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, M atthew 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 treatm ent 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 treatm ent 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 ential homogeneity espoused by Krifka (1992) and his predecessors. 2 8 3
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8/3/2019 Imperfective Paradox White
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T H E I M P E R F E C T I V E P A R A D O X A N D
T R A J E C T O R Y - O F - M O T I O N E V E N T S *
M i c h a e l W h i t e
D e p a r t m e n t o f C o m p u t e r a n d I n f o r m a t i o n S c i e n c e
U n i v e r s i t y o f P e n n s y l v a n i a
P h i l a d e l p h i a , P A , U S A
m w h i t e © l i n c . c i s. u p e n n , e d u
A b s t r a c t
I n t h e f ir s t p a r t o f t h e p a p e r , I p r e s e n t a
n ew t r e a t m en t o f THE IM P ERF I~CTIVE P ARADOX
( D o w t y 1 9 79 ) fo r t h e r e s t r i c t e d c a s e o f t r a j e c t o r y -
o f - m o t i o n e v e n t s. T h i s t r e a t m e n t e x t e n d s a n d r e -
f in e s t h o s e o f M o e n s a n d S t e e d m a n ( 1 98 8 ) a n d
J a c k e n d o f f ( 1 9 9 1 ) . I n t h e s e c o n d p a r t , I d e s c r i b e
a n i m p l e m e n t e d a l g o r i t h m b a s e d o n t hi s t r e a t m e n t
w h i c h d e t e r m i n e s w h e t h e r a s p e c if i ed s e q u en c e o f
s u c h e v e n t s i s o r i s n o t p o s s i b l e u n d e r c e r t a i n s i t -
u a t i o n a l l y s u p p l i e d c o n s t r a i n t s a n d r e s t r i c t i v e a s-
s u m p t i o n s .
I n t r o d u c t i o n
B a c h ( 1 9 8 6 : 1 2 ) s u m m a r i z e s T H E I M P E R F E C T I V E
P AR AD O X ( D o w t y 1 9 7 9 ) a s f o l lo w s : " . . . h o w c a n
w e c h a r a c t e r i z e t h e m e a n i n g o f a p r o g r e s s i v e s e n -
t e n c e l i k e ( l a ) [ 17 ] o n t h e b a s i s o f t h e m e a n i n g o f
a s i m p l e s e n t e n c e l i k e ( l b ) [ 1 8 ] w h e n ( l a ) c a n b e
t r u e o f a h i s t o r y w i t h o u t ( l b ) e v e r b e i n g t ru e ? "
( l a ) J o h n w a s c r os s i ng t h e s tr e e t .
( l b ) J o h n c r o ss e d t h e s t r e e t.
C i t i n g p a r a l le l s i n t h e n o m i n a l d o m a i n , B a c h g o e s
o n t o p o i n t o u t t h a t t h i s p u z z le i s s e e m i n g l y m u c h
m o r e g e n e r a l , i n s o f a r a s i t a p p e a r s w h e n e v e r a n y
s o r t o f p a r t i t i v e i s e m p l o y e d . I n s u p p o r t o f t h is
v i e w , w e m a y o b s e r v e t h a t t h e s tar t v - ing c o n -
s t r u c t i o n e x h i b i t s t h e s a m e b e h a v i o r :
( 2 a ) J o h n s t a r t e d j o g g i n g t o t h e m u s e u m .
( 2 b ) J o h n j o g g e d to t h e m u s e u m .
H e r e w e s e e t h a t ( 2 a ) d o e s n o t e n t a i l ( 2 b) - - w h i l e( 2 b ) a s s e r t s t h e o c c u r r e n c e o f a n e n t i r e e v e n t o f
J o h n j o g g i n g t o t h e m u s e u m , ( 2 a ) o n l y as s e rt s t h e
*The au thor g ratefu l ly acknowledges the helpfu lcom me nts o f Ma rk Steedm an, Jef f S isk ind , Chr is tyDo ran , M at th ew S to n e , an d t h e an o n y mo u s re f e r -ees, as wel l as the s upp or t o f DARP A N00014-90-J-1863, AI ~O DAA L03-89-C-0031, NS F IRI 90-16592,Ben Franklin 91S.3078C-1.
o c c u r r e n c e o f t h e b e g i n n i n g o f s u c h a n e v e n t , l e a v -
i n g o p e n t h e e x i s t e n t i a l s t a t u s o f it s c o m p l e t i o n .
C a p i t a l i z i n g o n B a c h ' s i n s i g h t , I p r e s e n t i n
t h e fi rs t p a r t o f t h e p a p e r a n e w t r e a t m e n t o f
t h e i m p e r f e c t i v e p a r a d o x w h i c h r e li e s o n t h e p o s -
s i b i l it y o f h a v i n g a c t u a l e v e n t s s t a n d i n g i n t h e
p a r t - o f r e l a t io n t o h y p o t h e t i c a l s u pe r - e v e n t s . T h i s
t r e a t m e n t e x t e n d s a n d r e f in e s th o s e o f M o e n sa n d S t e e d m a n ( 1 9 8 8 ) a n d J a c k e n d o f f ( 19 9 1 ), a t
l e a s t fo r th e r e s t r i c t e d c a s e o f t r a j e c t o r y - o f - m o t i o n
e v e n ts . 1 I n p a r t i c u l a r , t h e p r e s e n t t r e a t m e n t c o r -
r e c t l y a c c o u n t s n o t o n l y f o r w h a t ( 2 a ) f a i l s t o e n -
t a i l - - n a m e l y , t h a t J o h n e v e n t u a l l y r e a c h e s th e
m u s e u m - - b u t a l so fo r w h a t ( 2 a ) d o e s i n f a c t e n -
t a i l - - n a m e l y , t h a t J o h n f o ll o w s ( b y j o g g i n g ) a t
l e a s t a n i n i t i a l p a r t o f a p a t h t h a t l e a d s t o t h e
m u s e u m . I n t h e s e c o n d p a r t o f t h e p a p e r , I b ri e f ly
d e s c r i b e a n i m p l e m e n t e d a l g o r i t h m b a s e d o n t h i s
t h e o r e t i c a l t r e a t m e n t w h i c h d e t e r m i n e s w h e t h e r a
s p e c if i ed s e q u e n c e o f t r a j e c t o r y - o f - m o t i o n i s o r i s
n o t p o s s i b l e u n d e r c e r t a i n s i t u a t i o n a l l y s u p p l i e d
c o n s t r a i n t s a n d r e s t r i c t i v e a s s u m p t i o n s .
T h e o r y
T h e p r e s e n t t r e a t m e n t b u i l d s u p o n t h e a p -
p r o a c h t o a s p e c t u a l c o m p o s i t i o n d e v e l o p e d i n
W h i t e ( 1 9 9 3 ), a b r i e f s k e t c h o f w h i c h f o l lo w s .
W h i t e ( 1 99 3 ) a r g u e s t h a t s u b s t a n c e s , p r o c e ss e s
a n d o t h e r s u c h e n t i t i e s s h o u l d b e m o d e l e d a s a b -
s t r a c t k i n d s w h o s e r e a l i z a t i o n s ( t h i n g s , e v e n t s ,e t c . ) v a r y in a m o u n t . 2 T h i s i s a c c o m p l i s h e d f o r -
m a l l y t h r o u g h t h e u s e o f a n o r d e r - s o r t e d l o g ic
w i t h a n a x i o m a t i z e d c o l l e c t io n o f b i n a r y r e l a ti o n s .T h e i n t e n d e d s o r t h i e r a r c h y i s m u c h l i k e t h o s e
o f E b e r l e ( 1 9 9 0 ) a n d J a c k e n d o f f ( 1 99 1 ) ; i n p a r -t i c u l a r , b o t h s u b s t a n c e s a n d t h i n g s a r e t a k e n t o
b e s u b s o r t s o f th e m a t e r i a l e n t i t i e s, a n d s i m i l a r l y
1These are elsewhere cal led 'd i rected-mot ion 'even ts .
2This move is in tended to reso lve cer tain emp ir icalan d co m p u ta t i o n a l p ro b l ems w i th t h e v i ew o f r e f e r -en t ial homogenei ty espoused by Kri fka (1992) and h ispredecessors .
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b o t h p r o c e s s e s a n d e v e n t s a r e t a k e n t o b e s u b -
s o r t s o f t h e n o n - s t a t i v e e v e n t u a l i ti e s . W h a t i s n e w
i s t h e a x i o m a t i z a t i o n o f J a c k e n d o f f ' s c o m p o s e d - o f
r e l a t i o n ( c o m p ) - - w h i c h e f fe c ts t h e a f or e m e n -
t i o n e d k i n d - t o - r e a l iz a t i o n m a p p i n g - - i n t e r m s o f
K r i f k a ' s ( 1 9 9 2 ) p a r t - o f r e l a t i o n ( _U ). O f p a r t i c u l a r
i n t e r e s t i s t h e f o l lo w i n g s u b p a r t c l o s u r e p r o p e r t y :
(3 ) V x y l y 2 [ c o m p ( x ) ( y x ) A y2 C_ yl - -~ c o m p ( x ) ( y 2 ) ]
P o s t u l a t e ( 3 ) s t a t e s t h a t a l l s u b p a r t s o f a r e a li z a -
t i o n o f a g i v e n k i n d a r e a l s o r e a l i z a t i o n s o f t h a t
k i n d . 3 F r o m t h i s p o s t u l a t e i t fo l l ow s , f o r e x a m p l e ,
t h a t i f e is a p r o c e s s o f J o h n r u n n i n g a l o n g t h e
r i v e r w h i c h h a s a r e a l i z a t i o n e l l a s t i n g t e n m i n -
u t e s , a n d i f e 2 i s a s u b e v e n t o f e l - - t h e f i r s t h a l f ,
s a y - - t h e n e 2 i s a ls o a r e a l i z a t i o n o f e. A s s u c h ,
t h is p o s t u l a t e m a y b e u s e d t o m a k e J o h n r a n a lo n g
t h e r i v e r f o r t e n m i n u t e s e n t a i l J o h n r a n a l o n g t h e
r i v e r f o r f i v e m i n u t e s , i n c o n t r a s t t o t h e p a i r J o h n
r a n t o l h e m u s e u m i n t e n m i n u t e s a n d J o h n r a n
t o lh e m u s e u m i n f i v e m i n u l e s .
I n o r d e r t o r e s o lv e t h e i m p e r f e c t i v e p a r a d o x ,w e m a y e x t e n d W h i t e ( 1 9 93 ) b y a d d i n g a m a p p i n g
f r o m e v e n t s t o p r o c e s s e s ( w h o s e r e a l i z a t i o n s n e e d
n o t t e r m i n a t e i n th e s a m e w a y ) , a s w e ll a s a m e a n s
f o r d is t i n g u i s h in g a c t u a l a n d h y p o t h e t i c a l e v e n ts .
T o d o t h e f o rm e r , w e m a y a x i o m a t i z e c o m p ' s i n-
v e r s e m a p p i n g - - J a c k e n d o f f ' s g r o u n d - f r o m ( g r )
- - a g a i n i n t e r m s o f K r i f k a ' s p a r t - o f re l a t io n . T h i s
i s s h o w n b e l o w :
(4) V x y l Y 2 [ g r ( y l ) ( X ) A c o m p ( x ) ( y 2 ) ---*y 2 C _ y l ]
P o s t u l a t e ( 4 ) s i m p l y r e q u i r e s t h a t a l l t h e r e a l iz a -t i o n s e 2 o f a p r o c e s s e w h i c h i s ' g r o u n d f r o m ' a n
e v e n t e l m u s t b e s u b e v e n t s o f e l ( a n d l i k ew i s e ,
m u t a t i s m u t a n d i s , f o r s u b s t a n c e s a n d t h i n g s ) . A s
t h e r e a l i z a t i o n s e2 o f e m a y b e p r o p e r s u b e v e n t s o f
e l , t h e r e l a t i o n g r p r o v i d e s a m e a n s f o r a c c e s s i n g
s u b e v e n t s o f e l w i t h a l t e r n a t e t e r m i n a t i o n s .
T o d i s t i n g u i s h t h o s e e v e n t s w h i c h a c t u a l l y o c -
c u r f r o m t h o s e t h a t a r e m e r e l y h y p o t h e t i c a l , w e
m a y s i m p l y in t r o d u c e a s p ec i al p r e d ic a t e A c t u a l ,
w h i c h w e r e q u i r e t o p r e s e r v e t h e p a r t - o f re l a t i o n
o n l y in t h e d o w n w a r d s d i r e c t i o n :
( 5 ) V x y [ A c t u a l ( z ) A y U _z - -* A c t u a l ( y ) ]
P o s t u l a t e ( 5 ) i s n e c e s s a r y t o g e t J o h n s l o p p e d r u n -
n i n g to t h e m u s e u m a f t e r t e n m i n u t e s t o e n t a i l
J o h n r a n f o r t e n m i n u t e s a s wel l a s J o h n r a n f o r
n i n e m i n u t e s , b u t n o t J o h n r a n f o r e l e v e n m i n -
u t e s .
A t t h i s p o i n t w e a r e r e a d y t o e x a m i n e i n s o m e
d e t a i l h o w t h e a b o v e m a c h i n e r y m a y b e u s e d i nr e s o l v in g t h e i m p e r f e c t i v e p a r a d o x . L e t u s a s s u m e
3For the sake of simplicity I will not address the
min imal p a r t s p ro b l em h ere .
t h a t s e n t e n c e s s u c h a s ( 6 ) r e c e i v e c o m p o s i t i o n a lt r a n s l a t i o n s a s i n ( 7 ) :
( 6 a ) J o h n r a n t o t h e b r i d g e .
( 6 b ) J o h n s t o p p e d r u n n i n g t o th e b r i d g e .
( 7 a ) 3 e l .
r u n ' ( j ) ( e l ) A t o ' ( t h e ' ( b r id g e ' ) ) ( r ~ ( e l ) ) A
A c t u a l ( e l )
(7 b ) 3 e e l e 2 e 3 .
r u n ' ( j ) ( e l ) A t o ' ( t h e ' ( b r id g e ' ) ) ( r s ( e l ) ) A
g r ( e l ) ( e ) A c o m p ( e ) ( e 2 ) A s t o p ' ( e 2 ) ( e a ) A
A c t u a l ( e 3 )
I n ( 7 ), e l i s a n e v e n t o f J o h n r u n n i n g t o t h e
b r i d g e . 4 I n ( T a ) , th i s e v e n t i s a s s e r t e d t o b e a c t u a l ;
i n ( 7 b ) , in c o n t r a s t , t h e p r o g r e ss i v e m o r p h o l o g y o n
r u n t r ig g e r s t h e i n t r o d u c t i o n o f g r , w h i c h m a p s
e l t o t h e p r o c e s s e . 5 I t i s t h i s p r o c e s s w h i c h e 3 is
a n e v e n t o f s t o p p i n g : f o l lo w i n g J a c k e n d o f f ( 1 9 9 1 ),
t h i s is r e p r e s e n t e d h e r e b y i n t r o d u c i n g a n e v e n t e~
c o m p o s e d o f e w h i c h h a s e a a s i t s s t o p p i n g p o i n t .
N a t u r a l l y e n ou g h , w e m a y e x p e c t t h e a c t u a l i t y
o f e3 t o e n t a i l t h e a c t u a l i t y o f e2 , a n d t h u s a l l
s u b e v e n t s o f e2 . N e v e r t h e le s s , t h e a c t u a l i t y o f e t
d o e s n o t f o l lo w , a s P o s t u l a t e ( 4 ) p e r m i t s e 2 t o b e
a p r o p e r s u b p a r t o f e l ( w h i c h is p r a g m a t i c a l l y t h e
m o s t l i k e l y c a s e ) .
T o m a k e t h e s e m a n t i c s d e v e l o p e d s o f a r m o r e
c o n c r e te , w e m a y n o w i m p o s e a p a r t i c u l a r i n t e r -
p r e t a t i o n o n t r a j e c t o r y - o f - m o t i o n e v e n t s , n a m e l y
o n e i n w h ic h t h e s e a r e m o d e l e d a s c o n t i n u o u s f u n c -
t i o n s f r o m t i m e s t o l o c a t i o n s o f t h e o b j e c t i n m o -
t i o n . D e p e n d i n g o n ho w w e m o d e l o b j e c t s a n d
l o c a t io n s , w e o f c o u r s e a r r iv e a t i n t e r p r e t a t i o n s o f
v a r y i n g c o m p l e x i t y . I n w h a t f o ll o w s w e f o c u s o n l yo n t h e s i m p l e s t s u c h i n t e r p r e t a t i o n , w h i c h t a k e sb o t h t o b e p o i n t s .
N o t e t h a t b y a s s u m i n g t h e p r e c e d i n g i n t e r -
p r e t a t io n o f t ra j e c t o r y - o f - m o t i o n e ve n ts , w e m a y
i n t e r p r e t t h e r e l a t i o n _ a s t h e r e l a t i o n c o n t i n u o u s -
s u b s e t. F u r t h e r m o r e , w e m a y a ls o i n t e r p r e t p r o -
c e s se s a s s e t s o f e v e n t s c l o s e d u n d e r t h e v - r e l a -
t i on ; t hi s t h e n p e r m i t s c o m p t o be in t e r p r e t e d
a s e l e m e n t - o f , a n d g r ( f o r e v e n t s ) a s m a p p i n g a n
e v e n t t o t h e s m a l l e s t p r o c e ss c o n t a i n i n g i t . B e f o r e
c o n t i nu i n g , w e m a y o b s e r v e t h a t t h i s i n t e r p r e t a -
t i o n d o e s i n d e e d s a t i s f y P o s t u l a t e s ( 3 ) a n d ( 4 ) .
Applicat ionW h i l e t h e a b o v e i n t e r p r e t a t i o n o f t r a j e c t o r y - o f -
m o t i o n e v e n ts f o rc e s on e t o a b s t r a c t a w a y f r o m
*The spat ial t race funct ion r~ ma ps even tua l i t ies tothei r t rajector ies (cf . Whi te 1993) .
5Much as in Moens and Stee dm an (1988) and Jack-endoff (1991), the in t rod uct ion of g r i s neces sary toavoid hav ing an i l l - sor ted fo rmula.
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t h e m a n n e r o f m o t i o n s u p p l i e d b y a v e rb , i t d o es
n e v e r t h e le s s p e r m i t o n e t o c o n s i d e r fa c t o r s s uc h a s
t h e n o r m a l s p e e d a s w e l l a s t h e m e a n i n g s o f th e
p r e p o s i t i o n s 1 0 , lowards, e t c . B y m a k i n g t w o a d -
d i t i o n a l r e s t r i c t iv e a s s u m p t i o n s , n a m e l y t h a t t h e s e
e v e n t s b e o f c o n s t a n t v e l o c i ty a n d i n o n e d i m e n -
s i on , I h a v e b e e n a b l e t o c o n s t r u c t a n d i m p l e m e n t
a n a l g o r i t h m w h i c h d e t e r m i n e s w h e t h e r a s p e c i-
f i ed s e q u e n c e o f s u c h e v e n t s i s o r i s n o t p o s s i b l eu n d e r c e r t a i n s i t u a t i o n a l l y s u p p l i e d c o n s t r a i n t s .
T h e s e c o n s t r a i n t s i n c l u de t h e l o c a t i o n s o f v a r i ou s
l a n d m a r k s ( a s s u m e d t o r e m a i n s t a t io n a r y ) a n d t h e
m i n i m u m , m a x i m u m , a n d n o r m a l r a t e s a s so c ia t e d
w i t h v a r i o u s m a n n e r s o f m o t i o n ( e . g . r u n n i n g , j o g -
g i n g ) f o r a g i v e n i n d i v i d u a l .
T h e a l g o r i t h m t a k e s a n i n p u t s t r in g a n d c o m -
p o s i t i o n a l l y d e r i v e s a s e q u e n c e o f l o g i c a l f o r m s
( o n e f o r e a c h s e n t e n c e ) u s i n g a s i m p l e c a t e g o r i a l
g r a m m a r ( m o s t o f w h i c h a p p e a r s i n W h i t e 1 9 9 3 ).
A s p e c i a l - p u r p o s e p r o c e d u r e i s t h e n u s e d t o i n -
s t a n t i a t e t h e d e s c r i b e d s e q u e n c e o f e v e n t s a s a c o n -s t r a i n t o p t i m i z a t i o n p ro b l e m ; n o t e t h a t a l t h o u g h
t h i s p r o c e d u r e i s q u i t e a d - h o c , t h e c o n s t r a i n t s a r e
r e p r e s e n t e d i n a d e c l a r a t i v e , h i e r a r c h i c a l f a s h i o n
( c f. W h i t e 1 9 9 3 ). I f t h e c o n s t r a i n t o p t i m i z a -
t i o n p r o b l e m h a s a s o l u t i o n , i t i s f o u n d u s i n g a
s l i g h t l y m o d i f i e d v e r s i o n o f t h e c o n s t r a i n t s a t i s -
f a c t i o n p r o c e d u r e b u i l t i n t o S C a E A M E R , S i s k i n d
a n d M c A l l e s t e r ' s ( 1 9 9 3 ) p o r t a b l e , e f f ic i e n t v e r s i o n
o f n o n d e t e r m i n i s t i c C o m m o n L i sp . 6
A s a n e x a m p l e o f a n i m p o s s i b le d e s c r i p t io n ,
l e t u s c o n s i d e r t h e s e q u e n c e o f e v e n t s d e s c r i b e d
b e l o w :
( 8 ) G u y s t a r t e d j o g g i n g e a s tw a r d s M o n g t h e r iv e r .2 5 m i n u t e s l a t e r h e r e a c h e d { t h e c a f e / t h e
m u s e u m } .
I f w e a s s u m e t h a t t h e u s e r s p e c if i e s t h e c a f e a n d
t h e m u s e u m t o b e 5 a n d 1 0 k m , r e s p e c t iv e l y , f r o m
t h e i m p l i c i t s t a r t i n g p o i n t , a n d t h a t t h e r a t e s s p e c-
i f ie d fo r G u y a r e t h o s e o f a s e r i o u s b u t n o t s u p e r -
h u m a n a t h l e t e , t h e n t h e a l g o r i t h m w i ll o n l y f in d
a s o l u t i o n f o r t h e f i r s t c a s e ( 1 0 k m i n 2 5 m i n u t e s
is t o o m u c h t o e x p e c t . ) N o w , b y re a s o n i n g a b o u t
s u b e v e n t s - - h e r e , s u b s e g m e n t s o f l i ne s i n s p a c e -
t i m e - - t h e p r o g r a m e x h i bi ts t h e s a m e b e h a v i o r
w i t h t h e p a i r i n ( 9 ) :
( 9 ) G u y s t a r t e d j o g g i n g t o t h e b a r. 2 5 m i n u t e s
l a t e r h e r e a c h e d { t h e c a f e / t h e m u s e u m } .
S i n c e " G u y j o g g i n g t o t h e c a f e is a c c e p t e d a s a
p o s s ib l e p r o p e r s u b e v e n t o f G u y j o g g i n g t o t h e
6The const rain t op t imizat ion prob lem is sp l i t in totwo const rain t sat i sfact ion prob lems, namely f ind-ing the sm al les t consis ten t value of a cost var iab leand then f ind ing consis ten t values fo r the res t o f thevariables.
b a r ( a s s u m i n g t h e b a r i s f u r t h e r e a s t t h a n t h e
o t h e r l a n d m a r k s ) , e x a m p l e ( 9) s ho w s h o w t h e
p r e s e n t a p p r o a c h s u c c e s s f u l l y a v o i d s t h e i m p e r f e c -
t i r e p a r a d o x ; s i n c e G u y j o g g i n g to t h e m u s e u m ( in
2 5 m i n u t e s ) i s n o t a c c e p t e d a s a p o s s i b l e s u b e v e n t ,
e x a m p l e ( 9 ) li k e w is e s h o w s h o w t h e p r e s e n t a p -
p r o a c h e x t e n d s a n d r e fi n es t h o s e o f M o e n s a n d
S t e e d m a n a n d 3 a c k e n d o f f v i s -a - v is t h e s u b e v e n t
r e l a t i o n . 7
F u t u r e W o r k
T h e a l g o r i t h m a s i m p l e m e n t e d f u n c t i o n s o n ly u n -
d e r a n u m b e r o f q u i t e r e s t r ic t i v e a s s u m p t i o n s , a n d
s u f fe r s f r o m a r a t h e r a d - h o c u s e o f t h e d e r i v e d l o g i -
c a l f o r m s . I n f u t u r e w o r k I i n t e n d t o e x t e n d t h e a l -
g o r i t h m b e y o n d t h e u n i d i m e n s i o n a l a n d c o n s t a n t
v e l o c i t y c a s e s c o n s i d e r e d s o f a r , a n d t o i n v e s t i g a t e
i n c o r p o r a t i n g t h e p r e s e n t t r e a t m e n t i n t o t h e I n -
t e r p r e t a ti o n a s A b d u c t i o n a p p r o a c h a d v o c a t e d b y
H o b b s e t. a l . ( 1 9 9 3 ).
R e f e r e n c e s
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[2] David R. Dowty. W o r d M e a n i n g a n d M o n t a g u e G r a m -m a r . Reidel, 1979.
[3] Kurt Eberle. Eventualities in natural language under-standing systems. In S o r t s a n d T y p e s i n A r t i f i c i a l I n t e l -l i gence . Springer Verlag, 1990.
[4] Christopher Habel. Propositi onal and depictorial rep-resentations of spatial knowledge: The case of p a t h -concepts. In N a t u r a l L a n g u a g e a n d L o g i c . Springer Ver-lag, 1990. Lecture Notes in Artificial Intelligence.
[5] Erhard Hinrichs. A C o m p o s i t i o n a l S e m a n t i c s f o r A k-t i o n s a r t e n a n d N P R e f e r e n c e i n E n g l is h . PhD thesis,The Ohio State University, 1985.
[61 Jerr y Hobbs, Mark Stickel, Dougla s Appelt, and Paul
Martin. Inte rpret atio n as abduction, 1993. To appearin Artificial Intelligence J ournal.
[7] Ray Jackendoff. Parts and boundaries. C o g n i t i o n , 41:9-45, 1991.
[g] Manfred Krifka. Thematic relations as links between nom-
inal reference and temporal constitution. In Ivan A, Sag
and Ann a Szabolesi, editors, L e x i c a l M a t t e r s . CSLI, 1992.
[9] Marc Moens and Mark Steedman. Temporal ontologyand temporal reference. C o m p u t a t i o n a l L i n g u i st i c s , June1988.
[10] Jeffrey Mark Siskind and David Allen McAllester. Non-deterministic lisp as a substrate for constraint logic pro-gramming. To appear in AAAI-93, 1993.
[11] H. J. Verkuyl. Aspectual classes and aspectual composi-tion. L i n g u i s t i c s a n d P h i l o s o p h y , 12(1), 1989.
[12] Michael White. Delimitedness and traject ory-of- motion
events. In P r o c e e d i n g s o f t h e S i x t h C o n f e r e n c e o f t h eE u r o p e a n C h a p t e r o f th e A s s o c i a t i o n f o r C o m p u t a t i o n a lL i n g u i st i c s ( E A C L ' 9 3) , 1993.
7I t i s wor th no t ing that the constan t veloci ty re-s t r ic t ive assumpt ion makes start running to a n d startrunning towards synonymous, which i s no t the case ingeneral (cf. Habel 1990).
2 8 5
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