DELIMITEDNESS AND TRAJECTORY-OF-MOTION EVENTS * 
Michael White 
Department of Computer and Information Science 
University of Pennsylvania 
Philadelphia, PA, USA 
mwhit e@linc, cis. upenn, edu 
Abstract 
The first part of the paper develops a novel, 
sortally-based approach to the problem of 
aspectual composition. The account is ar- 
gued to be superior on both empirical and 
computational grounds to previous seman- 
tic approaches relying on referential homo- 
geneity tests. While the account is re- 
stricted to manner-of-motion verbs, it does 
cover their interaction with mass terms, 
amount phrases, locative PPs, and dis- 
tance, frequency, and temporal modifiers. 
The second part of the paper describes an 
implemented system based on the theoret- 
ical treatment which determines whether 
a specified sequence of events is or is not 
possible under varying situationally sup- 
plied constraints, given certain restrictive 
and simplifying assumptions. Briefly, the 
system extracts a set of constraint equa- 
tions from the derived logical forms and 
solves them according to a best-value met- 
ric. Three particular limitations of the sys- 
tem and possible ways of addressing them 
are discussed in the conclusion. 
1 Introduction 
Ever since Verkuyl (1972) first observed that the as- 
pectual class of a sentence depends not only on its 
main verb (as in Vendler, 1967) but also on its verbal 
*The author gratefully acknowledges the helpful com- 
ments of Jeff Siskind, Mark Steedman, Matthew Stone, 
and Christy Doran, as well as the support of DARPA 
N00014-90-J-1863, ARO DAAL03-89-C-0031, NSF IRI 
90-16592, Ben Franklin 91S.3078C-1. 
arguments and modifiers, numerous researchers have 
proposed accounts of this, the problem of ASPEC- 
TUAL COMPOSITION. Of course, the ultimate aims 
of these studies have never been to determine the 
aspectual class of an expression per se -- clearly a 
theory-internal notion -- but rather to predict the 
outcomes of certain aspect-related syntactic and se- 
mantic tests (cf. Dowty, 1979, Verkuyl, 1989). Like- 
wise, the present paper focuses on these empirical 
issues, in particular the compatibility of a given ex- 
pression with for- and in-adverbials and the result- 
ing existential and downward entailments. As an ex- 
ample of this temporal adverbial test, consider (1) 
below: 
(1) (a) John drank beer * in ten minutes. 
(*for} (b) 3ohn drank a pint of beer in ten 
minutes. 
In example (1) we may observe that the appropriate 
temporal adverbial is determined by the object of the 
verb drink -- at least as long as we exclude from con- 
sideration iterative, partitive, and other non-basic 
readings (cf. Moens and Steedman, 1988). 
Central to previous approaches to aspectual com- 
position have been attempts to explain the puzzling 
parallels between count noun phrases and telic sen- 
tences on the one hand, which have inherently "de- 
limited" extents, and mass nouns, bare plurals, and 
atelic sentences on the other, which do not. In con- 
nection with this intuitive notion of delimitedness, it 
has often been observed that mass terms (e.g. beer) 
and bare plurals (e.g. margaritas) are similar to 
atelic expressions (e.g. John drink beer / margari- 
tas), insofar as they share the property of REFER- 
ENTIAL HOMOGENEITY (reviewed below). This sets 
412 
them apart from count noun phrases (eg. a pint of 
beer) and teiic expressions (e.g. John drink a pint of 
beer), which do not generally do so. 
Observations such as these led Dowty (1979), Hin- 
richs (1985) and Krifka (1989, 1992) to incorporate 
various tests for referential homogeneity into their 
logical forms in order to account for the temporal 
adverbial variations. I argue against this move here 
by showing that it engenders a problem which I shall 
call THE ACCIDENTAL REFERENTIAL HOMOGENEITY 
PROBLEM (defined below). As an alternative, I de- 
velop in the first part of the paper a novel, sortally- 
based approach to aspectual composition. The ac- 
count is argued to be superior not only on empirical 
grounds, insofar as it dissolves this particular prob- 
lem, but also on computational grounds, insofar as it 
justifies employing a feature-based approach. While 
the account is restricted to manner-of-motion verbs 
(e.g. run), it does cover their interaction with mass 
terms, amount phrases, distance and locative modi- 
fiers, and temporal adverbials. 
In the second part of the paper, I describe an 
implemented system based on the theoretical treat- 
ment which determines whether a specified sequence 
of events is or is not possible under varying sit- 
uationally supplied constraints, given certain re- 
strictive and simplifying assumptions. These as- 
sumptions include requiring the sentences to spec- 
ify trajectory-of-motion events (e.g. Guy jogging 
from the inn to the bar) which are modeled as con- 
tinuous constant rate changes of location in one 
dimension. Briefly, the system extracts a set of 
constraint equations from the derived logical forms 
and solves them according to a best-value metric. 
The system is implemented in SCREAMER, Siskind 
and McAllester's (1993) portable, efficient version 
of nondeterministic Common Lisp augmented with 
a general-purpose constraint satisfaction package. 
Three particular limitations of the system and pos- 
sible ways of addressing them are discussed in the 
conclusion. 
2 The Accidental Referential 
Homogeneity Problem 
REFERENTIAL HOMOGENEITY is the conjunction of 
the properties of REFERENTIAL DIVISIVENESS and 
REFERENTIAL CUMULATIVITY. An expression refers 
divisively if whenever it applies to a given entity, it 
also applies to all subentities of that entity, down 
to a certain limit in size. For example, if there is a 
material entity to which beer applies, then beer also 
applies to all its (macroscopic) subparts; the same is 
clearly not true of a pint of beer. Cumulativity works 
in the other direction: an expression refers cumula- 
tively if whenever it applies to two entities, it also 
'applies to their collection. Here again, if we collect 
two entities to which beer applies then we get some- 
thing to which beer also applies; in contrast, if we 
collect two entities to which a pint of beer applies, 
we get an entity to which two pints of beer applies 
instead. Similarly, we may observe that the atelic 
expression John drink beer refers homogeneously to 
situational entities (eventualities), unlike the telic ex- 
pression John drink a pint of beer. 
With these properties in mind, THE ACCIDEN- 
TAL REFERENTIAL HOMOGENEITY PROBLEM may be 
stated as follows: some expressions which on intu- 
itive and syntactic grounds should be in the hetero- 
geneous class "happen" to refer homogeneously (cf. 
Schubert and Pelletier 1989). This problem has been 
noted in passing by Mittwoch (1982), Moens (1987), 
and Krifka (1989), but to my knowledge has not been 
systematically addressed by those focusing on the se- 
mantics of aspect. The easiest examples to construct 
involve lexical or quantificational vagueness, though 
more insidious examples exist involving self-similar 
objects. For instance, consider Mittwoch's example 
below: 
(2) John wrote something in ten minutes which it 
took me half an hour to translate. 
The problem here is that the expression John write 
something refers homogeneously, but nevertheless 
combines with an in-adverbial -- if there is an event 
of John writing something, then all the subevents 
of that event (down to a certain limit in size) will 
also be events of John writing something (Mbeit not 
the same thing), and thus the expression refers divi- 
sively; similar considerations show that it refers cu- 
mulatively as well. To take another example, con- 
sider the following sentence: 
(3) John typed a sequence of characters in thirty 
seconds (which it took me two minutes to write 
by hand). 
In (3) the problem is that subsequences of charac- 
ters are also sequences of characters (albeit different 
ones), and thus the expression John type a sequence 
of characters happens to refer homogeneously too. 
Since the indicated expressions in (2) and (3) turn 
out referentially homogeneous rather than heteroge- 
neous, their compatibility with in-adverbials (and 
not for-adverbials) is problematic for the theories of 
Dowty, Hinrichs and Krifka. 1 Now, as an alternative 
to the present approach, one might want to consider 
basing an account of this problem on differing scope 
possibilities for the expressions which "accidentally" 
and "truly" refer homogeneously -- that is, to some- 
how allow for different subquantities of beer but not 
different subsequences of characters. A serious prob- 
lem for any such approach, however, is the existence 
of readings where the temporal adverbial has wide 
scope, as in (4): 
1Showing this in detail is beyond the scope of the 
present paper. For a more detailed exposition of this 
problem as it relates to Ktifl~'s theory, see White (1993). 
413 
(4) Amazingly, John replied to every new email 
message in under two hours. 
The availability of such wide scope readings does 
not seem compatible with the idea of requiring the 
quantified phrase to outscope the temporal adver- 
bial, which would seem to be necessary in order to 
(always) correctly predict the appropriate temporal 
adverbial by means of a referential homogeneity test. 
Beyond the empirical problems engendered by ref- 
erential homogeneity tests, there appear to be sig- 
nificant computational ones as well. From the gen- 
eration standpoint, it seems quite unreasonable to 
test whether any or all subevents of an event to be 
described happen to meet the same description be- 
fore choosing a temporal adverbial to convey dura- 
tion. Likewise, from the standpoint of interpreta- 
tion, if one is to make use of aspectual information 
in processing successive sentences in discourse (as in 
the theories of Hinrichs, 1986, Moens and Steedman, 
1988, and Lascarides and Asher, 1991, for example), 
there is equally little time for performing such tests. 2 
3 Theory 
3.1 Ontology 
Various authors (including Link, 1983, Bach, 1986, 
Krifka, 1989, Eberle, 1990) have proposed model- 
theoretic treatments in which a parallel ontological 
distinction is made between substances and things, 
processes and events, etc. A similarly parallel dis- 
tinction is employed here, but in a rather different 
way: unlike the above treatments, the present ac- 
count models substances, processes, and other such 
entities as abstract kinds whose realizations vary in 
amount. As such, the approach developed here may 
be seen as building upon the work of Carlson (1977) 
and his successors; it also represents one way to fur- 
ther formalize the intuitions found in Moens and 
Steedman (1988) and Jackendoff (1991). 
Following Schubert and Pelletier (1987), the 
present account distinguishes individuals from kinds, 
but not from stages or quantities. Extending their 
ontology, the same distinction is assumed to hold 
not only in the domain of materials but also in the 
domain of eventualities, and derivatively in the do- 
mains of space and time as well. This extension sets 
the stage for taking a sortal approach to the seman- 
tics of aspect, in contrast to previous model-theoretic 
accounts. 
3.2 Semantics 
Let us assume a many-sorted higher-order logic with 
model structures consisting of the following elements, 
2A similar point was suggested by Manfred Krifka (p.c.). 
Entity 
• Material 
- Substance 
- Thing 
• Eventuality 
- Process 
- Event 
• Space 
- Trajectory 
• Time 
• Amount 
- Quantity 
- Distance 
- Duration 
• Number 
Figure 1: The (Abbreviated) Sort Hierarchy 
plus an interpretation function: 
• a set of entities: E 
• sorts: Material, Eventuality, Kind, ... 
• binary relations: p, comp, E_, r, amt, ... 
To structure the set of entities E, we require permis- 
sible models to satisfy various axioms on the binary 
relations. 
Roughly following Eberle (1990) and Jackend- 
off (1991), we assume postulates enforcing the (non- 
exhaustive) sort hierarchy shown in Figure 1. We 
also assume that certain sorts cut across the hier- 
archy, in particular the disjoint sorts Kind and In- 
dividual. These sorts partition the sorts Material, 
Eventuality, Space and Time. Some of the resultant 
sorts are named in Figure 1; these equivalences are 
shown below: 
• Kinds 
Substance = Kind f3 Material 
Process = Kind fl Non-State 
• Individuals 
Thing = Individual fl Material 
Event = Individual fl Non-State 
Following Schubert and Pelletier, we map pred- 
icates to kinds using the operator p. To map 
kinds to their realizations, we employ a relation 
comp(osed of) inspired by Jackendoff's (1991) con- 
ceptual function of the same name. As this relation 
is central to the present account, its sortal require- 
ments are shown below: 
414 
(5) Vxy. comp(z)(y) --* Kind(x) A Individual(y) 
(6) For all S in {Material, Eventuality,...}: 
Vxy. comp(z)(y) -* S(z) A S(y) 
In the spirit of Krifka (1989) and Eberle (1990), 
we also employ a partial order ff (part of) on the 
sort Individual, as well as total orderings ~ and < 
on the sorts Amount and Number, respectively. F\]-- 
nally, we employ spatio-temporal trace functions r 
mapping from Eventuality to Space and to Time, as 
well as a function am(oun)t mapping from Individual 
to Amount. 
We relate the preceding binary relations as follows. 
First, formal kinds and their realizations are required 
to satisfy the following axiom: 3 
(7) VPz. comp(p(P))(z) ~ P(z) 
Second, we require the spatio-temporal trace func- 
tions r to be homomorphisms preserving the part-of 
relation, as shown below: 
(8) Vele2 • el_e2 ~ v(el)__.v(e2) 
Third, we also require the spatio-temporal trace 
functions to preserve the composed-of relation, at 
least when they map processes to kind-level entities, 
as shown in (9); in the case of the temporal trace 
function rt, this requirement is strengthened to hold 
generally, as shown in (10): 
(9) Veal. comp(e)(el) ^ Kind(r(e)) 
--* comp(r(e))(r(el )) 
(10) Veel. comp(e)(ea) --* comp(rt(e))(rt(el)) 
Fourth, as a correlate of referential divisiveness, we 
assume that the set of individuals composed of a 
given kind is closed under the part-of relation; that 
is, whenever an individual y= is composed of a certain 
kind z, then all subparts Yl of y~ are also composed 
of z, as shown in (11). 4 
(11) Vxyly2. comp(z)(y2) A ylff_y2 -* ¢omp(z)(yl) 
Finally, we require the function amt and unit mea- 
sures such as minutes' to satisfy various fairly ob- 
vious postulates concerning the preservation of the 
orderings __, _ and _<. 
3.3 Syntax 
The rudimentary categorial grammar given in Fig- 
ure 2 suffices to derive all of the logical forms in the 
next section. Note that lexemes such as slime are 
paired with syntactic categories such as n and se- 
mantic functions such as slime ~ (where the category 
vp abbreviates s \ np). Three e-rules are also em- 
ployed, one for introducing p in a bare np, one for 
SNote that not all kinds need involve #; presumably, 
conventional kinds such as Coke or Heineken are named 
directly. 
4Because of the notorious MINIMAL PARTS PROBLEM 
(i.e., how little beer is still beer?), this postulate is not 
quite correct as stated; amending it would require adding 
a condition that yl be "large enough ~ for the kind z. 
lifting a vp to apply to a generalized quantifier, 5 and 
one for adding an existential quantifier to the sen- 
tence radical (ignoring tense and mood). 
3.4 Aspeetual Composition 
Manner-of-motion verbs such as run, wa&, etc. are 
interesting insofar as the telicity of the expressions 
in which they are used is dependent upon both the 
subject NP and an optional trajectory-specifying PP: 
(12) John ran along the river for 20 minutes. 
(13) John ran to the bridge in 20 minutes. 
(14) Slime oozed into the urn for 20 minutes. 
(15) Two liters of slime oozed into the urn in 20 
minutes. 
Let us assume that such verbs take material entities 
as arguments and describe eventualites (either events 
or processes). To capture their aspectual behavior, 
we stipulate the following preliminary postulate: 
For all A in {run', ooze',...} : 
(16) we. \[Individual(e) 
Individual(rs(e)) A Individual(z)\] 
Meaning postulate (16) states that if A(z) holds of 
an eventuality e, where A ranges over run e, ooze ~, 
etc., then e is an event (an individual eventuality) if 
and only if its spatial trace rs(e) is an individual tra- 
jectory and x is a thing (i.e., an individual material). 
If we assume that the expression to the bridge only 
describes individual trajectories, then postulate (16) 
forces John run to the bridge to describe an event. 
In contrast, if we assume that the expression along 
the river is not restricted in this way, then John run 
along the river may describe a process as well. To 
capture this formally, the following meaning postu- 
late is needed: 
(17) Vzp. to'(x)(p) --* Individual(p) 
Given the categories listed in Figure 2, the expres- 
sions John run along the river and John run to the 
bridge receive the following translations: 
(18) Ae. run'(j)(e ) A along'(the'(river'))(rs(e)) 
(19) Ae. run'(j)(e) A to'(the'(bridge'))(rs(e)) 
From meaning postulates (16) and (17), it follows 
that the latter expression must describe events; with 
no analogous meaning postulate for along, the former 
expression is free to describe processes as well. 
Before continuing, it is worth explaining why pos- 
tulate (17) is a reasonable one. Recall that a given 
process stands in the composed-of relation to mul- 
tiple events. If these events differ in their spatial 
extents, then the spatial trace of the process can- 
not sensibly be an individual-level entity, assuming 
SThis rule is a simplified version of a more general 
rule which introduces an existential quantifier over the 
eventuality variable. 
415 
John 
ten 
liters 
of 
slime 
the 
run 
e 
miles 
to 
for 
in 
minutes 
e 
:= np 
:= num 
:= gq \[ pp-of\ num 
:= pp-of/np 
:--~ n 
:---- np/ n 
:= np / n 
:-- s\np 
:= s \gq/vp 
:-- vp\vp\num 
:= vp\vp/tm 
:= vp\vp/tm 
:= vp\vp/tm 
:= tm \num 
:-" U/8 
: j 
: 10 
: ~nmP. B~. comp(m)(x) ^ amt(~) = liters'(n) ^ P(~) 
: Az. z 
: slime' 
: g 
: the' 
: run' 
: APQe.Q(Az.P(:r)Ce)) 
: AnPxe. P(z)(e) ^ amt(rs(e)) = miles'(n) 
: AyPze. P(z)(e) ^ to'(y)(rs(e)) 
: AdPxel. Be. P(z)(e) h comp(e)(el) ^ amt(rt(el)) = d 
: ,~dPze. P(z)(e) A amt(rt(e)) _ d 
: minutes' 
: AP.Be.P(e) 
Figure 2: Rudimentary Syntax 
unique amounts (distances) for individual trajecto- 
ries; instead, it should be a kind-level trajectory, 
standing in the composed-of relation to the various 
individual trajectories corresponding to these multi- 
ple events -- as per postulate (9). It is in this sense 
that the spatial trace of a process may not be "delim- 
ited" in extent. Of course, this does not mean that 
the spatial trace of a process cannot be bounded in 
any absolute sense; in the case of along the river, for 
example, no resultant trajectory is allowed to con- 
tinue (very far) past the river's end. l~eturning now 
to to the river, we may note that this expression de- 
scribes the end point of a trajectory; as such it is 
naturally restricted to describing individual trajec- 
tories, which always have defined endpoints. 
Next we turn to slime and two liters of slime. 
Given the categories listed in Figure 2, the expres- 
sions Slime ooze into the urn and Two liters of slime 
ooze into the urn receive the following translations: 
(20)),e. ooze'(u(slime'))(e) ^ 
into'(the'(urn'))(rs(e)) 
,Xe. Bz. comp(g(slime'))(z) ^ 
(21) amt(x) = liters'(2) ^ ooze'(z)(e) ^ 
into'(the'(urn'))(n(e)) 
Now, if we assume a sortal meaning postulate for 
into analogous to that of to, then it follows from the 
sortal requirements on p and comp that (20) can only 
describe processes, whereas (21) can only describe 
events. 
At this point we are ready to consider the temporal 
adverbials. Not surprisingly, the relation comp is 
crucial to the present account of the for- vs. in- 
adverbial test data, as can be seen from comparing 
their semantics: whereas for-adverbials measure out 
a process using comp and a given amount of time, 
in-adverbials simply require that an event take place 
within a given amount of time. 
Let us first consider how the machinery developed 
so far can be used to account for examples (14) and 
(15), augmented below: 
(22) Slime oozed into the urn {for} 
* in twenty minutes. 
(23) Two liters of slime oozed into the urn 
{*for t in twenty minutes. 
The respective translations of the two possibilities in 
(23) follow: 
3zeel. comp(p(slime'))(z) A 
amt(x) = liters'(2) ^ ooze'(x)(e) ^ 
(24) into'(the'(urn'))(rs(e)) ^ 
comp(e)(el) ^ amt(rt(el)) = minutes'(20) 
Bze. comp(p(slime'))(z) ^ 
amt(z) = liters'(2) ^ ooze'(x)(e) ^ 
(25) into'(the'(urn'))(rs(e)) ^ 
amt(r,(e)) -< minutes'(20) 
Since the entity e in (24) is required to be an event, 
comp(e)(el) turns out undefined, s making (24) se- 
mantically anomalous. In contrast, lacking comp, 
the translation in (25) is unproblematic. Simi- 
lar reasoning shows that (22) can only be compat- 
ible with for-adverbials, assuming durations (i.e., 
amounts of temporal traces) are not defined for pro- 
cesses. Furthermore, these same considerations lead 
to the correct predictions in examples (12) and (13) 
as well. T Finally, without further ado the theory 
makes the correct predictions in (26) below, as dis- 
tances (amounts of spatial traces) are only defined 
for events: 
sI-Iere I am assuming for expository purposes that the 
interpretation of a function is undefined if any of its ar- 
gument terms are not of the appropriate sort, or are un- 
denned themselves. 
ZNote, however, that the theory as it stands cannot 
rule out ? John ran along the river in £0 minutes, which 
comes out meaning the same thing as John ran some 
distance along the river in ~0 minutes. 
416 
* for ~ twenty (26) John ran four miles in j " 
minutes. 
Up until this point we have relied (in part) on the 
stipulated postulate (16) to capture the temporal ad- 
verbial data. We consider now how we may derive 
this postulate from more basic assumptions, begin- 
ning with the following one: 
For all A in {run', ooze' .... } : 
Wee~. A(x)(e) ^ comp(e)(e~) --. (27) \[3~. A(=l)(el) ^ comp(r.(e))(~.(ed)\] 
V \[:Ix1. A(xl)(el) A comp(z)(xl)\] 
Postulate (27) is meant to capture in a novel way 
the intuition that a A process e must be "measured 
out" either by its trajectory Ts(e) or by its material 
argument x (cf. Krifka, 1989, Dowty, 1991, Tenny, 
1992, Verkuyl and Zwarts, 1992). It does so by re- 
quiring that all individual events el composed of e be 
A events with either an individual trajectory %(el) 
composed of %(e) or an individual material argument 
x~ composed of x (or possibly both). From (27) fol- 
lows the only-if (~--) part of (16), as follows: if both 
x and rs(e) are individual-level entities, then neither 
of the alternatives in the. consequent of (27) can be 
true, since the composed-of relation is not defined for 
individual-level entities; therefore, by way of contra- 
diction, e cannot be a process (at least if we assume 
all kind-level entities are in the domain of comp). 
To make the if(--+) part of (16) follow too, we may 
employ the following postulate: 
For all A in {run', ooze',...} : 
(28) We. A(x)(e) ^ Individual(e) R(amt(rt(e) ) )(amtO'Je) ) )(amt(x) ) 
Postulate (28) relates the duration of a A event 
to the length of its trajectory and the quantity of 
its material argument by some unspecified relation 
R (which might limit speeds to acceptable ranges, 
for example). Since amounts are only defined for 
individual-level entities, this forces the trajectory 
and material argument of a A event to be individual- 
level as well. 
3.5 Referential Homogeneity Revisited 
While the property of referential homogeneity does 
not play a part in capturing the for- vs. in-adverbial 
test data in the present approach, it is nevertheless 
necessary to account for certain desired inferences. 
In particular, we shall need a version of referential 
divisiveness in order to make the first but not the 
second inference below a valid one: 
(29) John ran along the river for five minutes. 
John ran along the river for four minutes. 
(30) -, John ran to the bridge in five minutes. 
John ran to the bridge in four minutes. 
Given the translation of John ran to the bridge in n 
minutes in (31) below, it is easy enough to see why 
(30) is not a valid inference: all that is needed is a 
model in which there is an event of John running 
to the bridge that takes more than four minutes but 
takes place within five minutes. 
(31) 3e. run'(j)(e) A to'(the'(bridge'))(~',(e)) ^ 
amt(rt(e)) _ minuteg(n) 
Turning now to (29), consider the translations below: 
(32) 3ee2. run'(j)(e) A along'(the'(river'))(r.(e)) 
A comp(e)(e2) A amt(rt(e)) = minutes'(5) 
(33) 3eel . run'(j)(e) A along'(the'(river'))(rs(e)) 
A comp(e)(et) A amt(l"t(e)) = minutes'(4) 
Note here that the variables have been (equivalently) 
renamed to indicate which we shall take to be the 
same and which different: that is, we shall take e2 
and el to be two events of different durations com- 
posed of the same process e. To get (29) to follow in 
this way, we need the following two postulates: 
For all A in {run', ooze',...} : 
(34) Vze2dl . A(z)(e2) A dl _ amt(rt(e2)) ---, 
3el . elEe2 A amt(rt(el)) = dl 
For all r in {along', to',...} : 
(35) Vze. r(x)(rs(e)) A comp(e)(el) ---, r(x)(r,(el)) 
Postulate (34) states that if a A event e2 has du- 
ration amt(rt(e2)), then for all lesser durations dl, 
e2 has subevents el of that duration; postulate (35) 
states that r trajectory predicates are preserved by 
the composed-of relation. From postulate (34) it fol- 
lows that the running event e2 of duration five min- 
utes must have a subevent el of duration four min- 
utes, which we know by (11) to be composed of the 
same process e; finally, postulate (35) ensures that el 
is also located along the river, thus validating (29). 
In addition to accounting for the downward en- 
tailments above, the machinery developed so far also 
accounts for existential entailments such as the one 
in (36), assuming the translation of the consequent 
given in (37): 
Slime oozed into the urn for ten minutes. 
(36) Some amount of slime oozed into the urn 
in ten minutes. 
3zme. comp(/~(slime'))(z) A Amount(m) A 
(37) amt(x) = m ^ ooze'(x)(e) ^ into'(the'(urn'))(r.(e)) ^ 
amt(rt(e)) -< minutes'(10) 
The inference (36) follows by postulates (27) and 
(35). Since Some amount of slime ooze into the 
urn turns out to be referentially homogeneous, (36) 
concomitantly shows how the present approach dis- 
solves THE ACCIDENTAL REFERENTIAL HOMOGENE- 
ITY PROBLEM. 
417 
3.6 Repetitions 
So far we have been careful to exclude from consid- 
eration the iterative readings that for-adverbials can 
induce (cf. Moens and Steedman, 1988, Jackend- 
off, 1991). Here we consider some extensions to the 
approach developed above which permit these to be 
captured as well. 
Let us begin by adding retried sets to the do- 
main of individuals, along the lines of Link (1983) 
or Krifka (1989). We do so by partitioning the sort 
Individual using disjoint sorts Atom and Non-Atom 
and introducing a new relation __.i (individual part 
of) isomorphic to the subset relation over the power 
set of the atoms, minus the empty set (to avoid con- 
fusion, we might rename the other part of relation 
E_q, for quantity part of). We also add a cardinality 
function \[ • \] mapping individuals to numbers, and 
an operator plur(al) mapping predicates over atoms 
to predicates over non-atoms. Naturally enough, we 
require the operator plur to satisfy the following pos- 
tulate, where __.~i is equal to ___i with its domain re- 
stricted to the atoms: 
(38) VPzlz2. plur(P)(z2) A zl__.aiz2 -"* P(Zl) 
Given this additional machinery, we may account 
for the iterative readings induced by for-adverbials 
by simply positing a lexical ambiguity between the 
reading for for given in Figure 2 and the one below: 
(39) for: ~dPxel. 3e. t~(plur(P(z))) = e ^ 
comp(e)(el) ^ amt(rt(e)) = d 
Note that in reading (39), the process e measured out 
by the for-adverbial is not the one described by P(z), 
but rather the one equal to/~(plur(P(z))), which has 
as its realizations collections of P(z) events of vary- 
ing cardinalities; note also that the sortal require- 
ments on plur and comp ensure that the two readings 
off or-adverbials are in complementary distribution, 
insofar as only one can ever be defined for a given 
eventuality predicate p.8 
Finally, we may observe that these same extensions 
can be used to give a natural account of frequency 
adverbials such as twice or n times: 
(40) twice: APze. plur(P(x))(e) ^ l e 1= 2 
4 Application 
In this section we turn to an implemented system 
based on the above theoretical treatment which de- 
termines whether a specified sequence of events is or 
is not possible under varying situationally supplied 
constraints. The domain is limited to trajectory- 
of-motion events specified by the verbs run, jog, 
sit is worth noting that as an alternative to posit- 
ing a lexical ambiguity, one could just as easily invoke 
a coercion operator on an event predicate P(z) map- 
ping it to the process predicate he. #(plur(P(x))) = e, 
which would bring the present treatment more in line 
with Moens and Steedman (1988) and Jackendoff (1991). 
plod, and walk; the locative prepositions to, towards, 
from, away from, along, eastwards, westwards, and 
to and back; various landmarks; the distance adver- 
bials n miles; the frequency adverbials twice and n 
times; and finally the temporal adverbials for and 
in. Trajectory-of-motion events are modeled as con- 
tinuous constant rate changes of location in one di- 
mension of the TRAJECTOR relative to one or more 
LANDMARKS (following Regier 1992 in his use of Lan- 
gacker's 1987 terminology). 
Briefly, the system takes a set of landmark loca- 
tions (which are assumed to remain constant) and an 
input string from which it derives all possible logical 
forms for the given sentences; it then extracts a set of 
constraint equations from the derived logical forms 
and solves them according to a best-value metric. If 
a solution is found, it is displayed as a space-time 
diagram as shown in Figure 3. Note that distances 
are in miles, durations are in minutes, and the range 
of rates associated with the verbs are appropriate for 
a serious athlete. 
The best-value metric currently employed is prox- 
imity to the median rate for the given manner of 
motion, summed across successive events. Accord- 
ing to this metric, an event such as Guy running to 
the bar takes a default amount of time according to 
the distance and the median rate; however, an event 
of Guy running to the bar in n minutes may take 
less time if this duration is less than the default -- 
at least up to the point where the specified duration 
requires exceeding the given maximum running rate, 
thus making the constraint equations unsatisfiable. 
Likewise, an event of Guy running along the river 
(towards the bar, say) for n minutes will yield a de- 
fault distance according to the amount of time and 
the median rate; this distance may vary according to 
more demanding distance requirements imposed by 
succeeding sentences, again up to a certain point. 
The times of successive repetitive events are 
summed, so that scope differences between frequency 
and temporal adverbials may be adequately treated; 
that is, the system correctly determines when one 
but not the other of Guy jogged to the care and back 
in ten minutes twice and Guy ... twice in ten minutes 
is possible. The summing of the durations of succes- 
sive events also allows the system to determine an 
appropriate number of iterations for Guy jogged to 
the cafe and back for 30 minutes. 9 
The system is implemented in SCREAMER, Siskind 
and McAllester's (1993) portable, efficient version 
of nondeterministic Common Lisp augmented with 
a general-purpose constraint satisfaction package. 
Taking advantage of SCREAMER'S compatibility with 
the COMMON LISP OBJECT SYSTEM, constraints are 
specified in a declarative, hierarchical fashion. As 
an example, Figure 4 shows how variables associ- 
9Note that the system cannot find a solution for Guy 
ran to the bar \]or 30 minutes, since there is no provision 
for adding unspecified events (such as leaving the bar). 
418 
Guy's Journey 
Time 
120.00 - 
110.00 - 
100.00 - 
90.00 - 
80.00 - 
70.00 - 
60.00- 
50.00 - 
40.00- 
30.00 - 
20.00 - 
10.00 t ~ 
0.00 
I 
0.00 
I I 
5.00 10.00 
Guy 
- mouth 
bridge 
m 
care 
- museum 
bar 
inn 
-dam 
Location 
Figure 3: Program output for the following input string: "Guy walked eastwards along the river for 40 
minutes. Then he jogged from the cafe to the museum. Next he ran to the bar and back three times in 20 
minutes. Finally he plodded to the inn." Note that for 20 minutes could have been used instead of three 
times in 20 minutes. 
419 
(defclaes trajectory-event () 
;;; etc ... 
(del~aited :initformnil) 
;;; etc ... 
(defmethod initialize-instance :after 
((e trajectory-event) treat inits) 
;;; etc ... 
(assert! (=v dt (-v tl tO))) 
(assert! (=v d (*v r dr))))) 
(defclaeerun-event (trajectory-event) ()) 
(defmethod initialize-instance :after 
((e run-event) ~reet lairs) 
(declare (ignore inits)) 
(let ((r (slot-value • 'rate))) 
(assert! (<=v r (/ I 4.5))) 
(assert! (>=v r (/ i 6.5))))) 
Figure 4: Declarative, hierarchical constraint speci- 
fication in SCREAMER. 
ated with the trajectory-of-motion class of events 
are constrained according to the formula distance = 
rate x time; it also shows how a further constraint 
on rates is associated with the running specialization 
of this class. 
Because the domain is so simple, adequate con- 
straints on trajectories are trivial to specify. Some- 
what more imaginatively, processes are modeled by 
their constrained but unsolved-for realizations; they 
are distinguished from them solely (and efficiently!) 
by the value of the feature delimited, as justified 
by the sortal approach advocated in the last section. 
Likewise, kind- and individual-level trajectories are 
distinguished by the same feature, in such a way as 
to maintain postulate (16). Lest the reader miss the 
point for its simplicity, it is worth emphasizing (re- 
calling Figure 3) that this feature is crucial for de- 
termining whether single instances or repetitions are 
involved in sentences such as Guy walked eastwards 
along the river/or ~0 minutes and Guy ran to the 
bridge and back for ~0 minutes. 
5 Conclusion 
In this paper I have presented a novel, sortally-based 
approach to the problem of aspectual composition 
which I have argued to be superior on both em- 
pirical and computational grounds to previous ap- 
proaches relying on referential homogeneity tests. I 
have also described an implemented system based on 
the theoretical treatment which determines whether 
a specified sequence of trajectory-of-motion events is 
or is not possible under varying situationally speci- 
fied constraints. 
Beyond its obvious shortcomings, there are three 
specific limitations to the system worth mentioning. 
First, the range of discourses is limited to narrative 
sequences, which greatly simplifies the necessary rea- 
soning (el. Hwang and Schubert, 1991, Lascarides 
and Asher, 1991, Hobbs et. el. 1993). Second, the 
present approach does not lend itself well to flexibly 
accommodating new information. Third, in the case 
where a specified sequence of events turns out not 
to be possible, the constraint satisfaction approach 
does not provide any mechanism for explaining why 
this happens to be so. In order to address these prob- 
lems, in future work I intend to investigate to what 
extent the present approach can be meshed with the 
Interpretation as Abduction approach advocated by 
Hobbs et. al., which appears to be well suited to 
these issues. 
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