How to Represent Evidence - Aspects of Uncertain Reasoning

I n o r d e r t o r e p r e s e n t u n c e r t a i n knowledge e s p e c i a l l y i n f e r e n c e r u l e s m a t c h i n g u n c e r t a i n knowledge s t r u c t u r e s s e v e r a l e x t e n s i o n s o f a r e p r e s e n t a t i o n schema a r e d e s c r i b e d . F i r s t a t w o d i m e n s i o n a l e v i d e n c e space i s i n t r o d u c e d t o e x p r e s s p o s i t i v e and n e g a t i v e e v i d e n c e . I t i s shown how i n d i c a t o r s o f e v i d e n c e (as p a r t s o f N L u t t e r a n c e s ) m o d i f y e v i d e n c e v a l u e s b y m o d i f i c a t i o n r u l e s . Ev idence v a l u e s ( p o i n t s o f the e v i d e n c e space) can b e used i n p l a c e o f t r u t h v a l u e s i n o r d e r t o r e p r e s e n t i n f e r e n c e r u l e s . To do t h i s the e x t e n s i o n s d e a l w i t h the g o a l p o i n t concep t f o r r e d e f i n i n g ' v e r i f y ' ( t o g e t h e r w i t h the c o n c e p t s o f n e i g h b o r h o o d , d e v i a t i o n , and d i r e c t i o n ) , and the c o n c e p t o f r e l e v a n c e r e f l e c t i n g the r e l a t i o n s h i p between a s i n g l e p remise and the c o n c l u s i o n . I g i v e no d i r e c t answer to e i t h e r o f t hese q u e s t i o n s , bu t on the one hand I d e m o n s t r a t e how e v i d e n c e v a l u e s can a c c o m p l i s h the t ask o f t r u t h v a l u e s , and on the o t h e r hand I i l l u s t r a t e how to hand le the dependenc ies between e v i d e n c e v a l u e s . In the f o l l o w i n g I i n t r o d u c e a t w o d i m e n s i o n a l e v i d e n c e space and p o i n t ou t the consequences f o r the f o r m u l a t i o n o f i n f e r e n c e r u l e s . I I The e v i d e n c e space In c o n t r a s t to W a h l s t e r , who uses one d i m e n s i o n to r e p r e s e n t e v i d e n c e * * , I d i s t i n g u i s h between p o s i t i v e and n e g a t i v e e v i d e n c e as d i m e n s i o n s o f a t w o d i m e n s i o n a l e v i d e n c e space . I I n t r o d u c t i o n Research i n common sense r e a s o n i n g , n a t u r a l language s y s t e m s , e x p e r t sys tems (and a l m o s t a l l t he o t h e r s u b f i e l d s o f A I ) i s conce rned w i t h knowledge r e p r e s e n t a t i o n a s the main p rob lem o f A I . C e n t r a l t o p i c s a r e vagueness ( e . g . L e F a i v r e , 1 9 7 4 ) , u n c e r t a i n t y ( e . g . S h o r t l i f e , 1976; W a h l s t e r , 1 9 8 1 ) , and p a r t i a l ( o r i n c o m p l e t e ) knowledge ( e . g . Fox , 1 9 8 1 , J o s h i , 1 9 7 8 ) . I n t h i s paper I c o n c e n t r a t e on u n c e r t a i n t y and p r e s e n t a p r o p o s a l on how to r e p r e s e n t i n f e r e n c e r u l e s t h a t a re a b l e t o match u n c e r t a i n knowledge s t r u c t u r e s (macro s t r u c t u r e s ) p a r t i a l l y . S e v e r a l p r o p e r t i e s o f t he r e p r e s e n t a t i o n schema and the r u l e system a re under i n v e s t i g a t i o n . I p r e s e n t some b a s i c e x t e n s i o n s t h a t have a l r e a d y been imp lemented t o v a r y i n g deg rees ( i n PROLOG). The f i r s t and most i m p o r t a n t e x t e n s i o n i s t h e r e p r e s e n t a t i o n o f u n c e r t a i n t y w i t h p a i r s o f e v i d e n c e v a l u e s . Two i m p o r t a n t p rob lems a r i s e i f we e n r i c h a t r a d i t i o n a l r e p r e s e n t a t i o n schema ( e . g . l o g i c ) w i t h e v i d e n c e v a l u e s : f i r s t , how can t r u t h v a l u e s b e r e l a t e d t o e v i d e n c e v a l u e s and s e c o n d l y , what a re t h e dependenc ies between e v i d e n c e v a l u e s ? ( 1 ) Mary b e l i e v e s t h a t J im w i l l t r a v e l t o I J C A I 8 3 . W i t h ( 1 ) as p a r t of my p e r s o n a l knowledge base I can say t h a t I have some p o s i t i v e e v i d e n c e (name ly M a r y ' s b e l i e f ) f o r the p r o p o s i t i o n ' J i m w i l l t r a v e l t o I J C A I 8 3 ' . ( 2 ) John t e l l s me I J C A I 8 3 . t h a t J im w i l l n o t t r a v e l t o ( 2 ) g i v e s me some n e g a t i v e e v i d e n c e f o r t he same p r o p o s i t i o n . So I can have b o t h : p o s i t i v e and n e g a t i v e e v i d e n c e f o r one and the same p r o p o s i t i o n a t the same t i m e . Because I o f t e n am no t a b l e to r e s o l v e c o n f l i c t s i m m e d i a t e l y ( D o y l e , 1979) I w i l l answer "Maybe" * * * i f someone asks me in the meant ime whe the r o r n o t J im w i l l t r a v e l t o I J C A I 8 3 . Be ing a b l e t o g e n e r a t e answers l i k e t h a t i m p l i e s the a b i l i t y t o r e p r e s e n t such " c o n f l i c t s i t u a t i o n s " t o r e p r e s e n t u n c e r t a i n and c o n t r a d i c t o r y k n o w l e d g e . I t i s e s s e n t i a l i n t h i s * * W a h l s t e r uses a l i n e a r s c a l e l i k e the z v a l u e s c a l e o f Fuzzy ( Z a d e h , 1965) i n h i s f u z z y s o r t e d e v i d e n c e c a l c u l u s ( W a h l s t e r , 1 9 8 1 ) . * T h i s wo rk was s u p p o r t e d ( B u n d e s m i n i s t e r i u m f u e r Forschung under g r a n t PT 1 3 5 . 0 1 . by the BMFT und T e c h n o l o g i e ) * * * Many o t h e r answers a r e p o s s i b l e , b u t I am e s p e c i a l l y i n t e r e s t e d i n the m e n t a l a t t i t u d e t owa rds a p r o p o s i t i o n .