Unification of Idempotent Functions

For a l m o s t as l o n g as a t t e m p t s a t p r o v i n g theorems by mach ines have been made, a c r i t i c a l p r o b l e m has been w e l l known: E q u a t i o n a l a x i o m s , i f l e f t w i t h o u t p r e c a u t i o n s i n t he d a t a base o f a n a u t o m a t i c t heo rem p r o v e r (ATP) , w i l l f o r c e the ATP t o g o a s t r a y . Four approaches t o cope w i t h e q u a l i t y axioms have been p r o p o s e d : (1) T o w r i t e t h e axioms i n t o t h e d a t a b a s e , and use an a d d i t i o n a l r u l e o f i n f e r e n c e , such as p a r a m o d u l a t i o n . (2) T o use s p e c i a l " r e w r i t e r u l e s " . (3) T o d e s i g n s p e c i a l i n f e r e n c e r u l e s i n c o r p o r a t i n g t h e s e a x i o m s . (4) T o d e v e l o p s p e c i a l u n i f i c a t i o n a l g o r i t h m s i n c o r p o r a t i n g t h e s e a x i o m s . A t l e a s t f o r e q u a t i o n a l a x i o m s , the? l a s t approach (4) appears t o be most p r o m i s i n g . So f a r we have c o n c e n t r a t e d on t h e ax ioms o f asso c i a t i v i t y , c o m m u t a t i v i t y , i dempotence and v a r i o u s c o m b i n a t i o n s o f t h e s e . I n t h i s p a p e r , a l g o r i t h m s a re p r e s e n t e d f o r t h e o r i e s o f i dempotence and f o r idempotence t o g e t h e r w i t h c o m m u t a t i v i t y . I d e m p o t e n t f u n c t i o n s appear i n g roup t h e o r y , p r a c t i c a l exampl e s b e i n g p r o o f s abou t s u b s t i t u t i o n s wh i ch a re i dempoten t i f t h e y a re i n no rma l f o r m . See e . g . t h e p r o o f o f Theorem 3-2 i n t h i s p a p e r . A h i s t o r i c a l example i s how Luckham's p rogram v e r i f i e r f ound t h e c o r r e c t n e s s o f R o b i n s o n ' s u n i f i c a t i o n a l g o r i t h m o n l y a f t e r t h e idempotence f o r u n i f i e r s had been a d d e d . The main r e s u l t s a re t h a t t h e u n i f i c a t i o n p r o b l e m f o r i dempotence i s d e c i d a b l e , a n d t h e s e t o f a l l u n i f i e r s i s f i n i t e , b u t n o t a s i n g l e t o n i n g e n e r a l . 2 . A l g o r i t h m f o r I U n i f i c a t i o n 2 . 1 . I n t u i t i v e O v e r v i e w . Our a l g o r i t h m u n i f y i n g two te rms s and t i s s p l i t up i n t o two i n t e r l o c k i n g p a r t s : (1) C o l l a p s i n g p h a s e : In b o t h s and t we l o o k f o r sub te rms ( r ^ , ^ ) s . t . r and r can be u n i f i e d by a s u b s t i t u t i o n p to r . Then , a p p l y i n g p to s and t causes sub te rms ( r , r 2 ) t o " c o l l a p s e " t o r , g e n e r a t i n g a new u n i f i c a t i o n p r o b l e m . (2) R u n i f i c a t i o n phase,: U n i f i c a t i o n p rob lems r e s u l t i n g f rom t h e c o l l a p s i n g phase are s o l v e d b y t h e a l g o r i t h m RUNIFY. E s s e n t i a l l y , RUNIFY f o l l o w s the i d e a o f R o b i n s o n ' s u n i f i c a t i o n a l g o r i t h m e x c e p t f o r t h e way an atom and a n o n a t o m i c t e r m is u n i f i e d . (RUNIFY r e t u r n s a s u c c e s s / f a i l u r e message (SUCC/ FAIL) and a s u b s t i t u t i o n (empty upon f a i l u r e ) . ) For space l i m i t a t i o n s w e can n o t s t a t e t h e f u l l a l g o r i t h m s b u t t h e f o l l o w i n g examples may i n d i c a t e t h e b a s i c i d e a : Theorem P r o v i n g l : Kuhner 528