A TENSE AND ASPECT CALCULUS 
DIANA SANTOS 
INESC, R. Alves Redol, 9, 
P-1000 Lisbon, Portugal 
dms¢inesc, inesc .pt 
may be used for further reasoning, by assuming it 
in the absence of contradictory information. Fur- 
ther investigation should be done to connect these 
non-monotonic reasoning aspects. 
There may be more true statements than those 
that can be proved. However, this seems to be 
a correct property for a system modelling natu- 
ral language: Ordinarily, we only express relevant 
facts -- not all true facts. 
2 The method 
The departing points are aspectual classes (stored 
in the lexicon) and morpho-syntactic information 
(tenses, aspectualizers, adverbs, particles, sub- 
bordinated clauses, etc., which we presume are 
given by former syntactic processing). 
Aspectual classes will be translated by simple 
formulas talking about times (intervals, points or 
unspecified with regard to this distinction) or con- 
junctions or disjunctions of these. 
The other mechanisms will be described by im- 
plications. The formulas on both sides refer di- 
rectly to the aspectual classes instead of their re- 
spective translations, enabling us to express sig- 
nificant generalizations. 
Let us make some remarks on the intended 
meaning of the logical connectives. OR (V) links 
different properties of an object. That is, taken 
in isolation, that NL fragment or device has all 
those properties, irrespective of the fact that it 
may never exhibit them together in actual use. It 
will be the role of other constituents in the text 
to select the intended reading. However, that se- 
lection is not compulsive, which means that OR 
represents linguistic vagueness. 
Example: An acquisition (see below) like remem- 
ber may be used in an inehoative reading (-~ come 
to my mind~ "After some time, I remembered it"), 
or in a stative reading (= be in my mind, "~I still 
remember it"}. But none is enforced in UDead 
men do not remember ~. 
AND (A) connects a set of properties of a single 
occurrence, even if some do not occur (but they 
Abstract 
This paper focuses on a theory of tense and aspect 
(the representation of time in natural language) that 
attempts a formM representation of the relevant lilt- 
guistic devices as implication rules in first order logic. 
It presents a rich ontology as far as verb aspect is 
concerned, distinguishing between complex patterns 
and vagueness. 
Examples and conclusions are drawn from tile com- 
parison between English and Portuguese, pinpointing 
the importance of contrastive studies both for the un- 
derstanding and for the evaluation of a general theory 
of tense and aspect. 
In the paper, we present the actual translations of 
a wide range of phenomena, and a short example. 
1 Introduction 
The method we propose to represent and reason 
with tense and aspect (T&A) in natural language 
has several distinguished features: 
It is both used in the building of the seman- 
tic representation of natural language text, and 
in the inferences allowed from what was actually 
said. There is therefore no conceptual separation 
between parsing and reasoning. 
It is based on a two-step translation into first- 
order predicate calculus, that allows us to use 
standard logic while preserving certain general- 
izations, i.e., some (sets of) patterns of temporal 
relations are identified as lexicalizing an aspec- 
tual class, and T&A mechanisms are expressed in 
terms of the latter. 
Grammatical mechanisms are translated into 
simple implications. On one hand, they restrict 
/ identify the set of situations they can apply to, 
while they may also introduce lexical information 
of their own. This second property allows for an 
elegant treatment of redundancy, in addition to 
an obvious explanation for their stand-alone oc- 
currence (cf. begin or just). 
Being proof-theoretical, the meaning of a sen- 
tence or text is what can be proved. If other in- 
formation is brought to focus but not proved, it 
ACTeS nE COLING-92, NANTES, 23-28 hot3"r 1992 1 I 3 2 PROC. OI: COLING-92, NANTES, AUG. 23-28, 1992 
may). The several properties conjoined are re- 
lated for each actual occurrence; they are several 
faces of the same situation; and no vagueness is 
at stake. AND links the several ways people may 
choose to present a given situation (called view- 
point aspect in \[12\[). 
Example: An accomplishment like build may be 
used in a durative description ("He was building 
it slowly"), or in a resultative description ("He 
built it in two years')~ referring nevertheless to 
the same event. Or, in an achievement like leave, 
the period when a person is leaving surrounds the 
moment of leaving. 
The lefthand side of an implication represents the 
conditions which are required to the felicitous use 
of the mechanism. In case not all information 
is available, it may provide the sort of defeasible 
information that can be seen as the type coercion 
proposed in \[81 . 
A sentence has a meaning iff there is at lea~t 
one derivation including all components. If there 
is more than one, the alternatives will stand in an 
OR relation, and one will probably be selected by 
the following co-text. 
Example: "He was attending a course on seman. 
tics" could mean ~he would" at a future time point 
relative to the past in question, or that he was ac- 
tually attending it. 
Syntax defines what are the possible arguments 
to the operators, thus driving inference and disal- 
lowing certain combinations. One way to model 
this fact is to allow syntactic structure itself to 
introduce axioms or metaaxioms, restricting in- 
ference order. We stick to compositionality, but 
assign a non-trivial meaning to syntactic struc- 
ture, contrarily to e.g. \[1\]. 
The method thus reduces to applying simple 
resolution to get all possible derived information, 
and possibly get rid of different interpretations. 
The set of all asserted formulas in the system is 
what it can understand. 
We will now argue for this method's adequacy 
to handle natural language tense and aspect sys- 
tems by presenting linguistic motivation from two 
different languages (English and Portuguese), see 
\[10\] for more detail. At the same time, we 
give novel treatments of well-known phenomena. 
These two points, we think, should be considered 
independently. That is, one can accept the gen- 
eral calculus yet proposing totally different trans- 
lations, or on the contrary, import our linguistic 
solutions into a distinct framework. 
2.1 Aspectual classes 
Aspectual classes indicate the temporal con- 
stituency of the situation they describe. They 
range from states (without any restrictions what- 
soever, or better, with no internal temporal di- 
mension) to accomplishments or achievements 
that include a complex temporal pattern in their 
lexical ITleaning. They can~ moreover, represent 
vague concepts that materialize in different tem- 
poral patterns, with some common core meaning, 
as was described above. 
Our ontology is based solely in temporal prop- 
erties, thus cause or agentivity are considered 
separate information, contrary to the overwhelm- 
ing majority of aspectual classifications (cf. the 
overview in \[14\]). 
In the figure next page, T represents an unspec- 
ified temporal object, which can denote either an 
interval, I, or a time point, t. We define three 
'basic' classes, corresponding to three irreducible 
temporal patterns, and then five others in terms 
of the former, with some additional "low-level" 
conditions, linking the temporal variables among 
the elementary intervening aspectual class defi- 
nitions. The sole reason why we did not define 
other combinations is that we did not find, in the 
languages studied, examples of lexical items that 
covered such complex patterns. 
For a more detailed exposition of the classes 
and objective criteria for distinguishing between 
them in Portuguese, see \[11\]. Still, some remarks 
are in order here: 
1. We distinguish 
between changes and achievements since el- 
ements of the first class cannot be amplified 
in time, that is, they have to be punctual, 
like notice or discover, while the others also 
involve some encompassing time connected 
to them: leave, die, open. This shows in the 
acceptance of the progressive by the latter. 
2. We distinguish act-states from states, on 
one hand, for its well-formedness with the 
progressive, and from activities, on the 
other, because their present tense implies 
the progressive. 
Example: He lives there, it hangs from there 
=,~ he is living there, it is hanging there. 1 
~They were mentioned as problematic in \[4\] and ela~ed 
together in {3\]. 
ACTES DE COLING-92, NAICrEs, 23-28 AO~r 1992 1 1 3 3 PROC. OF COLING-92, NANTES, AUG. 23-28, 1992 
state(P) ~ P(T) 
act(P) ~ P(I) 
change(P) ~ P(t) A notQ(i) A Q(t) A initial(i, I) A final(t, 1) 
ace(P) ~-~ act(P) A change(R) A final(t, I) 
=- P(I1) A R(t) A notQ(i) A Q(t) A initial(i, I2) A final(t, I~) A final(t, I1) 
ach(e) ~-~ act(P) A change(P) A inside(t, 1) 
------ P(I1) A P(t) A notQ(i) A Q(t) A initial(i, 1~) A final(t, I~) A inside(Iz, It) 
act - state(P) ~ act(P) V state(P) A inside(T, \[) 
=-- P(I) V P(T) Ainside(T,I) 
acq(P) ~ change(P) V state(P) A initial(t,T) 
-- P(t) A notQ(i) A Q(t) A initial(,', I) A fi.al(t, I) v P(T) A initial(t, T) 
series(P) ~ change(P) V act(P) A initial(t, I) 
=- e(t) A notQ(i) A Q(t) A initial(i, I) A final(t, I) V P(I) A initial(t, I) 
3. We name acquisitions those verbs which ex- 
press both the change that takes place and 
the resulting state, such as remember, un- 
derstand, know. They can be operationally 
pinpointed by the simultaneous acceptance 
of present tense and of point adverbials like 
suddenly in the past. ~ 
4. Finally, series can mean both an individual 
change and a set of them. s They accept 
both frequency and duration adverbs in the 
simple past: 
Example: He coughed all night~once. 
The temporal indices have roughly the same sta- 
tus of those in the semantic representation of 
UCG(15\], except that we do not impose one (sole) 
index per formula. We differ also in that we are 
not committed to a basic ontological distinction 
between states and events, and that instead of 
those indexes we use plain temporal objects. 
2.2 Grammatical mechanisms 
Grammatical mechanisms are defined in terms of 
"simple" aspectual classes instead of a full trans- 
lation into predicates of time. We further need 
two basic operations, namely state creation (Slat) 
and activity creation (Act): 
Slat(P) = PO 
Act(P) = act(Q) 
The first abstracts from time, to model what we 
call temporal intensionality (i.e., non-dependence 
on temporal extension). The second creates a 
new activity ranging over an interval containing 
2In Portuguese, where this clals is bigger, there is 
clearer evidence~ namely the existence of a straightforward 
difference in meaning between the two "simple pasts", Per- 
felts and Imperfeito. References to this class (uanmned) 
can also be found in \[4} and \[6 I, 
aln this last sense, they correspond to the ones in \[5\]. 
\[9\] uses the name "actions" for our series. 
more than one instance of P, or, equivalently~ cov- 
ering an unhomogeneous temporal region where 
P is true. The introduction of these two opera- 
tions corresponds to our belief that there is more 
to sentence aspect than verbal aspect, contrarily 
to what most classifications and calculi have as- 
sumed without questioning. It seems to us that 
the latter is a proper subset of the former. Thus, 
it makes perfectly sense to let morphosyntactic 
(i.e., non-lexical) mechanisms introduce new as- 
pect properties that cannot be found at the lexical 
level. 
Let us now present some contrastive analyses: 
Progressive act(P) ~ P(t) A inside(t, I) 
Progressive P :=¢, Slat(Q) A Q(T)A 
P(t) A before(T, t) A short(T) 
Estar act(P) -~ P(t) A inside(t, I) 
By the first definition 4, progressive applies to ac- 
tivities, achievements and acquisitions, and is au- 
tomatically true of act-states, or rather: for that 
class, the progressive and the non-progressive ver- 
sions are equivalent. 
The second definition encompasses the fllturate 
reading and stative uses such as He is resem- 
bling his father more and more. With accomplish- 
rnents, progressive may be ambiguous between 
tile two. 
Estar, the Portuguese progressive, has only the 
first English reading. 
SirnplePres P =~. Slat(P) A P(now) 
Presente P ~ Slat(P) A P(T)A 
inside(now, T) 
The English definition allows for the timeless 
truths X are Y and property readings such as She 
dances, while the Portuguese one accounts for the 
fact that we say Estou em Lisbon hd £ anos ("I'm 
4Whenever there are several definitions of tile same op- 
erator~ an OR is intended. 
ACZF.S DE COLING-92, NANTES, 23-28 AOr~T 1992 1 1 3 4 PROC. OF COLING-92, NANTES. AUG. 23-28, 1992 
in Lisbon (since) two years ago"-- in English the 
present perfect should be used), or Estou em Lis- 
bon atd damingo ("I'm in Lisbon until Sunday"). 
lmperfeito P :~ state(P) A P(T)A 
before(7', now) 
Per felts ~ final(t, T) A before(t, now) 
SimplePast P ~ before(t, now) 
Perfect P ~ final(t,T) At <. T.~/',Q(T2) 
Tim above accounts for the following facts: I have 
run means I've finished (and similarly for any 
accomplishment/achievement), and that it gets 
translated by the Portuguese Perfeito in all those 
cases. Also, I had run means it was finished before 
the time I am talking abou, t. Finally, to have done 
definitely means completion. We can also derive 
tim meaning of the present and past perfects: 
PresPerf P ~ final(t,T) A t 5~_ T2A 
Q(7~) A Stat(Q) A inside(T2 ..... ) 
PastPerf P ~ final(t,7") A t < 7:~A 
Q(T2) A Stat(Q) A before(T2, now) 
Just final(t,T) 2, t < 7'.z ~ initial(t, I) 
A final(T2, I) A short(I) 
This definition allows one to apply just only fi~r 
perfect tenses, and also to make just a function 
of measure, thus domain dependent (by the use 
of the predicate short). Present perfect with just 
is translated in Portuguese by the aspectualizer 
aeabar (syntactically, a verb taking the preposi- 
tion de before the infinitive verb) 
Acabar act(P) A final(t, 11) ~ act(Q)A 
final(t, 12) A final(12, 11) 
Acabar P(t) ~ initial(t, I) A final(T:, l)A 
short(I) 
acabar (see \[2\]) may either specify the last in- 
terval of an activity with definite end, or a short 
interval after a point event (change). This makes 
it ambiguous for accomplishments. 
PC act(P) A final(t, T) -~ before(now, t)A 
initial(i, T) A before(i, now) 
PC P :> Act(P) A final(now, I) 
If Perfeito Composts (PC) takes an accomplish- 
ment, PC rueans the final is not yet over: Tenho 
eonstruido a minha easa (I have been building my 
house). If it takes an activity, it means a set of 
distinct actual activities up to now: Tenho cot- 
rids (I have run lately). If it applies to a state, 
PC may represent a set of distinct states or an 
homogeneous up-to-now state: Tenho estado aqul 
todas as quartas / desde as 3, just like its En- 
glish present perfect translation ("I've been here 
every Wednesday / since 3 o'clock"), it should 
be mentioned that apart from states, the English 
translation of PC requires the adverb lately. 
Lately flnal(t, 7") A t <_ T~ ^ inside(T~, now) 
initial(t, I)/~ final(now, I)/~ short(I) 
Tile formula above says that lately requires the 
present perfect, and implies t < T~. This is jus- 
tified if we note that even though states can be 
used with present perfect and lately in English, 
this use requires that tile state does not extend 
til now: Have you been here latelyf would other- 
wise not make sense. 
Andar P(t) ~ Act(Q) 
Andar act(P) :~- act(Q) A inside( lz, I1) 
Andar either creates a big activity out of small 
events, or selects a subactivity inside one. English 
renderings of the first meaning are keep -ing, now 
and then or tim iterative progressive. The second 
is usually translated by the progressive (check the 
similarity with our translation of the English pro- 
gressive). 
2.3 Connectors 
Connectors relate two different tense-aspectual 
descriptions and are realized ms temporal connec- 
tives or prepositions. 
To clearly indicate their double input, we will 
use the symbol & to separate the conditions from 
the first and the second arguments (the second is 
the one immediately following the connector). 
At P(t)&Q :~ Q(t) 
While act(P) v P(t)&act(Q) :> inside(Te, I0) 
While the translations above do not explain the 
different interpretations arising from different po- 
sitioning of the arguments, we believe that this is 
precisely the kind of information that should be 
brought by syntax, though for tile moment we do 
not have a precise formulation for it. 
3 Example 
Example: "Many people died yesterday". 
die: act(die) A change(die) A inside(t, ll) 
yesterday: P ~ before(T,,tow) A day(T) A 0 < 
T - now < 24h A inside(Tp, 7") 
simple-past: P ~ before(T, now) 
plural-NP: P => Act(P) 5 
die-yesterday: act(die) A 
change(die) A inside(t,I) A before(T, now) A 
day(T) A inside(t, T) A inside(I, T) 
died-yesterday: act(die) A 
change(die) A inside(t, I) A befo~e(T,.o,O A 
day(T) A inside(t, T) A inside(l, T) 
SSince we are dealing only with tente-~pectual phe- 
nomena, we ~implify plural noun phrases interpreting them 
;~s activity creation. 
AC:IEs DE COLING-92, NAN'rI~S, 23-28 Ao(Yr 1992 1 1 3 5 PROC. OF COLING-92, N^~rrl!s, AUG. Z3-28, 1992 
many-people-died-yesterday: 
Act(act(dle)) A Act(change(die)) A ... 
This example deliberately reflects an important 
issue, namely, the relevance of non-verbal con- 
stituents to the overall aspect, which is the sub- 
ject of most calculi (see \[13\] or \[7\]). We agree 
with Krifka \[7\] on that, as aspect marking on 
noun phrases in several languages demonstrates, 
there is no fundamental difference in the import 
brought by lexical aspect, be it of verbs or other 
parts of speech. When we postulate a verbal as- 
pect as opposed to sentence one, thus, the stress 
should be on lexical rather than verbal. What 
is particular to our approach is the belief that 
grammatical means (syntax or tense) may pur- 
port significant aspect properties not available (or 
not present) at the lexical level. 
4 Discussion 
From a descriptive point of view, the system pro- 
vides a liner characterization of verbs, drawing an 
important distinction: that between the parts of 
a same situation and a don't care or vagueness 
associated with a lexical item (the first is related 
to the perception of an action by a speaker of a 
language; the second is related to the language 
system / lexicon). 
Second, it is based on a linguistic comparison 
of two distinct languages, therefore making it ap- 
plicable at least to more than one. This makes it 
also of interest for machine translation research. 
Third, we expect a system thus formalized to 
be easy to extend, simply by adding new and sub- 
tler constraints in the form of implications and/or 
complex syntax import, and not bothering with 
the reasoning algorithm, by using standard logic. 
Finally, we think that this method has signifi- 
cant advantages compared to applicational mod- 
els in the treatment of redundancy. Redundancy 
is a pervasive property of natural language, but 
it is hard to model when one device is required to 
apply before another, each bringing different in- 
formation. In this account, on the contrary, simi- 
lar information brought in twice does nothing but 
assert it once. 
To conclude, we should acknowledge that given 
the early stage of this investigation, no implemen- 
tation and thus testing results are yet available. 
5 Acknowledgements 
I am indebted to JNICT who granted me a PhD 
scholarship, and to IBM Portugal for support of 
my studies. Most of the ideas present in this pa- 
per arose from stimulating discussions with Lauri 
Carlson. 
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ACRES DE COLING-92. NANTEs. 23-28 AOt'~" 1992 1 1 3 6 PROC. OV COLING-92. NANTES. AUG. 23-28, 1992 
