THE IMPERFECTIVE PARADOX AND 
TRAJECTORY-OF-MOTION EVENTS * 
Michael White 
Department of Computer and Information Science 
University of Pennsylvania 
Philadelphia, PA, USA 
mwhit e©l inc. c is. upenn, edu 
Abstract 
In the first part of the paper, I present a 
new treatment of THE IMPERFI~CTIVE PARADOX 
(Dowty 1979) for the restricted case of trajectory- 
of-motion events. This treatment extends and re- 
fines those of Moens and Steedman (1988) and 
Jackendoff (1991). In the second part, I describe 
an implemented algorithm based on this treatment 
which determines whether a specified sequence of 
such events is or is not possible under certain sit- 
uationally supplied constraints and restrictive as- 
sumptions. 
Introduction 
Bach (1986:12) summarizes THE IMPERFECTIVE 
PARADOX (Dowty 1979) as follows: "...how can 
we characterize the meaning of a progressive sen- 
tence like (la) \[17\] on the basis of the meaning of 
a simple sentence like (lb) \[18\] when (la) can be 
true of a history without (lb) ever being true?" 
(la) John was crossing the street. 
(lb) John crossed the street. 
Citing parallels in the nominal domain, Bach goes 
on to point out that this puzzle is seemingly much 
more general, insofar as it appears whenever any 
sort of partitive is employed. In support of this 
view, we may observe that the start v-ing con- 
struction exhibits the same behavior: 
(2a) John started jogging to the museum. 
(2b) John jogged to the museum. 
Here we see that (2a) does not entail (2b) -- while 
(2b) asserts the occurrence of an entire event of 
John jogging to the museum, (2a) only asserts the 
*The author gratefully acknowledges the helpful 
comments of Mark Steedman, Jeff Siskind, Christy 
Doran, Matthew Stone, and the anonymous refer- 
ees, as well as the support of DARPA N00014-90-J- 
1863, AI~O DAAL03-89-C-0031, NSF IRI 90-16592, 
Ben Franklin 91S.3078C-1. 
occurrence of the beginning of such an event, leav- 
ing open the existential status of its completion. 
Capitalizing on Bach's insight, I present in 
the first part of the paper a new treatment of 
the imperfective paradox which relies on the pos- 
sibility of having actual events standing in the 
part-of relation to hypothetical super-events. This 
treatment extends and refines those of Moens 
and Steedman (1988) and Jackendoff (1991), at 
least for the restricted case of trajectory-of-motion 
events. 1 In particular, the present treatment cor- 
rectly accounts not only for what (2a) fails to en- 
tail -- namely, that John eventually reaches the 
museum -- but also for what (2a) does in fact en- 
tail -- namely, that John follows (by jogging) at 
least an initial part of a path that leads to the 
museum. In the second part of the paper, I briefly 
describe an implemented algorithm based on this 
theoretical treatment which determines whether a 
specified sequence of trajectory-of-motion is or is 
not possible under certain situationally supplied 
constraints and restrictive assumptions. 
Theory 
The present treatment builds upon the ap- 
proach to aspectual composition developed in 
White (1993), a brief sketch of which follows. 
White (1993) argues that substances, processes 
and other such entities should be modeled as ab- 
stract kinds whose realizations (things, events, 
etc.) vary in amount. 2 This is accomplished for- 
mally through the use of an order-sorted logic 
with an axiomatized collection of binary relations. 
The intended sort hierarchy is much like those 
of Eberle (1990) and Jackendoff (1991); in par- 
ticular, both substances and things are taken to 
be subsorts of the material entities, and similarly 
1These are elsewhere called 'directed-motion' 
events. 
2This move is intended to resolve certain empirical 
and computational problems with the view of refer- 
ential homogeneity espoused by Krifka (1992) and his 
predecessors. 
283 
both processes and events are taken to be sub- 
sorts of the non-stative eventualities. What is new 
is the axiomatization of Jackendoff's composed-of 
relation (comp) -- which effects the aforemen- 
tioned kind-to-realization mapping -- in terms of 
Krifka's (1992) part-of relation (_U). Of particular 
interest is the following subpart closure property: 
(3) Vxyly2\[comp(x)(yx) A y2C_yl --~ comp(x)(y2)\] 
Postulate (3) states that all subparts of a realiza- 
tion of a given kind are also realizations of that 
kind. 3 From this postulate it follows, for example, 
that if e is a process of John running along the 
river which has a realization el lasting ten min- 
utes, and if e2 is a subevent of el -- the first half, 
say -- then e2 is also a realization of e. As such, 
this postulate may be used to make John ran along 
the river for ten minutes entail John ran along the 
river for five minutes, in contrast to the pair John 
ran to lhe museum in ten minutes and John ran 
to lhe museum in five minules. 
In order to resolve the imperfective paradox, 
we may extend White (1993) by adding a mapping 
from events to processes (whose realizations need 
not terminate in the same way), as well as a means 
for distinguishing actual and hypothetical events. 
To do the former, we may axiomatize comp's in- 
verse mapping -- Jackendoff's ground-from (gr) 
-- again in terms of Krifka's part-of relation. This 
is shown below: 
(4) VxylY2\[gr(yl)(X ) A comp(x)(y2) ---* y2C_yl\] 
Postulate (4) simply requires that all the realiza- 
tions e2 of a process e which is 'ground from' an 
event el must be subevents of el (and likewise, 
mutatis mutandis, for substances and things). As 
the realizations e2 of e may be proper subevents of 
el, the relation gr provides a means for accessing 
subevents of el with alternate terminations. 
To distinguish those events which actually oc- 
cur from those that are merely hypothetical, we 
may simply introduce a special predicate Actual, 
which we require to preserve the part-of relation 
only in the downwards direction: 
(5) Vxy\[Actual(z) A yU_z --* Actual(y)\] 
Postulate (5) is necessary to get John slopped run- 
ning to the museum after ten minutes to entail 
John ran for ten minutes as well as John ran for 
nine minutes, but not John ran for eleven min- 
utes. 
At this point we are ready to examine in some 
detail how the above machinery may be used in 
resolving the imperfective paradox. Let us assume 
3For the sake of simplicity I will not address the 
minimal parts problem here. 
that sentences such as (6) receive compositional 
translations as in (7): 
(6a) John ran to the bridge. 
(6b) John stopped running to the bridge. 
(7a) 3el. 
run'(j)(el) A to'(the'(bridge'))(r~(el)) A 
Actual(el) 
(7b) 3eele2e3. 
run'(j)(el) A to'(the'(bridge'))(rs(el))A 
gr(el)(e) A comp(e)(e2) A stop'(e2)(ea) A 
Actual(e3) 
In (7), el is an event of John running to the 
bridge. 4 In (Ta), this event is asserted to be actual; 
in (7b), in contrast, the progressive morphology on 
run triggers the introduction of gr, which maps 
el to the process e. 5 It is this process which e3 is 
an event of stopping: following Jackendoff (1991), 
this is represented here by introducing an event e~ 
composed of e which has ea as its stopping point. 
Naturally enough, we may expect the actuality 
of e3 to entail the actuality of e2, and thus all 
subevents of e2. Nevertheless, the actuality of et 
does not follow, as Postulate (4) permits e2 to be 
a proper subpart of el (which is pragmatically the 
most likely case). 
To make the semantics developed so far more 
concrete, we may now impose a particular inter- 
pretation on trajectory-of-motion events, namely 
one in which these are modeled as continuous func- 
tions from times to locations of the object in mo- 
tion. Depending on how we model objects and 
locations, we of course arrive at interpretations of 
varying complexity. In what follows we focus only 
on the simplest such interpretation, which takes 
both to be points. 
Note that by assuming the preceding inter- 
pretation of trajectory-of-motion events, we may 
interpret the relation _ as the relation continuous- 
subset. Furthermore, we may also interpret pro- 
cesses as sets of events closed under the v- rela- 
tion; this then permits comp to be interpreted 
as element-of, and gr (for events) as mapping an 
event to the smallest process containing it. Before 
continuing, we may observe that this interpreta- 
tion does indeed satisfy Postulates (3) and (4). 
Application 
While the above interpretation of trajectory-of- 
motion events forces one to abstract away from 
*The spatial trace function r~ maps eventualities to 
their trajectories (cf. White 1993). 
5Much as in Moens and Steedman (1988) and Jack- 
endoff (1991), the introduction of gr is necessary to 
avoid having an ill-sorted formula. 
284 
the manner of motion supplied by a verb, it does 
nevertheless permit one to consider factors such as 
the normal speed as well as the meanings of the 
prepositions 10, lowards, etc. By making two ad- 
ditional restrictive assumptions, namely that these 
events be of constant velocity and in one dimen- 
sion, I have been able to construct and implement 
an algorithm which determines whether a speci- 
fied sequence of such events is or is not possible 
under certain situationally supplied constraints. 
These constraints include the locations of various 
landmarks (assumed to remain stationary) and the 
minimum, maximum, and normal rates associated 
with various manners of motion (e.g. running, jog- 
ging) for a given individual. 
The algorithm takes an input string and com- 
positionally derives a sequence of logical forms 
(one for each sentence) using a simple categorial 
grammar (most of which appears in White 1993). 
A special-purpose procedure is then used to in- 
stantiate the described sequence of events as a con- 
straint optimization problem; note that although 
this procedure is quite ad-hoc, the constraints are 
represented in a declarative, hierarchical fashion 
(cf. White 1993). If the constraint optimiza- 
tion problem has a solution, it is found using a 
slightly modified version of the constraint satis- 
faction procedure built into SCaEAMER, Siskind 
and McAllester's (1993) portable, efficient version 
of nondeterministic Common Lisp. 6 
As an example of an impossible description, 
let us consider the sequence of events described 
below: 
(8) Guy started jogging eastwards Mong the river. 
25 minutes later he reached {the cafe / the 
museum}. 
If we assume that the user specifies the cafe and 
the museum to be 5 and 10 km, respectively, from 
the implicit starting point, and that the rates spec- 
ified for Guy are those of a serious but not super- 
human athlete, then the algorithm will only find 
a solution for the first case (10 km in 25 minutes 
is too much to expect.) Now, by reasoning about 
subevents -- here, subsegments of lines in space- 
time -- the program exhibits the same behavior 
with the pair in (9): 
(9) Guy started jogging to the bar. 25 minutes 
later he reached {the cafe / the museum}. 
Since "Guy jogging to the cafe is accepted as a 
possible proper subevent of Guy jogging to the 
6The constraint optimization problem is split into 
two constraint satisfaction problems, namely find- 
ing the smallest consistent value of a cost variable 
and then finding consistent values for the rest of the 
variables. 
bar (assuming the bar is further east than the 
other landmarks), example (9) shows how the 
present approach successfully avoids the imperfec- 
tire paradox; since Guy jogging to the museum (in 
25 minutes) is not accepted as a possible subevent, 
example (9) likewise shows how the present ap- 
proach extends and refines those of Moens and 
Steedman and 3ackendoff vis-a-vis the subevent 
relation.7 
Future Work 
The algorithm as implemented functions only un- 
der a number of quite restrictive assumptions, and 
suffers from a rather ad-hoc use of the derived logi- 
cal forms. In future work I intend to extend the al- 
gorithm beyond the unidimensional and constant 
velocity cases considered so far, and to investigate 
incorporating the present treatment into the In- 
terpretation as Abduction approach advocated by 
Hobbs et. al. (1993). 

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