e4 C t4 ez < e4 
e4: ... 
Figure 5: 
7 Acknowledgments 
The work of the second author was partially sup- 
ported by a grant from the Israeli ministry of sci- 
ence "Programming languages induced computa- 
tional linguistics", and by the fund for the pro- 
motion of research in the Technion. The authors 
would like to thank Nirit Kadmon and Uwe Reyle 
for reading a preliminary version of this paper. 
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266 
