NUMERICAL QUANTIFIERS AND THEIR USE IN
REASONING WITH NEGATIVE INFORMATION
S t u a r t C. Shapiro
D e p a r t m e n t o f Computer S c i e n c e
S t a t e U n i v e r s i t y o f New Y o r k a t B u f f a l o
4226 Ridge Lea Road
A m h e r s t , New Y o r k
14226
N u m e r i c a l q u a n t i f i e r s p r o v i d e s i m p l e means o f f o r m a l i z i n g such s t a t e m e n t s a s , " a t l e a s t t h r e e
p e o p l e a r e i n t h a t r o o m " , " a t most f i f t e e n p e o p l e a r e i n t h e e l e v a t o r " , and " e v e r y b o d y has
e x a c t l y two p a r e n t s " .
A l t h o u g h n u m e r i c a l q u a n t i f i e r s g e n e r a l i z e the e x i s t e n t i a l q u a n t i f i e r ,
t h e y have d i f f e r e n t uses i n r e a s o n i n g .
The e x i s t e n t i a l q u a n t i f i e r i s most u s e f u l f o r s u p p l y -­
i n g r e f e r e n t s f o r d e s i g n a t i n g p h r a s e s w i t h n o p r e v i o u s l y e x p l i c i t l y m e n t i o n e d r e f e r e n t . Numer-­
i c a l q u a n t i f i e r s a r e most u s e f u l f o r r e a s o n i n g b y t h e p r o c e s s o f e l i m i n a t i o n .
Numerical
q u a n t i f i e r s w o u l d , t h e r e f o r e , be a u s e f u l a d d i t i o n to the o p e r a t o r s of a r e a s o n i n g program or
d e d u c t i v e q u e s t i o n -­ a n s w e r i n g s y s t e m .
They have been added t o SNePS, t h e Semantic N e t w o r k
P r o c e s s i n g S y s t e m , t o f u r t h e r enhance i t s i n f e r e n c e c a p a b i l i t i e s .
1.
INTRODUCTION
L o g i c based r e a s o n i n g p r o g r a m s , t h a t i s r e a s o n -­
i n g programs based o n o p e r a t o r s ( c o n n e c t i v e s ,
q u a n t i f i e r s , m o d a l s ) w h i c h have been s t u d i e d a s
p a r t o f f o r m a l l o g i c a l systems b e n e f i t f r o m t h e
f a c t t h a t the i n f e r e n t i a l p r o p e r t i e s o f t h e i r
o p e r a t o r s a r e c l e a r and w e l l known. They need
not be r e s t r i c t e d , however, to a minimal set of
o p e r a t o r s . Minimal sets of o p e r a t o rs are u s e f u l
f o r p r o v i n g p r o p e r t i e s o f l o g i c a l systems such
a s c o n s i s t e n c y and c o m p l e t e n e s s , b u t u s i n g a
l o g i c a l system f o r c a r r y i n g o u t i n f e r e n c e s i s
s i m p l i f i e d ( f o r people) by e n l a r g i n g the set of
b a s i c o p e r a t o r s . T h i s i s one r e a s o n t h a t n a t u -­
r a l d e d u c t i o n systems l i k e t h o s e o f [l] , [ 4 ]
and [ 1 1 ] , w i t h r e a s o n a b l e s e t s o f c o n n e c t i v e s
and two r u l e s o f i n f e r e n c e f o r each o n e , a r e
e a s i e r t o use t h a n a x i o m a t i c systems w i t h
m i n i m a l s e t s o f c o n n e c t i v e s , r u l e s and a x i o m s .
T h i s p a p e r i s m o t i v a t e d b y a n i n t e r e s t i n p r o -­
grams t h a t r e p r e s e n t k n o w l e d g e , i n c l u d i n g t h e
k n o w l e d g e o f r u l e s o f r e a s o n i n g , and t h a t use
those r u l e s to perform r e a s o n i n g . I b e l i e v e
t h a t s u c h programs a r e enhanced b y t h e a v a i l a -­
b i l i t y of a large set of operators that t y p i f y
and f o r m a l l y m o d e l as many of t h e modes of
human r e a s o n i n g a s p o s s i b l e . T h i s paper d i s -­
c u s s e s a s e t o f o p e r a t o r s , t h e n u m e r i c a l q u a n -­
t i f i e r s , which can b e implemented i n reasonin g
* T h i s m a t e r i a l i s based o n work s u p p o r t e d
i n p a r t b y a F a c u l t y R e s e a r ch F e l l o w s h i p f r o m
the Research Foundatio n o f S t a t e U n i v e r s i t y o f
New Y o r k , and i n p a r t b y t h e N a t i o n a l S c i e n c e
F o u n d a t i o n u n d e r G r a n t No. MCS78-­02274.
791
792
793
794
(M13)
9 MSECS
* * ; P a t i s i n the h a l l .
*(BUILD A PAT R IN 0 HALL)
(M14)
11 MSECS
* * ; N i c k i s i n the h a l l .
*(BUILD A NICK R IN 0 HALL)
(M15)
10 MSECS
**;Who i s i n t h e meeting?
*(DESCRIBE (DEDUCE A %X R IN 0 MEETING))
(M17 (MIN(O)) (MAX(O)) (ARG(M16)))
(M16 (A(PAT)) ( R ( I N ) ) (O(MEETING)))
; Pat i s n o t i n the m e e t i n g .
(M19 (MIN(O)) (MAX(O)) (ARG(M18)))
(Ml8 ( A ( N I C K )) ( R ( I N ) ) (0(MEETING)))
; N i c k is n o t i n the m e e t i n g .
(M20 (A(GABOR) ) ( R ( I N ) ) (O(MEETING)))
; Gabor i s i n the m e e t i n g .
(M21 (A(JOHN)) ( R ( I N ) ) (0(MEETING)))
; John is i n th e m e e t i n g .
(M22 (A(STU)) ( R ( I N ) ) (O(MEETING)))
; Stu i s i n the m e e t i n g .
(DUMPED)
1427 MSECS
*(BUILD MEM ROVER CLASS DOG
(M10)
9 MSECS
* * ; S p o t is a dog.
*(BUILD MEM SPOT CLASS DOG)
(M11)
12 MSECS
* * ; L a s s i e i s a dog
*(BUILD MEM LASSIE CLASS DOG)
(M12)
10 MSECS
* * ; J o h n owns Rover.
*(BUILD A JOHN R OWNS 0 ROVER)
(M13)
11 MSECS
* * ; J o h n owns S p o t .
*(BUILD A JOHN R OWNS 0 SPOT)
(M14)
11 MSECS
* * ; M a r y owns L a s s i e .
*(BUILD A MARY R OWNS 0 LASSIE)
(M15)
13 MSECS
* * ; J a n e owns S p o t .
*(BUILD A JANE R OWNS 0 SPOT)
(M16)
11 MSECS
* * ; J i m owns L a s s i e .
*(BUILD A J I M R OWNS 0 LASSIE)
(M17)
12 MSECS
* * ; J o h n s p o i l s Rover.
*(BUILD A JOHN R SPOILS 0 ROVER)
(M18)
10 MSECS
* * ; J o h n s p o i l s S p o t.
*(BUILD A JOHN R SPOILS 0 SPOT)
(M19)
12 MSECS
* * ; j a n e s p o i l s Spot.
*(BUILD A JANE R SPOILS 0 SPOT)
(M20)
12 MSECS
**;Mary spoils Lassie.
*(BUILD A MARY R SPOILS 0 LASSIE)
(M21)
11 MSECS
**;Who s p o i l s whom?
*(DESCRIBE (DEDUCE A %X R SPOILS 0 %Y))
(M18 vA(JOHN)) (R(SPOILS)) (0(ROVER)))
;John s p o i l s Rover.
(M19 (A(JOHN)) (R(SPOILS)) (O(SPOT)))
;John s p o i l s S p o t .
(M20 (A(JANE)) (R(SPOILS)) (O(SPOT)))
;Jane s p o i l s S p o t .
(M21 (A(MARY)) (R(SPOILS)) (O(LASSIE)))
;Mary s p o i l s L a s s i e .
(M23 (MIN(O)) (MAX(O)) (ARG(M22)))
(M22 ( A ( J I M ) ) (R(SPOILS)) ( O ( L A S S I E ) ))
; J i m does n o t s p o i l L a s s i e .
(DUMPED)
1341 MSECS
7 -­ .3 Example 2
We a s s e r t t h a t between two and f o u r dog owner
i n p r e l a t i o n s i n v o l v e s p o i l i n g , and a s s e r t
f o u r such s p o i l i n g r e l a t i o n s . SNePS deduces
t h a t Jim does n o t s p o i l L a s s i e .
* * ; O f 5 dog ownershi p r e l a t i o n s ,
* ,between 2 and 4 i n v o l v e s p o i l i n g .
* , 532Xy m e m b e r ( x , p e r s o n ) , M e m b e r ( y , d o g ) ,
Owns(x,y) : S p o i l s ( x , y ) ]
•* (BUILD ETOT 5 EMIN 2 EMAX 4 PEVB($X $Y)
* 6ANT ((BUILD MEM *X CLASS PERSON)
*
(BUILD MEM *Y CLASS DOG)
*
(BUILD A *X R OWNS 0 * Y ) )
CQ (BUILD A *X R SPOILS 0 * Y ) )
(M5)
81 MSECS
* * ; J o h n is a person.
* ( B U I L D MEM JOHN CLASS PERSON)
(M6)
10 MSECS
* * ; J a n e is a person.
* ( B U I L D MEM JANE CLASS PERSON)
(M7)
10 MSECS
**;Mary is a person.
*(BUILD MEM MARY CLASS PERSON)
(M8)
9 MSECS
* * ; J i m is a person.
*(BUILD MEM J I M CLASS PERSON)
(M9)
9 MSECS
* * ; R o v e r is a dog.
795
8.
SUMMARY
We have d i s c u s s e d the r o l e s of e x i s t e n t i a l and
n u m e r i c a l q u a n t i f i e r s i n r e a s o n i n g programs. The
r o l e s a r e d i f f e r e n t and b o t h a r e i m p o r t a n t . The
e x i s t e n t i a l q u a n t i f i e r i s most u s e f u l f o r s u p p l y -­
i n g r e f e r e n t s f o r d e s i g n a t i n g phrases w i t h n o
p r e v i o u s l y e x p l i c i t l y mentioned r e f e r e n t . Numeri-­
c a l l y q u a n t i f i e d r u l e s a r e c o n c i s e r e p r e s e n t a -­
t i o n s o f r u l e s t h a t govern r e a s o n i n g b y the
process of e l i m i n a t i o n and can i n t r o d u c e ex-­
p l i c i t n e g a t i v e s i n t o a data base or can use
n e g a t i v e statements f o r d e r i v i n g p o s i t i v e s t a t e -­
ments. The most g e n e r a l schema f o r n u m e r i c a l
q u a n t i f i e r s t h a t we have d i s c u s s e d is
which says t h a t
at l e a s t i and at most j of the n sequences
of individuals that satisfy
a l s o s a t i s f y Q(X). We showed how r u l e s of t h i s
form can be r e p r e s e n t e d in SNePS, t h e Semantic
Network P r o c e s s i n g System, and gave examples of
SNePS runs t h a t used such r u l e s f o r c a r r y i n g o ut
inferences.
Two aspects of n u m e r i c a l q u a n t i f i e r s m i g h t
l i m i t t h e i r immediate u s e f u l n e s s . One is the n
parameter, r e q u i r e d whenever the m i n i m a l p a r a -­
meter i s p r e s e n t . I n f o r m u l a t i n g n u m e r i c a l l y
q u a n t i f i e d s t a t e m e n t s , we have found s p e c i f y i n g
n to be bothersome. The parameter c o u l d be
e l i m i n a t e d i f the i n f e r e n c e system had a n o t h e r
means of d e t e r m i n i n g how many i n d i v i d u a l s
s a t i s f y t h e r e s t r i c t i o n , f o r example i f t h e
c l o s e d w o r l d assumption h e l d . The o t h e r problem
i s t h a t i n h e r e n t i n any c o u n t i n g argument i s the
assumption t h a t any two i n d i v i d u a l s a r e , i n
f a c t , d i s t i n c t . I f t h i s assumption does n o t h o l d ,
the use of n u m e r i c a l q u a n t i f i e r s depends on a
s o l u t i o n t o the i d e n t i t y problem mentioned i n
Sec. 3. Except f o r these two problems, n u m e r i -­
c a l q u a n t i f i e r s seem t o b e u s e f u l a d d i t i o n s t o
the t o o l k i t o f r e p r e s e n t a t i o n and i n f e r e n c e .
ACKNOWLEDGEMENTS
The a u t h o r i s g r a t e f u l t o Don McKay f o r h e l p i n
SNePS development and to him and Rich F r i t z s o n
f o r SNePS and L i s p system s u p p o r t . He is a l s o
g r a t e f u l to B r i a n Funt and Don McKay f o r comments
on an e a r l i e r d r a f t .
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