Negative Hyper-Resolution for Proving Statements Containing Transitive Relations

A m o d i f i e d v e r s i o n of PHR~r e s o l u t i o n c o m p r i s i n g n e g a t i v e h y p e r r e s o l u t i o n and pa ramodu la t i on i s i n t r o d u c e d t o reduce t he search o f s ta tements t h a t c o n t a i n t r a n s i ­ t i v e r e l a t i o n s . L e t R be a symbol of t r a n s i t i v e r e l a t i o n and l e t a nuc leus c o n t a i n t he l i t e r a l t R p , and l e t a f a c t o r o f an e l e c ­ t r o n be of t h e fo rm C V n t , R t 2 . M o r e o v e r l e t us suppose t h a t tU = t , U , where U is a s imu l taneous most g e n e r a l u n i f i e r f o r t he c o r r e s p o n d i n g c l a s h . Then the r e s o l v e n t o f t h i s c l a s h c o n t a i n s t he subc lause (C V p R t 2 ) U . Th i s r u l e o f d e d u c t i o n i s s a i d to Be TPHR r u l e . I t i s shown t h a t TPHR"r e s o l u t i o n i s comp le te . More p r e c i s e l y , i t i s shown t h a t t he empty c l ause a can be deduced f rom a se t G o f c l auses wh ich c o n t a i n s t h e axiom o f t r a n s i t i v i t y T r f o r t h e r e l a t i o n R by u s i n g PHR""r e s o l u t i o n i f f can be deduced f rom the se t G { T r } by u s i n g TPHR"r e s o l u t i o n . The e f f i c i e n c y of t h e use of TPHR~r u l e i s i l l u s t r a t e d b y examples. The p o s s i b i l i t y f o r t h e s t r a t e g y o f p o s i t i v e h y p e r r e s o l u t i o n was shown in L"3] to reduce t h e search to some e x t e n t . Some f u r t h e r ideas on t h i s s u b j e c t can be found in Cl ] . In t h e p r e s e n t paper we show how to a v o i d t h e e x p l i c i t use o f t r a n ­ s i t i v i t y axiom w i t h i n t h e s t r a t e g y o f neg ­ a t i v e h y p e r r e s o l u t i o n . T h i s s t r a t e g y i s chosen because i t can be combined w i t h pa ramodu la t i on as i t i s shown in C2] and because each d e r i v a t i o n o f t h e empty c l ause can b e e a s i l y t r a n s f o r m e d i n t o d i r e c t p r o o f o f t he s ta temen t i n q u e s t i o n b y u s i n g n e g a t i v e h y p e r r e s o l u t i o n ( c f . [4]. R e c a l l t h a t under n e g a t i v e hype r r e s o l u t i o n s t r a t e g y w e d i v i d e t h e s e t o f a l l c l auses i n t o two p a r t s : c l auses o f nuc leus ( wh ich c o n t a i n one or more pos ­ i t i v e l i t e r a l s ) and e l e c t r o n s ( w h i c h c o n s i s t o f o n l y n e g a t i v e l i t e r a l s ) . Re­ s o l v e n t s a re a lways e l e c t r o n s . The i dea deve loped below i s as f o l ­ l o w s . L e t some e l e c t r o n c l a u s e C c o n t a i n l i t e r a l t 1 R t 2 T h i s means t h a t s ta temen t t 1 R t 2 shou ld Be p r o v e d . I f a nuc leus c o n ­ t a i n s a p o s i t i v e l i t e r a l o f t h e fo rm t1.,Rt' and U is t h e most g e n e r a l u n i f i e r o r t h e terms t1 , and t ' t hen t h e s ta temen t ( t ' R t 2 ) U m igh t b e proved i n s t e a d o f t h e o r i g i n a l one. Now we f o r m a l l y deve lop a l l t h e c o n s t r u c t i o n s necessa ry .