foundations on knowledge representation and reasoning by lakemeyer and nebel

7/28/2019 Foundations on Knowledge Representation and Reasoning by Lakemeyer and Nebel

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F o u n d a t io n s o f K n o w l e d g eR e p r e s e n t a t i o n a n d R e a s o n i n gA G u i d e t o t h i s V o l u m e

G e r h a r d L a k e m e y e r 1 a n d B e r n h a r d N e b e l 21 In s t i t u t e o f C o m p u te r Sc ien ceUnivers i ty o f B o n nR6mers t r . 164D-53117 Bonn , Germanygerhard@cs .un i -bonn .deDe p a r tm en t o f C o m p u te r ScienceUnivers i ty o f UlmD-89069 Ulm, Germanynebel@informat ik , un i -u lm.de

1 I n t r o d u c t i o n

K n o w l e d g e r e p r e s e n t a t i o n ( K R ) is t h e a r e a o f A r t i f ic i a l I n t e ll i g e n c e t h a t d e a l sw i t h t h e p r o b l e m o f r e p r e se n t in g , m a i n t a i n i n g , a n d m a n i p u l a t i n g k n o w l e d g ea b o u t a n a p p l i c a t i o n d o m a i n . S i n c e v i r t u a l l y a l l A r t i f ic i a l I n t e ll i g e n c e s y s t e m sh a v e t o a d d r e s s t h i s p r o b l e m , K R is o n e o f t h e c e n tr M s u b f ie l ds o f A r t i f ic i a lI n t e l l i g e n c e .

M a i n r e s e a r c h e n d e a v o r s i n K R a r e- r e p r e s e n t i n g k n o w l e d g e a b o u t a p p l i c a t i o n a r e a s ( e. g. , m e d i c a l k n o w l e d g e ,

k n o w l e d g e a b o u t t i m e , k n o w l e dg e a b o u t p h y s ic a l s y st e m s ) ,- d e v e l o p i n g a p p r o p r i a t e r e p r e s e n t a t i o n la n g u a g e s ,- s p e c i f y i n g a n d a n a l y z i n g r e a s o n i n g o v e r r e p r e s e n t e d k n o w l e d g e , a n d- i m p l e m e n t i n g s y s t e m s t h a t s u p p o r t t h e r e p r e s e n t a t i o n o f k n o w l e d g e a n d

r e a s o n i n g o v e r t h e r e p r e s e n t e d k n o w l e d g e .W h i l e k n o w l ed g e a b o u t a n a p p l i c a ti o n d o m a i n m a y b e r e p r e s e n te d i n a v a r i-

e t y o f f o r m s , e .g ., p r o c e d u r a l l y i n f o r m o f p r o g r a m c o d e o r i m p l i c i t l y as p a t t e r n so f a c t i v a t i o n i n a n e u r a l n e t w o r k , r e s e a r c h i n t h e a r e a o f k n o w l e d g e r e p r e s e n -t a t i o n a s s u m e s a n expl ic i t a n d dec larat ive r e p r e s e n t a t i o n , a n a s s u m p t i o n t h a td i s t in g u i s h e s K R f r o m r e s e a r c h i n , e .g . , p r o g r a m m i n g l a n g u a g e s a n d n e u r a l n e t -w o r k s . E x p l i c i t n e s s o f r e p r e s e n t a t io n m e a n s t h a t t h e r e p r e s e n t e d k n o w l e d g e iss t o r e d i n a k now le dge base c o n s i st in g o f a s e t o f f o r m a l e n t i ti e s t h a t d e s c r i b et h e k n o w l e d g e in a d i re c t a n d u n a m b i g u o u s w a y . D e c l a r a t i v e n e s s m e a n s t h a t t h e( f o r m a l ) m e a n i n g o f t h e r e p r e s e n t a t i o n c a n b e s p ec i fi e d w i t h o u t r e f e r e n c e to h o wt h e k n o w l e d g e is a p p l i e d p r o c e d u r a l l y , im p l y i n g s o m e s o r t o f l o g ic a l m e t h o d o l o g yb e h i n d i t .


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A l t h o u g h t h e a b o v e t w o p o i n ts s e e m t o b e a l m o s t u n i v e r s a l l y a c c e p t e d b y r e -s e a r c h e r s w o r k i n g in K R , t h i s c o n s e n s u s h a s b e e n a c h i e v e d o n l y r e c e n t l y . B r a c h -m a n a n d L e v e s qu e m e n t i o n e d i n th e I n t r o d u c t i o n t o a c o l le c t io n o f p a p e r s i n1 98 5 t h a t t h e " r e s ea r c h a r e a o f K n o w l e d g e R e p r e s e n t a t i o n h a s a l o n g , c o m p l e x ,a n d a s y e t n o n - c o n v e r g e n t h i s to r y , " [ B r a c h m a n a n d L e v e s q u e , 1 9 8 5, p . x ii i] a ni m p r e s s i o n t h a t i s i n d e e d c o n f i r m e d b y t h e p a p e r s i n t h i s c o l l e c ti o n . A l a r g e p o r -t i o n o f t h e p a p e r s c o n t a i n m e t a - l e v e l d is c u s si o n s a r g u i n g a b o u t t h e r i g h t m e t h -o d s f o r r e p r e s e n t i n g k n o w l e d g e o r t h e y p r e s e n t a p p r o a c h e s t h a t a r e c o m p l e t e l yi n c o m p a t i b l e w i t h a l o g i ca l , d e c l a r a t i v e p o i n t o f v i ew .

N o w a d a y s , t h e p i c t u r e h a s c o m p l e t e l y c h a n g e d , h o w e v e r. L o g i ca l m e t h o d sp r e d o m i n a t e a n d m e t h o d o l o g i c a l p r o b l e m s a r e h a r d l y d i s cu s se d a n y l o n g e r [ B ra c h-m a n e t a l . , 1 9 8 9 , A l l en e t a l . , 1 9 9 1 , Neb e l e t a l . , 1 9 9 2 , B r a c h m a n , 1 9 9 0 ] . I n s t e a d ,r e s e a r c h p a p e r s f o c u s o n p a r t i c u l a r t e c h n i c a l r e p r e s e n t a t i o n a n d r e a s o n i n g p r o b -l e m s a n d a d d r e ss t h e s e p r o b l e m s u s in g m e t h o d s f r o m l o g ic a n d c o m p u t e r s ci en c e.

W h i l e th i s d e v e l o p m e n t in d i c a te s t h a t K R h a s b e c o m e a m a t u r e s c ie n ti fi cd i s c i p l i n e , i t a l s o l e a d s t o t h e s i t u a t i o n t h a t r e s e a r c h r e s u l t s i n K R a p p e a r t o b el es s a c c e ss i bl e t o t h e r e s t o f t h e A r t i f ic i a l I n t e l li g e n c e c o m m u n i t y . A s a m a t t e ro f f a c t , i t i s o f t e n a r g u e d t h a t t h e f o u n d a t i o n a l r e s u l t s t h a t a r e a c h i e v e d in t h eK R f i e l d a r e n o t r e l e v a n t t o A r t i f i c i a l I n t e l l i g e n c e a t a l l .

W e c o n c e d e ~ h a t a l a r ge a m o u n t o f K R r e se a r c h p r o b a b l y d o e s n o t h a v e a n yi m m e d i a t e i m p a c t o n b u i l d i n g A r t if i c ia l I n t e l li g e n c e s y s t e m s . H o w e v e r , th i s isp r o b a b l y a s k i n g f o r t o o m u c h . F o u n d a t i o n a l K R r e se a r c h a i m s a t p r o v i d i n g t h et h e o re t ic a l f o u n d a t i o n s o n w h i c h w e c a n b u i l d s y s t e m s t h a t a r e u s e f u l , c o m p r e -h e n s i b l e , a n d r e l i a b l e , i . e . , i t a i m s a t p r o v i d i n g t h e l o g i c a l a n d c o m p u t a t i o n a lf o u n d a t i o n s o f k n o w l e d g e r e p r e s e n t a t i o n f o r m a l i s m s a n d r e a s o n i n g p r o ce s s es .R e s u l t s in f o u n d a t i o n a l K R o f t e n " o n l y " p r o v i d e e x p l a n a t io n s w h y a p a r t i c u l a ra p p r o a c h w o r k s o r h o w a n a p p r o a c h c a n b e i n t e r p r e t e d l o g ic a ll y . A d d i t i o n a l l y ,t h e b o r d e r l i n e s o f w h a t c a n b e r e p r e s e n t e d a r e e x p l o r e d a n d i t i s a n a l y z e d h o we f f ic i e n tl y a r e a s o n i n g p r o c e s s c a n b e . W h i l e t h i s m a y n o t b e o f c e n t r a l c o n -c e r n w h e n b u i l d i n g A r t i f i c i a l I n t e l l i g e n c e s y s t e m s , s u c h r e s u l t s a r e n e v e r t h e l e s si m p o r t a n t w h e n w e w a n t t o u n d e r s t a n d s u c h s y s te m s , a n d w h e n w e w a n t t og u a r a n t e e t h e i r r e l i a b i l i t y .

P e r h a p s t h e m a i n m o t i v a t i o n a n d d r i v i n g f o rc e b e h i n d m o s t r e s e ar c h i n K Rh a s b e e n t h e d e s ir e t o e q u i p a r t i f a c t s w i t h c o m m o n s e n s e . T h i s i s l i t e r a l ly t r u e o f ap a p e r b y J o h n M c C a r t h y , fi rs t p u b l i s h e d in 1 9 58 a n d r e p u b l i s h e d a s [ M c C a r t h y ,1 9 68 ], w h i c h s t a r t e d t h e w h o l e K R e n t e r p r i s e , a n d i t is s t i ll t r u e , i f o n l y i m p l i c i t l y ,o f t h e p a p e r s i n t h i s b o o k . I n f a c t , w o r k o n t h e f o u n d a t i o n s o f K R c a n l a r g e l yb e i n d e n t i f i e d w i t h w o r k o n t h e f o u n d a t i o n s o f c o m m o n s e n s e r e a so n i n g , a p o i n to f v ie w w h i c h w e w i ll f ol lo w t h r o u g h o u t t h i s b r i e f s u r v e y .

I n t h e f o l lo w i n g s e c ti o n s , w e t o u c h o n s o m e b a s ic l o g i c al a n d c o m p u t a t i o n a la s p e c t s o f c o m m o n s e n s e r e a s o n in g . T h e r e a d e r i s w a r n e d t h a t t h i s is n o t a c o m -p r e h e n s i v e o v e r v i e w o f th e f i e ld , w h i c h w o u l d b e f a r o u t s i d e t h e s c o p e o f t h i sb o o k . I n s t e a d w e c o n f in e o u r s e lv e s m a i n l y to t h o s e a r e a s t h a t a r e a c t u a l l y c o v -e r e d b y p a p e r s i n t h i s b o o k .


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2 L o g i c a l F o u n d a t i o n s o f C o m m o n s e n s e R e a s o n i n gA s m e n t i o n e d a l r e a d y i n t h e b e g i n n i n g , t h e m a i n a s s u m p t i o n t h a t d i s t i n g u i s h e sk n o w l e d g e - b a se d s y s t e m s f r o m o t h e r a p p r o a c h e s i s t h a t k n o w l e d g e is r e p r e s e n t e dd e c l a r a t i v e l y i n s o m e lo g ic - li k e l a n g u a g e . T h i s i s o n e p a r t o f w h a t B r i a n S m i t hh a s c a l l e d the know ledge representation hypothesis [ S m i t h , 1 9 8 2 ] . T h e o t h e r p a r tp o s t u l a t e s t h a t t h e s e r e p r e s e n t a t io n s p l a y a c a u s al r o l e i n e n g e n d e r i n g t h e b e -h a v i o r o f t h e s y s t e m . W h i l e t h i s c a u s a l c o n n e c t i o n is p r e s e n t i n o n e f o r m o ra n o t h e r i n e v e r y k n o w l e d g e - b a s e d s y s t e m , i t is f a i r t o s a y t h a t s o f a r t h e r e a r ev e r y f e w , i f a n y , t h e o r e t i c a l r e s u l ts t h a t e x p l a i n t h i s c o n n e c t i o n .

H e n c e m o s t f o u n d a t i o n a l r e s ea r c h i n K R , i n c lu d i n g t h e w o r k r e p o r t e d i n th i sb o o k , d e a ls w i t h p r o b l e m s t h a t a r is e f r o m t h e f ir st p a r t o f t h e K R h y p o t h e s i sa n d w h i ch c a n b e d e a l t w i t h i n d e p e n d e n t l y f r o m t h e s e c o n d p a r t . I n t h i s c o n t e x t ,o n e c a n i d e n t if y t h r e e f u n d a m e n t a l q u e s t io n s :

1. W h a t i s t h e r i g h t r e p r e s e n t a t i o n l a n g u a g e ?2 . W h a t i n fe r e nc e s s h o u l d b e d r a w n f r o m a k n o w l e d g e b a se ?3 . H o w d o w e i n c o r p o r a t e n e w k n o w l e d g e ?

I n t h e r e s t o f t h i s se c t i o n , w e w i ll a d d r e s s e a c h q u e s t i o n i n t u r n w i t h a n e m p h a s i so n t h e r e l e v a n t p a p e r s i n t h i s b o o k .

2 .1 T h e R i g h t R e p r e s e n t a t i o n L a n g u a g eW h i l e t h e r e i s l i tt le d i s a g re e m e n t a n y m o r e a b o u t t h e a s s u m p t i o n t h a t a r ep -r e s e n t a t i o n l a n g u a g e i s o n e o f l og ic , w h e r e t h e s e n t e n c e s c a n b e i n t e r p r e t e d a sp r o p o s i t i o n s a b o u t t h e w o r l d , 3 d e s i g n in g a n a d e q u a t e l a n g u a g e i s n o t a n e a s yt a s k , s i n c e t h e v a r i o u s d e s i r a b l e f e a t u r e s a r e o f t e n i n c o m p a t i b l e . I n p a r t i c u l a r ,v e r y e x p r e s si v e l a n g u a g e s u s u a l l y h a v e p o o r c o m p u t a t i o n a l p r o p e r t i e s , a n i ss u et h a t h a s d r a w n c o n s id e r a b le in t e r es t s in c e a s e m i n a l p a p e r b y B r a c h m a n a n dL e v e s q u e [1 98 4] a n d w h i c h i s d i s c u s s e d i n m o r e d e t a i l i n t h e n e x t s e c t i o n . A tt h i s p o i n t w e o n l y m e n t i o n t h a t c o m p u t a t i o n a l c o n s i d e r a ti o n s h a v e l ed t o t h ed e v e l o p m e n t o f l a n g u a g e s t h a t a r e f a r le ss e x p r e ss i v e t h a n f u ll f i r s t -o r d e r l o g i c,m o s t n o t a b l y t h e s o - ca l le d concep t languages o r terminological logics. F o u r o f th ep a p e r s i n t h i s c o l l e c ti o n a r e d e v o t e d t o t h i s t o p i c [ B a n d e r a n d H o l l u n d e r , 1 99 4 ,B e t t i n i , 1 9 9 4 , A l l g a y e r a n d F r a n c o n i , 1 9 9 4 , D o n i n i e l a l . , 1 9 9 4 ] . F r o m t h e p o i n to f vi e w o f e x p re s s iv e n e s s, i t o f t e n s e e m s u s e fu l t o h a v e s p e c i a l e p i s t e m o l o g i c a l o ro n t o l o g i c a l p r i m i t i v e s b u i l t in t o t h e l a n g u a g e . S h o h a m a n d C o u s i n s [1 99 4] s u r v e yw o r k in A I o n a w h o l e r a n g e o f m e n t a l a t t i t u d e s l ik e b e l ie f s , d e s ir e s , g o a l s , o r i n -t e n t i o n s . T h e n e e d f o r m a k i n g s u c h n o t i o n s e x p l i c i t i s p r o b a b l y m o s t c o n v i n c i n gi n m u l t i - a g e n t s e t t i n g s , w h e r e a g e n t s n e e d t o r e a s o n a b o u t e a c h o t h e r ' s m e n t a la t t i t u d e s i n o r d e r to c o m m u n i c a t e a n d c o o p e r a t e s u cc e s sf u ll y . [ G o t t l o b , 1 9 9 4,K a l i n s k i , 1 9 9 4, N i e m e l ~ a n d R i n t a n e n , 1 99 4] c o n s i d e r t h e s p e c if i c c a s e o f belief,3 Unti l t h e l a t e seven t ies , many so -ca l led rep resen ta t ion languages ac tua l ly v io la tedthis fund am en tal assum ption and led to v ivid discussions such as [Hayes, 1977,McDermot t , 1978] .


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w h i c h , t o g e t h e r w i t h knowledge, is p r o b a b l y t h e b e s t u n d e r s t o o d a m o n g th e a t -t i t u d e s . I n t h e s e p a p e r s a v e r y s p ec if ic a s p e c t o f b e l i e f i s c o n s i d e r e d , n a m e l y t h ea b i l i t y t o m o d e l c e r t a i n f o r m s o f d e f e a s ib l e r e a s o n i n g b y r e f e r r i n g e x p l i c i t ly t ot h e s y s t e m ' s o w n e p i s te m i c s t a t e ( se e S e c t i o n 2 .2 b e l o w ) . B e t t i n i a n d L i n [B e t-t i n i , 1 9 9 4 , L i n , 1 9 9 4 ] , o n t h e o t h e r h a n d , a r e c o n c e r n e d w i t h a d d i n g e x p l i c i tn o t i o n s o f t i m e t o t h e l a n g u a g e . W h i l e B e t t i n i c o n s id e r s i n c o r p o r a t i n g a n e x -i s t i n g i n t e r v a l - b a s e d c o n c e p t o f t i m e t o a t e m p o r a l l o gi c , L i n p r o p o s e s a n e wa x i o m a t i z a t i o n o f t i m e , w h e r e t i m e i n s ta n c e s a r e d e f i ne d o n t h e b a si s o f e v e n ts .2 . 2 T h e R i g h t I n f e r e n c e sH a v i n g e x p l i c i t r e p r e s e n t a t i o n s o f k n o w l e d g e a l o n e i s n o t v e r y u s e f u l i n g e n e r a l .I n s t e a d o n e w a n t s t o r e a s o n a b o u t t h e s e r e p r e s e n t a t i o n s t o u n c o v e r w h a t i simpl ied b y t h e m . A f t e r a ll , w e u s e th e t e r m c o m m o n s e n s e r e a s o n i n g a n d n o tc o m m o n s e n s e r e p r e s e n t a t i o n . U n t i l t h e e a r l y s e v e n t i e s , deduct ion w a s t h e m a i nf o c u s o f a t t e n t i o n a s f a r a s i n f e re n c e m e c h a n i s m s a re c o n c e r n e d . I t b e c a m e c l e a r,h o w e v er , t h a t a l o t o f c o m m o n s e n s e re a s o n i n g is n o t d e d u c t i v e i n n a t u r e .I n p a r t ic u l a r , m a n y in f er e n ce s h u m a n s d r a w a l l t h e t i m e a r e u n c e r t a i n i n s o m es e ns e a n d m a y t h e re f o r e b e d e fe a si bl e if n e w i n f o r m a t i o n b e c o m e s a v a il a b le . T h ep r o t o t y p i c a l e x a m p l e is t h e a s s u m p t i o n t h a t b i r d s n o r m a l l y f ly a n d i f s o m e o n et e l l s m e a b o u t a b i r d c a l l e d T w e e t y , t h e n , k n o w i n g n o t h i n g e l s e , I c o n c l u d e t h a tT w e e t y f li es . L a t e r o n , i f I f i n d o u t t h a t T w e e t y i s i n d e e d a p e n g u i n , I w i t h d r a wm y e a r li e r c o n c lu s i o n w i t h o u t h e s i t a t i o n . T h e r e a r e e s s e n t i a l l y t w o m a i n r e s e a rc hf ie l d s t h a t t r y t o f o r m a l i z e s u c h r e a s o n i n g , o n e w h i c h i s b a s e d o n p r o b a b i l i t yt h e o r y ( se e, fo r e x a m p l e , [ P e ar l, 1 9 88 ]) a n d a n o t h e r w h i c h d i r e c t l y m o d e l s non-monotonic reasoning by m o d i f y i n g c l a ss i ca l l og i c i n o n e w a y o r a n o t h e r ( se e , f o re x a m p l e , [ B r e w k a , 1 9 9 1 ] ) . W h i l e p r o b a b i l i s t i c m e t h o d s a r e n o t d e a l t w i t h a t a l li n th i s v o l u m e , n o n m o n o t o n i c r e a s o n i n g r e ce iv e s a f a i r ly b r o a d c o v e r a ge [ B a a d e ran d H o l l u n d e r , 1 9 94 , G o t t l o b , 1 99 4, K ak as , 1 9 9 4 , K a l i n s k i , 19 94 , N i em e l ~ an dR i n t a n e n , 1 99 4, W e y d e r t , 1 99 4]. E x c e p t f o r M c C a r t h y ' s [ 19 8 0] C i r c u m s c r i p t i o n ,t h e m a i n f o r m a l i s m s o n n o n m o n o t o n i c r e a s o n i n g a r e r e p r e se n t e d i n t h is v o l-u m e . B a a d e r a n d H o l l u n d e r [ 19 9 4] d i s cu s s e x t e n d i n g t e r m i n o l o g i c a l l o g ic s u s i n gR e i t e r ' s [ 19 80 ] D e f a u l t L o g ic ( D L ) . K a k a s e x t e n d s D L b y a p p l y i n g i d e a s f r o ma b d u c t i v e l o gi c p r o g r a m m i n g t o it . G o t t l o b [1 99 4] r e l a t e s D L a n d M o o r e ' s [1 98 5]A u t o e p i s t e m i c L o g i c ( A E L ) b y s h o w i n g h o w t o f a i t h f u l l y t r a n s l a t e D L t h e o ri e si n t o A E L t h e o r i e s . B o t h K a l i n s k i [1 9 94 ] a n d N i em e l~ i a n d R i n t a n e n [1 99 4] a r ec o n c e r n e d w i t h c o m p l e x i t y i ss u es , t h e f o r m e r b y c o n s i d e r i n g a w e a k e r f o r m o fA E L a n d t h e l a t t e r b y c o n s i d e ri n g o n l y A E L t h e o r i e s o f a s p e ci al f o r m ( w i t ha p p l i c a ti o n s t o o t h e r n o n m o n o t o n i c f o r m a l i s m s a s w e l l) . F i n a l ly , W e y d e r t [1 99 4]p r e s e n t s r e s u l t s o n n e s t e d c o n d i t i o n a l s . T h i s w o r k i s i n t h e t r a d i t i o n o f m o d e l i n gn o n m o n o t o n i c i n f e re n c e s o n t h e b a s i s o f c o n d i t i o n a l l o gi c s s u c h a s [ L ew i s, 1 97 3,A d ams , 1 9 7 5 ] .

A p a r t f r o m p r o b a b i l is t ic a n d n o n m o n o t o n i c r e a s o n in g , t h e r e a r e m a n y o th e rf o r m s s u c h a s f u z z y , i n d u c t i v e , a b d u c t i v e o r a n a l o g i c a l r e a s o n i n g . O f t h o s e t h el a t t e r t w o a r e re p r e s e n t e d h e r e w i t h o n e p a p e r e a c h . C o n s o l e a n d D u p r e [ 19 94 ]a d d r e s s a b d u c t i o n , w h i c h is c o n c e r n e d w i t h f i n d i n g p l a u s i b l e e x p l a n a t i o n s f o r a


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g i v e n o b s e r v a t i o n . I n p a r t i c u l a r , t h e y a d d r e s s t h e p r o b l e m o f f i n d i n g e x p l a n a t i o n sa t d i f f e r e n t l e v e ls o f a b s t r a c t i o n . M y e r s a n d K o n o l i g e [ 19 9 4] d i s cu s s r e a s o n i n gw i t h a n a l o g i c a l r e p r e s e n t a t i o n s s u c h a s m a p s . T h e y a r e p a r t i c u l a r l y c o n c e r n e dw i t h i n t e g r a t i n g b o t h a n a l o g i c a l a n d s y m b o l i c ( s e n t e n t i a l ) r e p r e s e n t a t i o n s .

2 .3 E v o l v i n g K n o w l e d g eS i n c e k n o w l e d g e b a se s a r e h a r d l y e v e r s ta t i c , d e v i s i n g m e t h o d s f o r i n c o r p o r a t i n gn e w i n f o r m a t i o n i n to a k n o w l e d g e b as e i s o f g r e a t i m p o r t a n c e i n K R r e s ea r c h .T h i s p r o b l e m , o f t e n r e f e r r e d t o a s belief revis ion, i s p a r t i c u l a r l y c h a l l e n g i n g i ft h e n e w i n f o r m a t i o n c o n f li c ts w i t h t h e c o n t e n t s o f t h e o l d k n o w l e d g e b a s e . O v e rt h e p a s t d e c a d e , s u b s t a n t i a l p r o g r e s s h a s b e e n m a d e o n t h e t o p i c o f b e l i e f r e -v i s io n , p a r t i c u l a r l y s i n c e t h e g r o u n d - b r e a k i n g w o r k b y A l c h o u r r S n , G ~ i rd e n fo r s,a n d M a k i n s o n [ 1 9 8 5 ] , w h o p r o p o s e p o s t u l a t e s w h i c h a n y r a t i o n a l r e v i s i o n o p -e r a t o r s h o u ld o b e y ( n o w r e f er r ed t o as A G M - p o s t u l a t e s ) . L a t e r , K a t s u n o a n dM e n d e l z o n [ 1 9 9 1 ] i n t r o d u c e a n i m p o r t a n t d i s t i n c t i o n b e t w e e n r e v i s i n g a k n o w l -e d g e b a s e, w h i c h r e fe r s t o i n c o r p o r a t i n g n e w i n f o r m a t i o n a b o u t a s t a t i c w o r l d ,a n d u p d a t i n g i t, w h e r e t h e n e w i n f o r m a t i o n re f le c t s c h a n g e s i n t h e w o r l d . T h e ya ls o p r o p o s e a s e t o f r a t i o n a l i t y p o s t u l a t e s f o r u p d a t e o p e r a t o r s . I n th i s v o l u m e ,B o u t i l i e r [1 99 4] a n d N e j d l a n d B a n a g l [ 19 94 ] p r e s e n t n e w r e s u l t s f o l l o w i n g t h i sl in e o f r e s e a rc h . N e j d l a n d B a n a g l d e f in e s u b j u n c t i v e q u e r ie s f o r k n o w l e d g e b a s e si n t h e c a s e o f b o t h u p d a t e a n d r e v i s io n . I n p a r t i c u l a r , t h e y s h o w t h a t t h e i r q u e r ys e m a n t i c s f o r r e v i si o n a n d u p d a t e s a ti sf ie s p r e c is e l y t h e A G M - p o s t u l a t e s a n d t h eK a t s u n o - M e n d e l z o n - p o s t u l a t e s , r e s p e c ti v e ly . B o u t i l i e r s h o w s t h a t , i n t h e c o n t e x to f c o n d i t i o n a l l o g ic , b e l i e f r e v is i o n a n d n o n m o n o t o n i c r e a s o n i n g h a v e p r e c i s e l yt h e s a m e p r o p e r t i e s , f u r t h e r s u b s t a n t i a t i n g t h e c l a i m t h a t t h e t w o a r e a s a r ec l o s el y r e l a t e d .

W i t t e v e e n a n d 3 o n k e r [1 9 9 4] a d d r e s s r e v i s io n f r o m a s o m e w h a t d i f fe r e n t a n g l e .H e r e t h e e m p h a s i s i s o n f in d i n g p l a u si b l e e x p a n s i o n s o f l o gi c p r o g r a m s , w h i c ha r e i n c o h e r e n t u n d e r t h e w e l l- f o u n d e d s e m a n t i c s , s u c h t h a t t h e r e v i se d p r o g r a m sa r e n o l o n g e r i n c o h e r e n t .3 C o m m o n s e n s e R e a s o n i n g a s C o m p u t a t i o nO n c e a k n o w l e d g e re p r e s e n t a t io n s c h e m e t o g e t h e r w i t h i t s a s s o ci a te d c o m m o n -s e n se r e a s o n i n g t a s k h a s b e e n f o r m a l i z e d lo g ic a ll y , w e c a n i m m e d i a t e l y m a k eu s e o f t h e c o m p u t a t i o n a l m a c h i n e r y a s s o c i a te d w i t h l o gi c. F o r i n s ta n c e , o n c e w eh a v e i d e n t if i e d t h a t a p a r t i c u l a r r e p r e s e n t a t i o n f o r m a l i s m i s " s i m p l y " a s u b s e t o fs t a n d a r d f i r s t - o r d e r l o g i c , w e k n o w t h a t r e s o l u t i o n ( o r a n y o t h e r c o m p l e t e p r o o fm e t h o d ) i s a m e t h o d t o c o m p u t e a l l t h e v a l id c o n s e qu e n c e s o f a k n o w l e d g e b a s e .I n o t h e r w o r d s , i n s u c h a c as e , c o m m o n s e n s e r e a s o n i n g c o u l d b e r e d u c e d t o aw e l l- k n o w n c o m p u t a t i o n t e c h ni q u e .

H o w e v e r , t h i s p o i n t o f vi e w i s o v e r - s i m p l i f y in g . F i r s t o f M l, o f t e n o n e d e a l sw i t h non- s tandard l og ic s , e . g. , n o n - m o n o t o n i c o r m o d a l l o g ic s , f o r w h i c h s t a n d a r dt e c h n i q u e s d o n o t w o r k . S e c o n d ly , e v e n in t h e c a s e t h a t o n e o n l y h a s a s u b s e t o f


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s t a n d a r d f i rs t -o r d e r lo g ic , i t d o e s n o t m a k e s e n se t o u s e g e n e r a l p r o o f m e t h o d si f s p e c i a li z e d r e a s o n i n g t e c h n i q u e s , t a i l o r e d t o t h e r e s t r i c t e d l a n g u a g e , t u r n o u tt o b e m u c h m o r e ej~icient. I n p a r t i c u l a r , o n e m i g h t b e a b l e t o s p e c i f y m e t h o d st h a t a l w a y s t e r m i n a t e , i . e. , inference algori thms.E f f i c ie n c y i s i n d e e d o n e o f t h e m a j o r p r o b l e m s w h e n w e t u r n l o g ic a l f o r-m a l i z a t i o n i n t o c o m p u t a t i o n . A s is w e l l- k n o w n , e ve n p r o p o s i t i o n a l l o gi c re q u ir e sa l r e a d y s i g n i f ic a n t c o m p u t a t i o n a l r e s o u r c e s - r e a s o n i n g i n p r o p o s i t i o n a l lo g ic i sN P - h a r d . 4 O n t h e o t h e r h a n d , c o m m o n s e n s e re a s o n i n g a p p e a r s t o b e q u i t e f a s tw h e n h u m a n s p e r f o r m i t , a n d , m o r e o v e r , s h o u l d w o r k r e a s o n a b l y f a s t o n c o m -p u t e r s i f t h e s y s t e m i s r e q u i r e d t o b e o f a n y u s e [ L e v es q u e, 1 9 88 ]. I n p a r t i c u l a r ,i f i t is r e q u i r e d t h a t t h e r e a s o n i n g p r o c e s s i s computat ional ly tractable, w e a r eo f t e n f o r c e d to r e s t r i c t t h e e x p r e s s iv e n e s s o f t h e r e p r e s e n t a t i o n l a n g u a g e o r t og i v e u p o n t h e a c c u r a c y o f t h e a n s w e r [ L e v e sq u e a n d B r a c h m a n , 1 98 7].R e s e a r c h q u e s t io n s c o m i n g u p i n th i s c o n t e x t a r e:

1 . C a n w e s p e c i f y a n i n f e r e n c e a l g o r i t h m s f o r t h e r e a s o n i n g t a s k ?2 . W h a t i s t h e c o m p u t a t i o n a l c o m p l e x i t y o f t h e r e a s o n i n g t a s k ?3 . H o w c a n w e a c h i e v e t r a c t a b i l i t y ?

3 .1 I n f e r e n c e A l g o r i t h m sA s i s e v i d e n t f r o m m o s t p a p e r s , t h e f o r m a l i z a t i o n o f a c o m m o n s e n s e r e a s o n in gt a s k a s a f o r m o f lo g i c a l i n f e re n c e i s u s u a l l y n o t o v e r w h e l m i n g l y d if f ic u l t , p r o -v i d e d a p p r o p r i a t e f o r m a l t e c h n iq u e s a n d t o o l s a r e e m p l o y e d . F o r in s t a n c e , t h es e m a n t i c s o f a t e r m i n o l o g i c a l l o g ic e x t e n d e d b y o p e r a t o r s t o e x p r e s s c o l le c t iv ee n t i t i e s a n d r e l a t i o n s [ A l lg a y e r a n d P r a n c o n i , 1 9 94 ] c a n b e s p e c i f ie d o n l es s t h a nh a l f a p ag e . W h a t a p p e a r s t o b e m u c h m o r e i n v o lv e d i s t h e s p e c i f ic a t io n o f a na p p r o p r i a t e r e a s o n i n g t e c h n i q u e .

A s p o i n t e d o u t a b o v e , o n e c o u ld e m p l o y s t a n d a r d p r o o f t e c h n i q u e s i f t h ef o r m a l i s m u n d e r c o n s i d e r a t io n is (a n o t a t i o n a l v a r ia n t o f ) a s u b s e t o f s t a n d a r df i rs t -o r d e r l og ic . H o w e v e r, u s u a l ly w e d o n o t w a n t a n a r b i t r a r y m e t h o d , b u t a nalgori thm t h a t i s a s e f fi c ie n t a s p o s s i b le - a p r o b l e m t h a t i s a d d r e s s e d b y m o s to f t h e p a p e r s i n t h is v o l u m e .

A l l g a y e r a n d F r a n c o n i [ 19 94 ], fo r in s t a n c e , s h o w e d i n t h e i r p a p e r t h a t i t i sp o s s i b l e t o e x t e n d t h e t a b l e a u - b a s e d t e c h n i q u e i n t r o d u c e d b y S c h m i d t - S c h a u t 3a n d S m o l k a [ S c h m i d t - S c h a u B a n d S m o l k a , 19 91 ] t o t e r m i n o l o g i c a l l o gi c s c o n t a i n -i n g o p e r a t o r s f o r c o l le c t iv e e n t i ti e s , p r o v i d i n g u s w i t h a s o u n d , c o m p l e t e , a n dt e r m i n a t i n g m e t h o d f o r r e a so n i ng i n t h i s la n g u a g e .

B a n d e r a n d H o l l u n d e r [ 1 99 4 ] a l so s t a r t w i t h t e r m i n o l o g i c a l l o gi c s, b u t e x -t en d t h e s e b y i n co rp o ra t i n g d e fau l t l o g i c [Re i t e r , 1 9 8 0 ] , i . e . , i n t h i s ca s e i t i s n o tp o s s i b l e t o u s e s t a n d a r d f i r s t- o r d e r lo g i c m e t h o d s . H o w e v e r , a s t h e y a r e a b l e t os h o w , i t i s p o s s i b l e t o c o m b i n e t h e t a b l e a u - b a s e d r e a s o n i n g t e c h n i q u e s f o r t e r -m i n o l o g i c a l l og ic s w i t h r e ~ o n i n g t e c h n i q u e s d e v e l o pe d f o r d e f a u l t l og ic s [ J u n ke r4 Consu l t , e .g ., [Garey and Johnson , 1979] fo r an in t roduct ion to com puta t iona l com-plexi ty theory.


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a n d K o n o l i g e , 1 99 0 , S c h w i n d a n d R i s c h, 199 1] i n a n a l m o s t s t r a i g h t f o r w a r d w a y ,l e a d in g t o a n i n fe r e n c e a l g o r i th m f o r t h e c o m b i n e d f o r m a l i s m . I t s h o u l d b e n o t e dt h a t i n o r d e r t o g u a r a n t e e d e c i d a b i li t y , i t is n e c e s s a r y t o u s e a s o m e w h a t n o n -s t a n d a r d i n t e r p r e t a t i o n o f o p e n d e f a u l ts , t h o u g h . S in c e, a s s h o w n b y B a a d e r a n dH o l l u n d e r , t h e s t a n d a r d i n t e r p r e t a t i o n o f o p e n d e f a u l t s n o t o n l y l ea d s t o u n d e -e i d a b i l i t y b u t a l s o t o c o u n t e r - i n t u i t i v e r e su l ts , g i v i n g u p t h i s i n t e r p r e t a t i o n d o e sn o t s e e m t o b e m u c h o f a s a cr if ic e .

3 .2 C o m p u t a t i o n a l C o m p l e x i t y o f R e a s o n i n gA n i n fe r e n ce a l g o r i t h m fo r a p a r t i c u l a r c o m m o n s e n s e r e a so n i n g t a s k d e m o n -s t r a t e s t h a t t h a t t h e r e is o n e w a y t o tu r n t h is t as k i n t o c o m p u t a t i o n . H o w e v e r ,i t d o e s n o t a n s w e r t h e q u e s t i o n w h e t h e r t h i s is th e m o s t e f fi c ie n t w a y . I n o r d e r t oa n s w e r t h is q u e s t io n , c o m p u t a t i o n a l c o m p l e x i t y t h e o r y c a n b e u s e d f o r a n a l y z i n gt h e i n h e r e n t d i f fi c u lt y o f t h e p r o b l e m . S u c h a n a n a l y s i s c a n g u i d e t h e s e a r c h f o rm o r e e ff ic i en t a l g o r i t h m s o r f o r a r e f o r m u l a t i o n o f t h e r e as Q n i ng p r o b l e m i n aw a y t h a t r e n d e r s r e a s o n i n g m o r e e ff ic ie n t. F i n a ll y , a c o m p u t a t i o n a l c o m p l e x i t ya n a l y s i s c a n b e u s e d t o c o m p a r e a n d c o n t r a s t d i f f e r e n t r e a s o n i n g p r o b l e m s .

F o r i n s t a n c e , D o n i n i e t a l [1 99 4] s t u d y t h e e x t e n s i o n o f t e r m i n o l o g i c a l l o g ic sb y a n e p i s te m i c o p e r a t o r a n d s h o w t h a t t h is o p e r a t o r d o e s n o t i n c re a s e t h e c o m -p u t a t i o n a l c o m p l e x i t y o f r e a s o n in g in o n e o f t h e s t a n d a r d t e r m i n o l o g i c a l lo g ic s( t h e s o - c a ll e d .A /:C l a n g u a g e [ S c h m i d t - S c h a u B a n d S m o l k a , 1 9 9 1] ). F u r t h e r m o r e ,D o n i n i e t a l [ 1 9 9 4 ] a r e a b l e t o s h o w t h a t i n s o m e r e l e v a n t s p e c i a l c a s e s t h ec o m p l e x i t y g o es e ve n d o w n f r o m c o - N P - h a r d n e s s t o p o l y n o m i a l t i m e .

K a u t z a n d S e l m a n [ 1 9 9 4 ] a n a l y z e t h e c o m p u t a t i o n a l p r o b l e m s a r i s i n g w h e na p p r o x i m a t i n g a r b i t r a r y p r o p o s i t i o n a l th e o r ie s b y H o r n t h e o ri e s. T h e y s h o w t h a ts u c h a H o r n t h e o r y m a y s o m e t i m e s b e o f e x p o n e n t i a l s iz e a n d t h a t i t is u n l i k e l yt h a t a d e n s e r e p r e s e n t a t i o n c a n b e f o u n d i n a l l c a s e s .

A f in a l e x a m p l e f o r t h e u s e o f c o m p u t a t i o n a l c o m p l e x i t y t h e o r y is t h e p a p e rb y G o t t l o b [199 4]. A l t h o u g h t h i s p a p e r is n o t b y i ts e l f a p a p e r o n c o m p u t a t i o n a lc o m p l e x i t y a n a ly s is o f c o m m o n s e n s e r e a so n i ng , i t m a k e s u s e o f c o m p u t a t i o n a lc o m p l e x i t y r e s u l t s [ G o t t l o b , 1 992] t h a t s h o w t h a t t h e t h r e e m a i n f o r m s o f n o n -m o n o t o n i c r e a s o n i n g a ll h av e t h e s a m e c o m p l e x i t y , w h i c h i m p l ie s t h a t t h e r em u s t e x i s t ( p o l y n o m i a l ) t r a n s l a t io n s b e t w e e n t h e s e f o r m a l i s m s . B a s e d o n t h i so b s e r v a t i o n , G o t t l o b d e v e l o p s a t r a n s l a t i o n f r o m d e f a u l t l o g i c t o a u t o e p i s t e m i cl o g ic t h a t i s q u i t e i n t e r e s t i n g .

3 .3 T h e E x p r e s s i v e n e s s v s . E f f i c i e n c y T r a d e o f fI f a r e a s o n in g p r o b l e m c a n b e s h o w n t o r e q u i r e t i m e t h a t i s n o t p o l y n o m i a l i nt h e si ze o f t h e p r o b l e m d e s c r ip t io n ( u n d e r t h e a s s u m p t i o n t h a t N P c P ) , t h isi m p l i e s t h a t i n t h e wors t case w e w i l l n o t g e t a n a n s w e r i n t o l e r a b l e t i m e w h e nt h e p r o b l e m d e s c r i p t i o n g r o w s b e y o n d a c e r t a i n ( u s u a l l y m o d e r a t e ) s i z e . O fc o u r se , if t h e p r o b l e m d e s c r i p t io n s a r e a l m o s t a l w a y s s m a l l , s u c h c o m p u t a t i o n a lc o m p l e x i t y r e s u l t s a r e i rr e l e v a n t . H o w e v e r , w e u s u a l l y w a n t t o d e a l w i t h m o r e


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t h a n 2 0 c o n c e p t s o r 1 0 d e f a u l t r u l e s . S o w e s h o u l d c o n s i d e r t h e p o s s i b i l i t y o fw o r s t c a s e s f o r m o d e r a t e l y s i z e d p r o b l e m d e s c r i p t i o n s .O n e w a y t o e x c l u d e w o r s t c a s e s i s t o r e s t r i c t t h e e x p r e s si v e n e ss o f t h e r e p re -

s e n t a t i o n l a n g u a g e t h e r e a s o n in g t a sk h a s t o d e a l w i t h . B r a c h m a n a n d L e v e sq u e ,f o r e x a m p l e , s h o w e d t h a t e x c l u d i n g a p a r t i c u l a r o p e r a t o r f r o m a t e r m i n o l o g i c a ll og ic re d u c e s t h e c o m p l e x i t y o f r e a s o n in g f r o m N P - h a r d n e s s t o p o l y n o m i a l i t y[ B r a c h m a n a n d L e v e s q u e , 19 84 , L e v e s q u e a n d B r a c h m a n , 1 98 7]. S u b s e q u e n t i n -v e s t i g a t i o n s a l o n g t h i s l i n e [ D o n i ni et al . , 1 9 91 ] h a v e s h o w n t h a t r e q u i ri n g p o l y -n o m i a l i t y o f t h e i n f e re n c e a l g o r i t h m l e a d s t o a s e v e re r e s t r i c t i o n o n t h e p o s s i b lec o n s t r u c t s o n e c a n u s e .

A l t h o u g h t h e r e h a v e b ee n s t r o n g a r g u m e n t s a b o u t t h e u s e f u ln e s s o f a c h i e v i n ge f f i c ien cy b y r e s t r i c t i n g t h e ex p re s s i v en es s [D o y l e an d P a t i l , 1 9 9 1] , t h e re s eem st o b e n e v e r t h e l e s s a c o n s e n s u s t h a t i t i s u s e f u l t o a n a l y z e s p e c ia l c a s e s o f g e n e r a lr e a s o n i n g p a t t e r n s t h a t c a n b e s o lv e d m o r e e a si ly t h a n t h e g e n e r a l p r o b l e m ,p r o v i d e d t h e s p e c i a l c a s e s a r e r e l e v a n t . M o r e o v e r , r e s t r i c t i n g t h e ex p re s s i v en es sc a n m e a n a n u m b e r o f th i n g s t h a t a r e q u i t e d i ff e r e n t f r o m , f o r e x a m p l e , e x c l u d i n ga p a r t i c u l a r o p e r a t o r f r o m a r e p r e s e n t a t io n l a n g u a g e .

F o r in s t a n c e , i n s t e a d o f c o n s i d e r i n g a r e p r e s e n t a t i o n l a n g u a g e w i t h l e ss c on -s t r u c t s, i t m a k e s s o m e t i m e s s e ns e t o us e a l a n g u a g e w i t h m o r e c o n s t r u c t s b u tw i t h r e s t r i c t i o n s o n t h e s t r u c t u r e o f a ll o w e d e x p r e s si o n s . D o n i n i e t a l [ 19 94 ] s h o wt h a t e n l a r g i n g a t e r m i n o l o g i c a l l og ic w i t h a n e p i s te m i c o p e r a t o r f o r b u i l d in g c o n -c e p t s t h a t a r e u s e d a s q ue r ie s a n d r e s t ri c t in g t h e f o r m s o f t h e q u e r y c a n i n d e e dl e a d t o a m o r e n a t u r a l r e a s o n i n g t a s k w h i c h i s a l s o m o r e e f f i c i e n t .

T h e w o r k b y M y e r s a n d K o n o l i g e [ 19 94 ] a ls o e x t e n d s t h e r e p r e s e n t a t i o n a lf r a m ew o rk ( f i r s t -o rd e r l o g ic ) i n o rd e r t o ach i ev e e ff ic i en cy . In t h i s ca s e , h o w ev e r ,t h e a i m i s n o t t o g u a r a n t e e w o r s t - c a se e f f ic i e n cy i n a l l c a se s , b u t t o p r o v i d es p e c ia l m e a n s f o r r e p r e s e n t in g k n o w l e d g e a b o u t o n e p a r t i c u l a r d o m a i n - s p a t i a lk n o w l e d g e - t h a t c a n b e m o r e n a t u r a l l y r e p re s e n t e d a n d m o r e e f fi c ie n tl y r e a s o n e da b o u t u s i n g a n a l o g i ca l r e p r e s e n ta t i o n s , w h i c h a re a l s o m u c h m o r e r e s t r ic t e d t h a ng e n e r a l p r o p o s i t i o n a l r e p r e s e n t a t i o n s . T h e m a i n p r o b l e m M y e r s a n d K o n o l i g ei d e n t i f y a n d s o lv e i s th e i n t e g r a t i o n o f a n a l o g i c a l r e a s o n i n g w i t h t h e g e n e r a lf r a m e w o r k o f r e a s o n i n g i n fi r s t- o r d e r l o g ic .

T h e p a p e r b y N i em e l s a n d R i n t a n e n [19 94 ] a i m s a g a i n a t g u a r a n t e e i n g p o l y -n o m i a l r u n t i m e i n a l l c a se s b y r e s t r i c t in g e x p r e s s i v e p o w e r . A s i n t h e c a s es a b o v e ,h o w e v e r , t h e y d o n o t r e s t r i c t t h e e x p r e s s iv e p o w e r b y d i s a l l o w i n g l o g i ca l o p e r a -t o r s i n A E L t h e o ri e s, b u t t h e y c o n s id e r r e s t ri c t io n s o n t h e f o r m o f t h e t h e o ri e s.I n p a r t i c u l a r , t h e y s h o w t h a t r e a s o n i n g in s tr a t if i e d A E L H o r n t h e o r ie s c a n b ed o n e i n p o l y n o m i a l t im e .

3 .4 T h e A c c u r a c y v s . E f f i c i e n c y T r a d e o f fI f t h e e x p r e ss i v e n e ss o f a r e p r e s e n t a t i o n c a n n o t h e r e s t r i c t e d , o t h e r m e a n s fo rg e t t i n g t i m e l y a n s w e r s a r e c a l le d fo r . U s u a l ly , o n e g i v e s u p o n t h e q u a l i t y o ra c c u r a c y o f a n a n s w e r, f o r e x a m p l e , b y r e s t ri c t in g t h e p r o c e ss in g t i m e o r b ye m p l o y i n g i n c o m p l e t e r e a s o n in g m e t h o d s . W h i l e t h is m a y l e a d t o t h e d e s ir e dr u n t i m e b e h a v i o r , i t r a is e s t h e q u e s t i o n a s t o h o w f a r w e c a n s t i ll t r u s t a n s w e r s


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from a representation and reasoning system. In other words, we are seeking ap r in c ip led description of the reasoning capabilities of an incomplete reasoner.Kautz and Selman [1991] addressed this problem by a "knowledge compila-tion" technique. They propose to compute (off-line) Horn theories that approx-imate the logical contents of a given arbitrary theory. As mentioned above, thisapproximat ion can lead to computationa l problems in itself [Kautz and Selman,1994]. Kautz and Selman show that the approximating theory can become verylarge, and although there are sometimes ways around this problem, they canshow that it is very unlikely that dense representations of a approx imating Horntheory exist in all cases. Nevertheless, their approximation scheme appears to beinteresting since instead of general Horn theories one may aim for more restrictedforms of such theories which can be polynomial ly bounded in size.

Greiner and Schuurmans [1994] address the mu l t ip l e ex t en s io n problem of de-faul t reasoning [Reiter, 1987], which is known to be one source of computationalcomplexity in default reasoning [Gottlob, 1992, Nebel, 1991]. They propose toapproximate default reasoning by ordering the defaults linearly, where the par-ticular order chosen is intended to be "opt imally correct." As they show, it is notpossible to compute such an ordering in polynomial time, but they approx imatesuch an ordering by computing a locally optimal ordering.The paper by Witteveen and Jonker [1994] applies a similar method toachieve tractability for revising logic programs. They show that a globally min-imal revision cannot be computed in polynomial time, but a locally minimalrevision can well be computed in polynomial time.

4 Ou t lookThe collection of papers in this book does certainly not give a complete overviewof the research going on at providing foundations for knowledge representationand reasoning. For instance, probabilistic approaches are not represented at all.Nevertheless, the set of papers in this book covers a wide range of topics inthe area of foundational KR&R research and highlights the common researchmethodology, namely, to analyze representation and reasoning tasks from a log-ical and co mp u ta t io n a l perspective. As already mentioned in the Introduction,this research methodology does most probably not lead to any immediate ben-efit in the sense that we can build faster or better reasoning systems. However,by providing the theoretical underpinning for KR&R systems, this research willhelp us understand where and what the limits of representation and reasoningare and how we can guarantee a reasonable behavior of KR&R systems.

R e f e r e n c e s[AAAI-90, 1990] Proceedings of the 8th Na tional C onference of the A m erica n A ssoci-ation ]or Artificial Intelligence, Boston, MA, August 1990. MIT Press.[Adams, 1975] E. W. Adams. The Log ic of Conditionals. Reidel, Dordrecht, Holland,1975.


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]0[Alchourr6n et al., 1985] Car los E. Alchourr6n , Pe te r Gs and Dav id Makin -son . O n th e log ic o f theory change: Pa r t i a l m eet con t rac t ion and rev i sion func t ions .Journa l o f Symbol ic Log ic , 50(2):510-530, June 1985.[Allen e t a l . , 1991] J . A . Allen , R. Pikes , a nd E. Sandew all , edi tors . Princ ip les o fKnowledge R epresen ta t ion a nd Reason ing: Proceed ings o f the ~nd In tern a t iona l Con-ference, C am b r i d g e , M A, A p r il 1 9 9 1 . M o rg an Kau fm an n .[Allgayer a nd Franconi , 1994] J i i rgen Allgayer and En rico Franc oni . Col lect ive ent i t iesand re la t ions in concep t l anguages . In La kem eyer and Neb el [1994] .[Baader and HoUunder , 1994] Franz Bander and Bernhard Hol lunder . Comput ing ex-tens ions o f t e rmino log ica l defau l t theor ies. In L akem eyer and N ebel [1994] .[Bet t in i, 1994] C . Bet t in i . A fo rmal iza t ion o f in te rva l -based tem pora l subsum pt ion infi rs t order logic. In La kem eyer and Neb el [1994].[Bou t il i e r, 1994] Cra ig Bou t i l ie r . Norm at ive , sub junc t ive and au to ep is tem ic defau l t s.In L akem eyer and Nebel [1994].[B rach m an an d Lev esq u e , 1 9 8 4 ] R o n a l d J . B rach m an an d He c t o r J . Lev es q ue . Th et rac ta b i l i ty o f subsum pt ion in f rame -based descr ip t ion languages . In Proceedings ofthe th Na tiona l Conference of the Am eric an A ssoc iat ion for Ar t i f ic ia l In tel l igence,pages 34-37 , Aus t in , TX, 1984 .[Brachm an an d Levesque , 1985] Ronald J . Brach ma n and H ecto r J . Levesque , ed i to rs .Readings in Knowledge Represen ta t ion . Mo rgan K aufm ann , L os Al tos , CA, 1985 .[B rach m an et al., 1989] R . Brachman , H. J . Levesque , and R. Rei te r , ed i to rs . P r in c i -p les o f Knowledge Represen ta t ion an d Reason ing: Proceed ings o f the 1s t In terna t ion a l

Conference, To ro n t o , ON, M ay 1 9 8 9 . M o rg an K au fm an n .[B rach m an , 1 9 9 0 ] R o n a ld J . B rach m an . Th e fu t u re o f k n o wled g e r ep re s en t a ti o n . InAAAI-90 [1990], pages 1082-1092.[Brewka, 1991] Gerha rd Brewka. Nonmonoton ic Reason ing: Log ica l Founda t ions o fC o mmo n s en s e . Cambridge Univers i ty Press , Cambr idge , UK, 1991 .[Conso le and Dupr6, 1994] Luc a Conso le and Dan ie le Dupr6 . Ab duct ive reason ingwi th ab s t rac t ion ax ioms . In Lake me yer and N ebel [1994] .[Donini e t a l . , 1991] Franc esco M. Donini , M aurizio L enzerin i , D aniele N ardi , a ndW ern e r Nu t t . T rac t ab l e co n cep t l an gu ag es. In Proceedings of the 1Pth Interna-t iona l Jo int Conference on Art i f ic ia l In tel l igence, pages 458-465 , Sydney , Aus t ra l ia ,

Au g u s t 1 9 9 1 . M o rg an Kau fm an n .[Donini et al., 1994] Francesco Donin i, Manr iz io L enzer in i , Dan ie le N ard i , A ndr eaSchaerf , and W erner Nut t . Quer ies , ru les and def in i tions as ep i s temic sen tencesin conc ep t l anguages. In Lake me yer and Nebel [1994].[Doyle and Pa t t i , 1991] Jon D oyle and R ame sh S . Pa t i l . Tw o theses o f knowledge rep -resen ta t ion : Language res t r ic t ions , t axono mic c lass if i ca tion , and th e u t i l ity o f rep re-sen ta t ion serv ices . Arti f ic ia l In tel l igence, 48(3):261-298, Apri l 1991.[Garey and Johnson , 1979] Michael R . G arey an d D av id S. Johnson . C o mp u te r s a n dIn t r a c ta b i l i t y - -A Gu id e to t h e Th eo r y o f N P -C o m p le t en ess . Freeman , San Franc i sco ,CA, 1979.[Got t lob , 1992] Georg Got t lob . Complex i ty resu l t s fo r nonmonoton ic log ics . Jo u r n a lfor Log ic and Computa t ion , 2(3), 1992.[Got t lob , 1994] Georg G ot t lob . T he pow er o f bel ie fs o r t rans la t ing defau l t log ic in tos tan dard au toep is tem ic log ic . In Lakem eyer and N ebel [1994].[Gre iner and Schuurmans , 1994] Russel l G re iner and Dale Schuurm ans . Learn ing anop t imal ly ac cura te rep rese n ta t ion sys tem . In L akem eyer and N ebel [1994] .


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A u g u s t 1 9 7 7 .[ J u n k e r a n d K o n o l i g e , 1 9 9 0 ] U l r i c h J u n k e r a n d K u r t K o n o l i g e . C o m p u t i n g e x t e n s i o n so f a u t o e p i s t e m i c a n d d e f a u l t lo g ic s w i t h a t r u t h m a i n t e n a n c e s y s t e m . I n A A A I - 9 0

[ 1 9 9 0 ] , p a g e s 2 7 8 - 2 8 3 .[K a k a s , 1 9 9 4 ] A . C . K a k a s . D e fa u l t r e a s o n in g v i a n e g a t i o n a s f a il u r e. I n L a k e m e y e rand Nebe l [1994] .[K a l i n s k i , 1 9 9 4 ] J i i r g e n K a l i n s k i . W e a k a u t o e p i s t e m i c r e a s o n i n g a n d w e l l - fo u n d e d s e -m a n t i c s . I n L a k e m e y e r a n d N e b e l [1 9 94 ].[ K a t s u n o a n d M e n d e lz o n , 1 9 9 1 ] H i r o f u m i K a t s u n o a n d A l b e r t o O . M e n d e l zo n . O n t h ed i f fe rence be tween upda t ing a knowledge base and rev i s ing i t . In Al len e t a J . [1991] ,

p a g e s 3 8 7 -3 9 4 .[ K a u t z a n d S e lm a n , 1 9 9 4 ] H e n r y K a u t z a n d B a r t S e lm a n . F o r m i n g c o n c e p t s f o r f a s ti n f e re n c e . I n L a k e m e y e r a n d N e b e l [ 19 94 ].[L a k e m e y e r a n d N e b e l , 1 9 9 4 ] G e rh a rd L a k e m e y e r a n d B e rn h a rd N e b e l , e d i t o r s. F o u n -da t ions o f Knowledge Represen ta t ion and Reason ing . Spr inger -Ver lag , Ber l in , He i -de lbe rg , New York , 1994 .[L e v e sq u e a n d B ra c h m a n , 1 9 8 7 ] H e c t o r J . L e v e s q u e a n d R o n a l d J . B ra c h m a n . E x -p re s s i v e n e s s a n d t r a c t a b i l i t y i n k n o w l e d g e r e p re s e n t a t i o n a n d r e a s o n i n g . C o mp u -tat ional In tel l igence, 3:78-93 , 1987.[L e v es q u e , 1 9 8 8 ] H e c t o r J . L e v e s q u e . L o g ic a n d t h e c o m p l e x i t y o f r e a s o n i n g . Jo u r n a lof Phi losophica l Logic, 17:355-389, 1988.[Lewis, 1973] Da vid K. Lewis . Counterfactuals . H a rv a rd U n i v e r s i t y P re s s , C a m b r i d g e ,MA, 1973 .[Lin , 1 9 9 4 ] Y u e n Q . L in . A c o m m o n -s e n s e t h e o ry o f t i m e . I n L a k e m e y e r a n d N e b e l[1994].[M c C a r t h y , 1 9 6 8 ] J o h n M c C a r t h y . P ro g ra m s w i t h c o m m o n s e n se . I n M . M i n s k y , e d i -

t o r , Semant ic In format ion Process ing , p a g e s 4 0 3 -4 1 8 . M I T P r e ss , C a m b r i d g e , M A ,1968.[ M c C a r t h y , 1 9 8 0 ] J o h n M c C a r th y . C i r c u m s c r i p t i o n - - a f o r m o f n o n - m o n o t o n i c r e a so n -ing. Arti f ic ia l In tel l igence, 13(1-2) :27-39 , 1980 .[ M c D e r m o t t , 1 9 7 8 ] D r e w V . M c D e r m o t t . T a r s k i a n s e m a n t ic s , o r n o n o t a t i o n w i t h o u td e n o t a t i o n ! Cognit ive Science, 2(3) :277-282 , Ju ly 1978 .[M o o re , 1 9 8 5 ] R o b e r t C . M o o re . S e m a n t i c a l c o n s i d e ra ti o n s o n n o n m o n o t o n i c l og ic .Arti f ic ia l In tel l igence, 25:75-94, 1985.[M y e r s a n d K o n o l i g e , 1 9 9 4 ] K a re n M y e r s a n d K u r t K o n o l i g e . R e a s o n i n g w i t h a n ~ d o g -ica~ repres en ta t ion s . In Lak em eye r and Ne be l [1994].[N e b e l et al., 1992] B . Nebe l , W. Swar tou t , and C . R ich , ed i to rs . P r in c ipl e s o f K n o w l -edge Representat ion and Reasoning: Proceedings of the 8rd Internat ional Conference,C a m b r i d g e , M A , O c t o b e r 1 9 9 2 . M o r g a n K a n f m a n n .[N e b e l, 1 9 9 1 ] B e rn h a rd N e b e l . B e l ie f r e v is i o n a n d d e fa u l t r e a s o n in g : S y n t a x -b a s e dapp roach es . In Al len e t a l. [1991] , pag es 417-428 .[N e j d l a n d B a n a g l , 1 9 9 4 ] W o l fg a n g N e j d l a n d M a rk u s B a n a g l . A s k i n g a b o u t p o s s i b i l -i t ie s - r e v i si o n a n d u p d a t e s e m a n t i c s f o r s u b j u n c t i v e q u e r i e s ( e x t e n d e d r e p o r t ) . I n

L a k e m e y e r a n d N e b e l [ 1 9 9 4 ] .[N i e m e l~ a n d R i n t a n e n , 1 9 9 4 ] I l k k a N i e m e l /i a n d J u s s i R i n t a n e n . O n t h e i m p a c t o fs t r a ti f ic a t i o n o n t h e c o m p l e x i t y o f n o n m o n o t o n i c r e a s o n in g . I n L a k e m e y e r a n d N e b e l

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]2[Pear l , 1988] Judea Pear l . Probabilistic Reasoning in Intelligent Systems: Networks ofPlausible Inference. Morgan Kaufmann , San Mateo , CA, 1988 .[Rei te r , 1980] Ray mo nd R ei te r . A log ic fo r defau l t reason ing . Artificial Intelligence,

t3(1):81-132, Apri l 1980.[Rei te r , 1987] Ray mo nd Rei te r . N onm onoton ic reason ing . Annual Review of Comput-ing Sciences, 2, 1987.[Schmidt-SchauB and Smolka, 1991] M anfre d Schmidt-SchauB and G er t Smolka. At-t r ibu t ive concep t descr ip t ions wi th complements . Artificial Intelligence, 48:1-26,1991.[Schwind and Risch, 199 1] (7. Schw ind an d V. Risch. A tab lea u- ba sed chaxa~:terisa-t ion for defaul t logic. In R. Kruse and P. Siegel , edi tors , Symbolic and QuantitativeApproaches to Uncertainty, Proceedings of the European Conference EGSQA U,pages310-317, Marsei lles , Franc e, 19 91. Springer-Verlag.[Se lman a nd K au tz , 19 91] Ba r t Se lman and Henry Ka u tz . Know ledge com pi la t ion us-ing Horn approx imat ions . In Proceedings of the 9th National Conference of the Amer-ican Association for Artificial Intelligence,pages 904-909 , Anaheim, CA, Ju ly 1991 .M I T P r es s.[ S h o h a m a n d C o u s i n s , 1 9 9 4 ] Y o a v S h o h a m a n d S t e v e B . C o u s i n s . L o g i c s o f m e n t a l

a t t i t u d e s i n A I - - a v e r y p r e l i m i n a r y s u r v e y . I n L a k e m e y e r a n d N e b e l [ 1 9 94 ].[Smi th , 1982] Br ian C . Smi th . Reflection and Semantics in a Procedural Language.Ph D thesi s , Massachuse t ts In s t i tu te o f Techno logy , Cam bridge , MA , 1982 . Re por tMIT/LCS/TR-272.[Weyd er t , 1994] Em i l W eyder t . Hy perra t ion a l cond i t iona l s - mo noton ic reason ingabo u t ne s ted defau l t cond i tiona l s. In Lak emey er and Nebel [1994] .[W i t teveen and Jonker , 1994] Cees W i t teveen and Catho l i jn M. Jonker . Rev is ion byexpa nsion in logic programs. In Lak em eyer and Neb el [1994].


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C o l l e c t i v e E n t i t i e s a n d R e l a t i o n sin C o n c e p t L a n g u a g es *J i i rgen Al lgayer 1 and Enr ico Franco ni2

x IBM Germany, Software Architectures and TechnologiesSchlo6-Str. 70, D-7000 Stuttgart 1, Germany

(allg ayer@ vnet.i bm.com)2 I stit uto per la Ricerca Scientifica e Tecnologica (IRST )

1-38050 Povo TN, Italy([email protected])

A bs tr a c t. Collective entities and collective relations play an import antrole in natural language. In order to capture the full meaning of sen-tences like "The Beatles sing 'Yesterday' ", a knowledge representationlanguage should be able to express and reason about plural entities - like"the Beatles" and their relationships - like "sing" with any possiblereading (cumulative, distributive or collective).In this paper a way of including collections and collective relations withi na concept language, chosen as the formalism for representing the seman-tics of sentences, is presented. A twofold extension of the A/ZC conceptlanguage is investigated : (1) special relations introduce collective en-tities either out of their components or out of other collective entities,(2) plural quantifiers on collective relations specify their possible read-ing. The formal syntax and semantics of the concept language is given,together with a sound and complete algorithm to compute satisfiabilityand su bsum ptio n of concepts, and to comput e recognition of individuals.An advantage of this formalism is the possibility of reasoning and step-wise refining in the presence of scoping ambiguities. Moreover, manyphenomena covered by the Generalized Quantifiers Theory are easilycaptured within this framework. In the final part a way to include a the-ory of parts (mereology) is suggested, allowing for a lattice -theoretica lapproach to the treatment of plurals.

1 I n t r o d u c t i o n

I n t h i s p a p e r i t i s s h o w n h o w a c o n c e p t l a n g u a g e , i .e , a k n o w l e d g e r e p r e s e n -t a t i o n l a n g u a g e o f t h e K L -O N E f a m i l y - a ls o c al le d F r a m e - B a s e d D e s c r i p t i o n* This paper is a a reduced version of a paper to appear in Minds and Machines,

special issue on Knowledge Representation for Natural Language Processing. Thiswork has been partially supported by the Italian National Research Council (CNR),project "Sistemi Informatici e Calcolo Parallelo", and by the IRST M A I A project.We would like to thank also Alessandro Artale, Werner Nutt and Achille Varzi forthe helpful and incisive discussions we had with them.


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14L a n g u a g e s , T e r m S u b s u m p t i o n L a n g u a g e s , T e r m i n o l o g i c a l L o g i c s , T a x o n o m i cL o g i c s o r D e s c r i p t i o n L o g i c s [3 0] - c a n b e e x t e n d e d i n o r d e r t o r e p r e s e n t a n dr e a s o n a b o u t collective enti t ies o r collections [1 , 9 ] . T h e en r i ch ed co n cep t l an -g u a g e p r o p o s e d h e r e is i n t e n d e d t o f o r m t h e s e m a n t i c a l a n d c o m p u t a t i o n a l m e a n sf o r t h e r e p r e s e n t a t i o n o f p l u r a l s a n d p l u r a l q u a n t i f ie r s in n a t u r a l l a n g u a g e ; o t h e ra p p r o a c h e s i n t h e l i t e r a t u r e i n c l u d e [ 3 , 1 7 , 20, 22, 2 6 , 2 7 ] . A l t h o u g h t h i s w o rkh a s b e e n c o n c e i v e d f o r c o n c e p t l a n g u a g e s , i t c a n b e a l so a p p l i e d t o o t h e r k n o w l -e d g e r e p r e s e n t a t i o n f o r m a l i sm s , s u c h a s C o n c e p t u a l G r a p h s [26 ] a n d I n t e n s i o n a lP r o p o s i t i o n a l S e m a n t i c N e t w o r k s S N e P S [2 4].

A n a n a l y s is o f p l u ra l s i n n a t u r a l l a n g u a g e l e a d s u s t o d i s t in g u i s h a m o n g t w od i f f e r en t ca t eg o r i e s o f p l u r a l en t i t ie s : c l a s s e s an d co l l ec t io n s . C l a s s e s a r e i n v o l v edi n s e n te n c e s l ik e " M e n a re p e r s o n s " , w h e r e t h e N P " m e n " is r e p r e s e n t e d b y m e a n so f t h e c la s s p r e d i c a t e M A N :V x. M A N ( x ) -- . P E R S O N ( x ) .

O n t h e o t h e r h a n d , c o l le c t io n s a r e c o n t i n g e n t a g g r e g a t e s o f o b j e c t s , a n d t h e ys h o u l d b e r e p r e s e n t e d a s t e r m s i n s t e a d o f p r e d i c a t e s , i .e . t h e y s h o u l d b e i n t e r -p r e t e d a t t h e s a m e l e v e l o f i n d i v i d u a l s a s si n gl e e l e m e n t s o f t h e d o m a i n . F o re x a m p l e , t h e l o gi c al f o r m o f t h e s e n t e n c e s " T h e B e a t l e s a r e J o h n , P a u l , G e o r g ea n d R i n g o " a n d " J o h n i s t h e l e a d e r o f t h e B e a t l e s " i s t h e f o ll o w i n g:~ ( b e a t l e s , p a u l ) , ~ ( b e a t l e s , j o h n ) , ~ ( b e a t l e s , r i n g o ) , ~ ( b e a t l e s , g e o r g e ) ,L E D - B Y ( b e at le s, j o h n ) .T h e p l u r a l e n t i t y B e a t l e s i s i n t e rp r e t e d a s a co l l ec t io n , an d i n t h e l o g i ca l f o r mi t d o e s n o t a p p e a r a s a p r e d i c a t e , b u t a s a t e r m , a t t h e s a m e l e v el a s t h e o b j e c t si t i s c o m p o s e d b y .

I n o r d e r t o g i v e a m e a n i n g t o t h e t e r m s d e n o t i n g c o l l ec t io n s , a w e a k e n e df o r m o f S e t T h e o r y - c a ll ed Col lec tion Theory - i s a d o p t e d , a n d t h e d a n g e r o u sl e a p i n t o a s e c o n d o r d e r t h e o r y i s a v o i d e d . I t t u r n s o u t t h a t t h e c o l l e c ti o n t h e o r yis m o r e a d e q u a t e t o r e p re s e n t p l u r a ls t h a n s e t th e o r y , b e c a u s e th e e x t e n s i o n a l i t yp r i n c i p l e d o e s n o t h o l d .

W i t h i n t h e c o l l e c t i o n t h e o r y , plural quanti f iers a r e i n t r o d u c e d , i n o r d e r t oc a p t u r e t h e d i f f e r e n t r e a d i n g s o f a r e l a t i o n w h e n a p p l i e d t o a c o l le c t io n . T h i sa p p r o a c h a ll o w s f o r t h e r e p r e s e n t a t i o n o f ambiguous r e a d i n g s , s o t h a t i n t h ep r e s e n c e o f i n c o m p l e t e i n f o r m a t i o n a c o m p l e t e r e a s o n i n g c a n s t i ll b e c a r r i e d o n .

I n t e r e s t i n g c o n n e c t i o n s w i t h t h e G e n e r a l i z e d Q u a n t i f i e r s T h e o r y [ 5 ] c a n b ed r a w n ; f o r a d e e p a n a l y s i s o f t h e r e l a t i o n s b e t w e e n c o l l e c ti o n t h e o r y a n d G e n e r -a l iz e d Q u a n t i f i e r s T h e o r y p l e a s e r e f e r t o t h e f u l l p a p e r [ 10 ]. I n t h e f u l l p a p e r t h eu s e o f t h i s f o r m a l i s m w i t h i n a n a t u r a l l a n g u a g e d i a lo g u e s y s t e m is e x t e n si v e lyd i s cu s s ed .

A m o r e r a d i c a l d e p a r t u r e f r o m s e t t h e o r y t o r e p r e s e n t c o ll e ct io n s is p r o p o s e di n th e l a s t p a r t o f t h i s p a p e r , i n t r o d u c i n g a n non-ex tens iona l mereo logy [25]. Int h i s w a y , a l a t t i c e - t h e o r e t i c a l a p p r o a c h f o r th e t r e a t m e n t o f p l u r a l s a s in [ 18 ] isp o s s i b l e .


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1 5

T h e p a p e r i s o r g a n i z e d a s f o ll ow s . A t t h e b e g i n n i n g t h e C o l l e c t i o n T h e o r ya n d t h e P l u r a l Q u a n t i f i e r s a re i n t r o d u c e d i n a g e n e r i c l o g ic a l f r a m e w o r k . T h e n i tis p r e s e n t e d h o w t h e s e t h e o r ie s c a n b e m e r g e d i n t o t h e p r o p o s i t io n a l l y c o m p l e t ec o n c e p t l a n g u a g e .A ~:C ; m a n y e x a m p l e s w i ll c l a r if y t h e e x p r e s s i v e p o w e r o f t h en e w l y o b t a i n e d l a ng u a g e A s A n a c c o u n t o f t h e c o m p u t a t i o n a l p r o p e r t i e s o f. A /: C S is g iv e n , a n d a s o u n d a n d c o m p l e t e d e c i si o n p r o c e d u r e fo r a w e a k e r f o r mo f t h e l a n g u a g e is d e v is e d . F i n a ll y , t h e c o l l e ct i o n t h e o r y i s a b a n d o n e d i n f a v o u ro f a M e r e o l og y , i.e . a t h e o r y o f t h e p a r t - w h o l e r e l a t io n .

2 T h e C o l l e c t i o n T h e o r yI n t h i s s e c t i o n a s i m p l e f o r m a l w a y t o m o d e l c o l le c t io n s o f o b j e c t s i s i n t r o d u c e d .A collection i s f o r m e d b y s e l e c t i n g c e r t a i n o b j e c t s , c a l l e d members o r e lemen tso f t h e c o l l e c t io n .

W i t h i n t h i s m o d e l , b o t h o b j e c t s a n d c o l l e c t io n s o f o b j e c t s a r e d e n o t e d b yt e r m s , i .e . t h e y a r e i n t e r p r e t e d a s e n t i t i e s o f t h e d o m a i n . L i k e i n t h e s t a n d a r ds e t t h e o r y , a p r i m i t i v e membership b i n a r y r e l a t i o n ( d e n o t e d b y 9 ) i s i n t r o d u c e d ,t o r e l a t e t h e c o l l e c ti v e e n t i t ie s w i t h t h e i r e l e m e n t s . F o r e x a m p l e , t h e f o r m u l a s2(bea t le s , pau l ) , ~ (bea t lc s , john) , ~ (bea t le s , r ingo) , ~ (bea t le s , george ) ,

a r e i n t e n d e d t o m e a n t h a t t h e e n t i t i e s john , pau l , r ingo a n d george a r e e l e m e n t so f t h e c o l le c t iv e e n t i t y n a m e d beatles; t h u s , t h e i r c o n j u n c t i o n r e p r e s e n t s t h em e a n i n g o f t h e s e n te n c e " T h e B e a t le s a r e J o h n , P a u l , G e o r g e a nd R i n g o " .

A c o l l e c t i o n c a n b e r e l a t e d w i t h o t h e r c o l l e c t i o n s : i t c a n s h a r e s o m e c o m p o -n e n t s o r i t c a n i n c l u d e a l l th e c o m p o n e n t s o f o t h e r c o l l e c ti o n s. T h e sub-collectiona n d overlapping r e l a t i o n s b e t w e e n c o l le c t io n s a r e d e f i n e d b y m e a n s o f t h ep r i m i t i v e r e l a ti o n :D e f i n i t i o n 1 . ( S u b - c o l l e c t i o n a n d O v e r l a p p i n g )C_(a,b) i f fV x . ~ ( a , x ) ~ ~ ( b , x )N ( a , b ) i f f 3 x . O ( a , x ) A ~ ( b , x ) .

F o r e xa m p l e , t h e f o r m u l a sC_ (b ea tle s, a p p l e C h a r t e r M e m b e r s ) , f l (b e a t l e s, m o s tP o p u la r S in g e r s )e x p r e s s t h a t e a c h o f t h e beatles is al so a f o u n d e r o f t h e A p p l e R e c o r d s c o m p a n y ,a n d t h a t s o m e o f t h e m a r e a m o n g t h e m o s t p o p u l a r s in g er s.T h e C r e l a t i o n is d e fi n e d a s t h e u s u a l subset o f t h e o r d i n a r y s e t t h e o r y , w h e r e a st h e Cl r e l a t i o n i s d e fi n e d t o b e t r u e f o r a ll t h e n o n - e m p t y s e t - t h e o r e t i c i n t e r s e c -t ions N s , i .e . C ' l(a , b ) i f and on ly i f Cl'(a,b) # O.S i n ce a n y o t h e r a x i o m f o r s u c h r e la t i o n s is n o t i n t r o d u c e d , i t t u r n s o u t t h a t t h eC_ a n d Cl r e l a t i o n s h a v e w e a k e r p r o p e r t i e s t h a n t h e i r c o u n t e r p a r t s i n s e t t h e o r y .F o r e x a m p l e , i t f o l lo w s f r o m t h e d e f i n i t i o n t h a t t h e C_ r e l a t i o n i s r e f l e x i v e a n dt r a n s i t i v e b u t n o t a n t i - s y m m e t r i c - i .e . i t is a q u a s i - o r d e r i n g - s i n c e t h e e x t e n -s i o n a l i t y a x i o m d o e s n o t h o l d . I n s e t t h e o r y , t h e e x t e n s i o n a l i t y a x i o m s a y s t h a t


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16two sets are equal if and only if they have the same elements. In our framework,two collections having the same elements are not necessarily equal. So, fromC _( app l e C har t e r M e m be r s , be a tl es ), C _ ( be a t l e s , app l e C h ar t e r M e m be r s )does not follow thata p p l e C h a r t e r M e m b e r s = b e a t l e s ,because the entity beat les could have different attributes from the entity repre-senting the group of people who founded the Apple Records company, like, forinstance, legal liability or taxes to pay.

This simple framework will be referred as the Collection Theory. A richertheory would include also negated relations, such as "~", "/l" (disjointness)and "qs - see [28]; they are needed to properly cover many natural languagephenomena. However, it is argued (though not yet proved) that, in the con-text of concept languages, negated relations may lead to undecidability. As atrivial consequence of this choice, it is worth noting that paradoxes - like theRussel paradox - are avoided and well-foundedness of the Collection Theory isguaranteed, if the negation of the ~ relation can not be expressed.

3 P l u r a l Q u a n t i f ie r sGeneric relations which apply to collective entities can be quantified in differentways. So, representational means are introduced to capture the semantics of the(possibly underdetermined) reading variants of NL expressions. As an example,take the possible readings involving the plural subject of the sentence "John,Paul, George and Ringo sing 'Yesterday' ". For the collective reading all the mentogether sing the song; in the case each man sings separately, we speak of thedistributive reading; and, finally, the cumulat ive reading can describe the mixedsituations in which, say, one sings alone, and separately the others sing together,with the proviso that all of them are involved in some action of singing. The


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17" J o h n i s t h e l e a d e r o f t h e B e a t l e s "L E D - B Y ( b ea tle s, j o h n ) ," T h e B e a t l e s w e r e b o r n i n L i v e r p o o l "< 1 B O R N - I N ( b e a t l e s , l i v e r p o o l ) ," T h e B e a t l e s si n g ' Y e s t e r d a y ' "~_ S IN G (b e a t l e s , y e s t e r d a y ) .

T h e r e l a t i o n L E D - B Y h a s a "c olle ctiv e" r e a d i n g - i .e . j o h n i s t h e l e a d e r o f t h ew h o l e co l l ec t i o n beatles. T h e r e l a t i o n B O R N - I N is "left distributive" o v e r t h ec o m p o n e n t s o f th e beat les - i . e . e a c h m e m b e r w a s b o r n i n L i v e r p o o l :B O R N - I N ( j o h n , liv er po ol) , B O R N - I N ( p a u l , liv er po ol) ,B O R N - I N ( g e o r g e , l iv er po o l) , B O R g - I N ( r i n g o , liv er po o l) .T h e r e l a ti o n S I N G has a " le ft cumula tive" r e a d i n g w i t h r e s p e c t t o t h e beatles -i .e . it i s po s s ib l e t h a t a n y c o l le c t io n o f c o m p o n e n t s o f beatles s i n g s ' Y e s t e r d a y ' ,r a n g i n g f r o m s i n g l e i n d i v i d u a l s t o t h e e n t i r e beatles c o l l e c t i o n , w i t h t h e p r o v i s ot h a t t h e u n i o n o f su c h c o l l e c ti o n s s h o u l d i n c l u d e a t l e a s t a ll th e beat les m e m b e r s .F o r e .x a m p le , t o g e t h e r w i t h t h e c o l le c t iv e i n t e r p r e t a t i o nS I N G ( b e a t l e s , y e s t e r d a y ) ,a n d t h e d i s t r i b u t i v e i n t e r p r e t a t i o nS I N G ( j o h n , y e s t e r d a y ) , S I N G ( p a u l , y e s t e r d a y ) ,S I N G ( g e o r g e , y e s t e r d a y ) , S I N G ( r i n g o , y e s t e r d a y ) ,a p o s s i b l e v a l i d i n t e r p r e t a t i o n f o r t h e c u m u l a t i v e r e a d i n g i s t h e f o l l o w i n g :~ ( C 1 , pau l ) , ~ ( C 1 , j o h n ) , ~ ( C 1 , clv is) ,~ (C2 , pau l ) , 2 (C2 , george ) , 2 ( C 2 , r ingo) ,S I N G ( C 1 , y e s t e r d a y ) , S I N G ( C 2 , y e s t e r d a y ) .I n t h i s i n t e r p r e t a t i o n , t h e r e l a t i o n S I N G h o l d s b e t w e e n t w o c o l l e c t i v e e n t i t i e sC1 , C2 an d y e s t e r d a y . T h e ' i n c l u s i o n ' c o n d i t i o n o f t h e c u m u l a t i v e r e a d i n g iss a t is f ie d : e a c h m e m b e r o f t h e beatles b e l o n g s t o a t l e a s t o n e o f t h e co l l ec t io n sp a r t i c i p a t i n g t o t h e r e l a t i o n . T h e p l u r a l q u a n t i f i e r s a r e f o r m a l l y d e f i n e d as fo l-l o w s .D e f i n i t i o n 2 . ( P l u r a l Q u a n t i f ie r s )R(a, b) i f f Vx. ~(b , x ) --* R(a , x )

b ) i fr V x . x) b ) V ( 3 s . A R ( s , b )) ):>R(a, b) i f f Vx . ~(b , x ) --+ (R (a , x ) V ( 3 s . ~ ( s , x ) A R(a , s ) ) ) .

I t i s e a s y t o c h e c k t h a t t h e c u m u l a t i v e p l u r a l q u a n t i f i e d e x p r e s s i o n s s u b s u m e


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18th e n o n -q u an t i f i ed - i .e . t h e co l l ec t iv e - ex p re ss io n s an d th e d i s t r ib u t iv e p lu r a lq u an t i f i ed o n es :W , y . R ( x , y ) -~ ~ R ( x , y ), W , y . ~ R ( x , y ) - ~ ~ R ( x , y ),W , y . R ( x , y ) --* ~ n ( x , y ) , W , y . ~ R ( ~ , y ) ~ ~ R ( ~ , y ).I n th i s way , t h e n o n -d i sam b ig u a ted lo g ica l f o rm o f a sen ten ce w i th m u l t ip l ei n t e r p r e t a t i o n s c a n b e r e p r e s e n t e d :" T h e B e a t l e s s i n g ' Y e s t e r d a y ' " ,


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19

C~ D --~ A [ (pr imi t ive conce pt)T I ( t o p )-l- I (b otto m )-~C I (general com plem ent)C ~ D I (conjunc t ion)C U D I (d is junct ion)V R . C [ (universal quan t i f ier)3 R . C (exis tent ial quant i f ier)

R --+ P I (prim itive role)9 I (has elem ent relat ion)C_ I (sub-collection re la tio n)n I (overlaps relat io n)


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7 2 :.1_2: =

(C [q D )2: ---( C U D ) z =( ~ c ) 2 : =

( V R . C ) 2 : =( ~ R . C ) 2 : =

C I =CI2:

( ,~R ) ~ =( ~ R) z =( ~ R)2: =( ~ R)2: =

20A Z0

C z n D2:C2: u D2:A 2: \ C2:{ a 9 A 2 :{a 9 z~2:{ ( ~ , b ) 9{ ( a , b ){ ( a , b ) 9{ ( a , b ) 9

IV b '(a ,b ) 9 z - - ~ b 9 z }1 3 b -(a ,b ) 9 1 4 9 z }z ~2 : x ~ 2 : I V x . ( a , x ) 9 9 ~ - - . ( b , x ) 9 9 2 : }a 2 : a s 1 3 x . (a , x ) 9 9 ~ ^ ( b , ~ ) ~ 9 2 : }~ 2 : ~ 2 : I W . ( ~ , x ) ~ 9 ~ - ~ ( ~ , b ) 9 R ~ }a 2 : ~ 2 : I W . ( b , ~ ) e 9 ~ - ~ ( a , ~ ) 9 R D{ ( a , b ) 9 A z x A z [

V x . ( a , x ) 9 9 z ---+ ( ( 3 s ~ ( s , x ) 9 9 z A ( s , b ) 9 R z ) V ( x , b ) 9 R Z ) }{ ( a , b ) 9 , ~ 2 : a s I

V x . ( b , x ) 9 9 2 : -'- + ( ( 3 s . ( s , x ) 9 9 2 : A ( a , s ) 9 R 2 :) V ( a , x ) 9 R z ) }

Fi g . 2 . Th e sem ant ic in te r p re ta t io n fo r concep ts and ro les in ~4Z:C,~.

T h e s e s t a t e m e n t s m a t c h o u r i n t u i t i o n s a b o u t c o l l e c t i o n s , a n d r e f l e c t t h e c o r r e -s p o n d i n g v a li d a x i o m s i n se t t h e o r y , w h e r e E C O L L is i n t e r p r e t e d a s t h e e m p t ys e t a n d C O L L is i n t e r p r e t e d a s a n y n o n - e m p t y s e t :V y . C s ( 0 , y ) ,V x , y . ( x # 0 ^ y : / : O ) ~ ( c _ s ( x , y ) ~ n ' ( x , y ) r 0 ) ,~ 3 x , y . n ' ( x , y ) r 0 ^ ( x = O v y = O ) .

L e t u s i n t r o d u c e n o w s o m e m o r e c o m p l e x c o n c e p t d e f i n i t i o n s , i n o r d e r t ou n d e r s t a n d b e t t e r t h e e x p r e s si v e p o w e r o f .Af_.CS.T h e c o n c e p t ( V r ' I . C O L L ) d e n o t e s a n y e n t i t y w h i c h p o s s i b l y h a s a n o v e r -

l a p p i n g w i t h s o m e n o n - e m p t y c o ll ec ti on ; t h e c o n c e p t ( V C _ . C O L L ) d e n o t e s a n ye n t i t y w h i c h p o s s i b l y c o n t a i n s le ss e l e m e n t s t h a n s o m e n o n - e m p t y c o l l ec t io n ;f in a ll y, t h e c o n c e p t ( 3 n . T ) d e n o t e s a n y c o l l e ct i o n h a v i n g a t l ea s t a c o m m o ne l e m e n t w i t h s o m e t h i n g e l s e . I t c a n b e s h o w n t h a t t h e f i r s t c o n c e p t i s e q u i v a -l e n t t o t h e t o p c o n c e p t , t h e s e c o n d i n c l u d e i n i t s d e n o t a t i o n a l l t h e n o n - e m p t yc o l le c t io n s , a n d t h e l a t t e r d e n o t e s o n l y n o n e m p t y c o l le c t io n s :V n . C O L L =_ T , C O L L C V C . C O L L , 3 N . T E C O L L .

I t is w o r t h n o t i n g t h a t e x p r e s s i o n s c o n t a i n i n g t h e


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21# Vm.(3R.C) .

T h i s c a n b e u n d e r s t o o d b y c o n si d e r in g t h a t t h e c o n c e p t in t h e ri g h t h a n d s i d ei n t r o d u c e s f o r e a c h e l e m e n t a p o s s ib l e n e w r e l a ti o n , w h e r e a s t h e c o n c e p t i nt h e l e ft h a n d s id e i n t r o d u c e s t h e s a m e r e l a t i o n f o r e a c h e l e m e n t ; t h e r e f o r e , t h ef o r m e r h a s a l a r g e r d e n o t a t i o n . I n fa c t , t h e b e s t t h a t c a n b e p r o v e d is a n i n c l u s io nr e l a t i o n :3 ( 4 R ) . C E_ V g . ( 3 R . C ) .

L e t u s c o n s i d e r n o w asser t ions , i .e . p r e d i c a t i o n s o n i n d i v i d u a l o b j e c t s ; u s u a ll y ,t h e y a r e r e f e r r e d t o a s A B o x s t a t e m e n t s . L e t O b e t h e a l p h a b e t o f s y m b o l sd e n o t i n g individuals; a n a s s e r t io n i s a s t a t e m e n t o f t h e f o r m C ( a ) o r R (a , b ) ,w h e r e C is a c o n c e p t , R is a ro l e , a n d a a n d b d e n o t e i n d i v i d u a l s i n O . I n o r d e rt o a s s ig n a m e a n i n g t o t h e a s s e rt i o n s , t h e i n t e r p r e t a t i o n f u n c t i o n z i s e x t e n d e dt o in d i v i d u a l s , s o t h a t a z E A z f o r e a c h i n d i v i d u a l a E O a n d a z 7s b~ i f a r b( U n i q u e N a m e A s s u m p t i o n ) . T h e s e m a n t i c s o f t h e a s s e r t io n s i s t h e f o ll o w in g :C ( a ) i s s a t i s f i e d b y a n i n t e r p r e t a t i o n Z i f f a z E C z , a n d R ( a , b) i s s a t i s f i e d b y Zi f f (a s, b ) E R z .

A s e t ~ o f a s s e r t i o n s i s c a l l e d a n knowledge base . A n i n t e r p r e t a t i o n Z i s am o d e l o f Z i ff e v e r y a s s e r t i o n o f ~ is s a t is f i e d b y I . I f Z h a s a m o d e l , t h e n i t i ssatisf iable . Z logically im pl ie s a n a s s e r t i o n a ( w r i t t e n Z ~ a ) i f a i s s a t i sf i e d b ye v e r y m o d e l o f ~ .

G i v e n a k n o w l e d g e b a s e Z , a n i n d i v i d u a l a i n O i s s a i d t o b e a n i n s t a n ceo f a c o n c e p t C i f ~ ' ~ C ( a ) . T h e ins tance recogn i t ion prob lem, i . e . c h e c k i n gw h e t h e r Z ~ C ( a ) , c a n b e r e d u c e d t o s a t is f i a b i l i ty s i n c e a is a n i n s t a n c e o f Cw i t h r e s p e c t t o a k n o w l e d g e b a s e Z i f a n d o n l y if Z U { ~ C ( a ) } is u n s a t i s fi a b l e[13).

C o m i n g b a c k to o u r e x a m p l e r e g a r d i n g t h e B e a t l e s , l e t u s s e e h o w t h e c o n c e p tr e p r e s e n t i n g a n y p o p g r o u p c a n b e d e f i n e d u s i n g t h e . A s l a n g u a g e :P O P - G R O U P - V ~ . P E R S O N fq V L E D - B Y . P E R S O N I q

V


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225 T h e calculus for Az . c , s

I n t h i s s e c t io n a n a l g o r i t h m f o r d e c i d in g s a t i sf i a b i li t y o f . A f _ . g S - c o n c e p t s i s p ro -p o s e d . T h i s a l g o r i t h m c a n b e u s e d a l s o t o d e c i d e s u b s u m p t i o n b e t w e e n t w oc o n c e p t s a n d i n s t a n c e r e c o g n i t io n b e t w e e n a n i n d i v i d u a l a n d a c o n c e p t . T h es o u n d n e s s o f t h e a l g o r i t h m a n d i ts c o m p l e t e n e s s w i t h r e s p e c t t o . A L g S - , w h i c hi s a w e a k e r v a r i a n t o f . A s w i ll b e p r o v e d . T h e a l g o r i t h m is s o u n d b u t n o tc o m p l e t e f o r A s

I n o r d e r t o o b t a i n A L : g S - f r o m .A / :C S , t h e C o l l e c ti o n T h e o r y i s c h a n g e d b yr e l a x i n g t h e s e m a n t i c s o f t h e C , n , ( 4 R ) , ( t> lZ ), (~_ R ) a n d ( _ R ) col lec t iveroles.D e f i n i t i o n 3 . ( C o l l e c t i v e R o l e s f o r A L :g ,~ - )

C _(a , b ) ~ V x . ~ ( a , x ) ---+ O ( b , x )f 3 ( a , b )

. R ( a , b )

~_R(a,b)I>R(a,b)

::~ 3x . ~ ( a , x ) A ~ ( b , x )V x. ~(a, x ) ---* R (x , b)W . ~(a , ~) --+ ( (3s. ~(s, ~) ^ R (s, b)) V R (~, b))W . ~(b, ~) -~ ( (3s. ~(s, ~) ^ R (a, s)) V R (a, ~)).

W i t h r e s p e c t t o t h e d e f i n i ti o n o f t h e c o l le c t iv e r o le s f o r . A Z C S , t h e i f f ' rh a s b e e n r e p l a c e d w i t h a si m p l e i m p l i c a t i o n ' ~ ' . I n t h is w a y , t h e c o l l ec t iv er o l e s b e c o m e p r i m i t i v e ; n o n e t h e l e s s , t h e y s ti ll i n d u c e a s t r u c t u r e b e t w e e n t h ee l e m e n t s . E v e n i f a t f i r s t s i g h t t h i s s i m p l i f ic a t i o n m a y g i v e t h e i m p r e s s i o n t h a tt h e o b t a i n e d t h e o r y i s t o o w e a k , w e c la i m t h a t t h i s is n o t a c t u a l l y t h e c a s e . T h e r ea r e t w o r e a s o n s : f ir s t, in a o p e n w o r l d s e m a n t i c s - w h i c h i s u s u a l l y a d o p t e d f o rN L P s e m a n t i c s - t h e lo s t d e d u c ti o n s t h a t c a n b e p e r f o r m e d i n . 4 / 3 g 8 f r o m t h ee l e m e n t s t o t h e c o l le c t iv e r o le s a r e v e r y fe w ; s e c o n d , f r o m t h e n a t u r a l l a n g u a g ep o i n t o f v i e w s u c h d e d u c t i o n s a r e n o t t h e i n t u i t i v e o n e s . I n t h e f o l l o w i ng w e w i llr e f e r t o t h e l a n g u a g e . A / : C S - .

T h e r u l e - b a s e d c M c u l u s t o d e c i d e t h e s a t i s f i a b i l i t y o f . A / :C , 5 - a n d A / :C , Sc o n c e p t s o p e r a t e s o n c o n s t r a i n t s [1 4]. A c o n s t r a i n t c a n b e o f t h e t y p e x : C o rx R y , w h e r e C i s a c o n c e p t , R i s a r o le , a n d x , y a r e v a r i a b l e s b e l o n g i n g t o ap r e d e f i n e d a l p h a b e t o f v a r i a b l e s y m b o l s .T h e i n t e r p r e t a t i o n o f c o n s t r a i n t s i s d e f i n e d a s fo ll o w s. L e t Z b e a n i n t e r p r e t a t i o no f t h e c o n c e p t l a n g u a g e . A n Z - a s s i g n m e n t r is a f u n c t i o n t h a t m a p s e v e r y v ar i -a b le t o a n e l e m e n t o f A z . W e s a y t h a t c~ s a t i s f i e s x : C i f a n d o n l y i f a ( x ) E C z ,a n d ~ s a t i s f i e s x R y i f a n d o n l y if ( a ( x ) , ~ ( y ) ) E R z .A c o n s t r a i n t s y s t e m S is a f i n it e , n o n - e m p t y s e t o f c o n s t r a i n t s . A n Z - a s s i g n m e n to~ sa t is f ies a c o n s t r a i n t s y s t e m S i f ~ s a t is f ie s e v e r y c o n s t r a i n t i n S . A c o n s t r a i n ts y s t e m S i s s a t i s f i ab l e i f t h e r e is a n i n t e r p r e t a t i o n Z a n d a n Z - a s s i g n m e n t c~ s u c ht h a t ~ s a t i s f i e s S .A c lash i s a s y s t e m h a v in g o n e o f t h e fo rm s : {x : _l_}, {~: : A , 9 : -~A}.


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23P r o p o s i t i o n 4 . ( R e d u c t i o n t o a c o n s t r a i n t s y s t e m ) A . A E C S - c o n ce p t Ci s s a t i sf i a b le i f a n d o n l y i f t h e c o n s t r a i n t s y s t e m { x : C } i s s a t is f ia b l e .P r o o f . Follows from the definition s. []

For the calculus, we consider only s i m p l e . A R C S - concepts. A concept iscalled simple if i t co ntain s only com ple men ts of the f orm ~ A, where A is apr imi t ive concept . An arbi t ra ry A R C S - ( A E C S ) concept can be t ransformed inl inear t ime into an equivalent s imple concept by means of the fol lowing rewri t ingrules:

-~T - - . _L -~ (C n D ) --~ -~C u -~D -~V R. C --~ 3R . -~ C- 1 1 - -, T - ~( C U D ) - -. -~ C n -~ D ~ 3 R . C - ~ V R . - ~ C

-~-~C -~ CSta rt i ng from the system S = {x : C}, the p r o p a g a t i o n ~ u l e s are applied,

unti l a co ntra dic t ion is gene rated or a model of C is explici t ly obta ined : theprop agat ion rules preserve sat is f iabi li ty. I t i s imp or tan t to remar k tha t they areint rod uce d in order to prove sat isfiabil ity, and tha t they are n o t in tended to beused as d edu ctio n rules. We have the following rules:S - * n { x : C 1, x : C 2 } u Si fx :Cl [TC 2 in S, and bo thx :C1 and x:C 2 not in SS - * u { x : D } u S

if x: Cl UC 2 in S, and neither x: C' l nor x: C2 in S, and D----C1 or D= 6' 2S - + 3" { x R y , y : C } U S

if x : S R . C in S, and there is no z s.t. both x R z and z : C in S,and y is a new variable

S - + v { y : C } U Sif x : V R . C and x R y in S, and y : C not in S

s - * c _ { y ~ z } u sif x C y and x ~ z in S, and y O z not in SS--~r { x S k , y g k } u Sif x N y in S, and there is no z s.t. both x g z and y S z in S,and k is a new variable

S - - * ~ { z R y } U Si f x ( < l R ) y and x g z in S, and z R y not in SS -+ I> {* R z} u Sif x( ~>R)y and y 2 z in S, and x R z not in SS - ~ T u Sif x ( ~ R ) y and y g z in S,and there is no t s.t. both

foundations on knowledge representation and reasoning by lakemeyer and nebel

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