Modal Theorem Proving: An Equational Viewpoint

We propose a new method f o r automated theorem p r o v i n g i n f i r s t o r d e r modal l o g i c . E s s e n t i a l l y , the method c o n s i s t s i n a t r a n s l a t i o n o f modal l o g i c i n t o a s p e c i a l l y des igned t y p e d f i r s t o r d e r l o g i c c a l l e d Path L o g i c , such t h a t c l a s s i c a l modal systems ( f i r s t o r d e r Q, T , 0 4 , S4, S5) can be c h a r a c t e r i zed by s e t s o f e q u a t i o n s . The q u e s t i o n of modal theorem p r o v i n g t hen amounts to c l a s s i c a l theorem p r o v i n g i n some e q u a t i o n a l t h e o r i e s . D i f f e r e n t methods can be i n v e s t i g a t e d and in t h i s paper we con s i d e r R e s o l u t i o n . We may use R e s o l u t i o n w i t h Pa ra m o d u l a t i o n , or a c o m b i n a t i o n o f R e s o l u t i o n and R e w r i t i n g t e c h n i q u e s . I n b o t h cases , known r e s u l t s p r o v i d e " f r e e o f cha rge " a f ramework immed ia te l y a p p l i c a b l e t o Path L o g i c , w i t h completeness t h e o rems. C o n s i d e r i n g e f f i c i e n c y , the R e w r i t i n g method seems b e t t e r and we p r e s e n t here in d e t a i l s i t s a p p l i c a t i o n to Pa th L o g i c . I n p a r t i c u l a r we show how i t i s p o s s i b l e t o d e f i n e a s p e c i a l k i n d o f s k o l e m i s a t i o n and d e s i g n a u n i f i c a t i o n a l g o r i t h m wh ich i n s u r e s t h a t two c lauses w i l l a lways have a f i n i t e se t o f r e s o l v e n t s .