Improving the Efficiency of Higher Order Unification

The sources o f i n e f f i c i e n c y i n c u r r e n t l y e x i s t i n g h i g h e r o rder u n i f i c a t i o n a l g o r i t h m s are i n v e s t i g a t e d . Aside from such t h e o r e t i c a l d i f f i c u l t i e s as the u n d e c i d a b i l i t y o f u n i f i c a t i o n i n t h i r d o rder l o g i c , and the ex i s t ence o f i n f i n i t e u n i f i e r s and the lack of a po l ynomia l bound on the number o f a p p l i c a t i o n s o f the " i m i t a t i o n " r u l e even in the monadic subcase o f second o rder u n i f i c a t i o n , the c u r r e n t a l g o r i t h m s s u f f e r from a b u i l t i n i n e f f i c iency due to t h e i r i n t r o d u c t i o n and subsequent e l i m i n a t i o n o f many a u x i l i a r y f u n c t i o n a l v a r i a b l e s , and t o the n o n d i r e c ted na tu re of the s u b s t i t u t i o n s made by the " p r o j e c t i o n " r u l e . I t i s argued t h a t a procedure based on a t t emp t i ng to match the argument or arguments of a f u n c t i o n a l o r p r e d i c a t e v a r i a b l e w i t h the subterms o f the o the r fo rmu la i n the u n i f i c a t i o n can dec ide the p o s s i b i l i t y o f u n i f i c a t i o n and genera te the r e s u l t i n g u n i f i e r s much more d i r e c t l y than the t h e o r e t i c a l l v comolete a l g o r i t h m . D e s c r i p t i v e Terms Higher o rder l o g i c , r e s o l u t i o n , theorem p r o v i n g , u n i f i c a t i o n The recen t i n t e r e s t i n deve lop ing l i n e a r and n e a r l i n e a r u n i f i c a t i o n a l g o r i t h m s f o r f i r s t o rde r languages (see f o r example Huet 1976, Paterson and Wegman 1976, and work r e f e r r e d to by them) has , w i t h few e x c e p t i o n s , not been matched by a co r respond ing e f f o r t to improve the e f f i c i e n c y o f h i ghe r o rder u n i f i c a t i o n . A l i n e a r u n i f i c a t i o n a l g o r i t h m i s o f cou rse -ou t o f t he q u e s t i o n f o r h ighe r o r der l o g i c i n g e n e r a l , f o r no t on l y i s u n i f i c a t i o n known to be undec idab le in t h i r d o rder l o g i c (Huet 1973; Lucchesi 1972) , bu t even in t he monadic subcase of second order l o g i c i t has been shown (by W i n t e r s t e i n 1976) t h a t t h e r e e x i s t s no po l ynomia l upper bound on the number of a p p l i c a t i o n s o f the " i m i t a t i o n " r u l e which t oge the r w i t h t he " p r o j e c t i o n " r u l e p lays an e s s e n t i a l r o l e in a complete h ighe r order u n i f i c a t i o n a l g o r i t h m . A l though l i n e a r bounds on the number of t h e i r app l i c a t i o n s do e x i s t in some cases (Wint e r s t e i n 1976) , t he two above-mentioned r u l e s are i n h e r e n t l y i n e f f i c i e n t , (a) b e cause o f t h e i r i n t r o d u c t i o n and subsequent e l i m i n a t i o n o f many a u x i l i a r y