E-Resolution: Extension of Resolution to Include the Equality Relation

The E-Resolut ion inference p r i n c i p l e described in t h i s paper is a s ing le in fe rence l og i c system f o r the f i r s t o r d e r predicate ca lcu lus w i t h equality. Special axioms f o r equa l i t y ( i . e . , axioms f o r symmetry, r e f l e x i v i t y , t r a n s i t i v i t y , and s u b s t i t u t i v i t y ) are not requi red to be added to the o r i g i n a l set of c lauses. Other advantages of E-Resolut ion are the r e l a t i v e l y small number of in termediate clauses which must be re ta ined in a proof and the d i s t i n c t p o s s i b i l i t y that search s t ra teg ies s u i t able f o r Resolut ion w i l l also be su i tab le f o r EResolu t ion. Although i t is not known whether or not E-Resolut ion is complete, t h i s top ic is cur r e n t l y being inves t iga ted by the author. Key terms: automatic theorem-proving, equality s u b s t i t u t i o n , E-Resolut ion, f i r s t order predicate ca lcu lus w i t h e q u a l i t y , theorem-proving. I . I n t r oduc t i on E-Resolut ion Is a technique f o r automat ica l ly generat ing proofs of theorems in the f i r s t o r d e r pred icate ca lcu lus w i t h equa l i t y . The mechanism by which s u b s t i t u t i o n of equals fo r equals is handled is au tomat ica l ly par t o f E-Resolut ion, o r " b u i l t i n . " The completeness of E-Resolution has no t , as y e t , been proven, and it is therefore not known whether or not the system is complete. However, the author i s c u r r e n t l y i nves t i ga t i ng t h i s t op i c . E-Resolution incorporates and combines some ideas s im i l a r to those used In Paramodulation, k -modu la t ion , 1 0 and S i b e r t ' s t h e s i s , 9 but is d i f fe ren t from a l l th ree. Some of the features which enhance i t s usefulness are: (a) I t is not necessary to add specia l axioms fo r equa l i t y to the o r i g i n a l set o f c lauses, i . e . , axioms fo r symmetry, r e f l e x i v i t y , t r a n s i t i v i t y , and s u b s t i t u t i v i t y . (b) The mechanism of equa l i t y s u b s t i t u t i o n is not ac t i va ted u n t i l i t s success could help produce a reso lven t . This causes the number of i n t e r mediate clauses which must be re ta ined in the proof of a theorem to be held to a minimum. (c) The system is a d i r e c t extension of J. A. Robinson's Resolut ion p r i n c i p l e 4 and is designed so tha t (hopefu l ly ) search techniques developed f o r Resolut ion can be used, w i thou t extensive m o d i f i c a t i o n , in E-Resolut ion. At leas t t h i s seems to be the case f o r set of support s t ra tegy , 11 maximal c lash resolut ion,5 , 7 and merging. 1 This i n ves t i ga t i on was supported in par t by Publ ic Health Service Research grant GM-15769 (NIH) Morr is* iputer Sciences Texas at Aust in The system is programmed in LISP and is running on the CDC 6600 as the main p o r t i o n of a theorem proofchecker f o r Set Theory.2 It has been used to prove several modest theorems invo l v ing equality s u b s t i t u t i o n . The fo l l ow ing sect ions serve to fo rma l l y descr ibe E-Resolut ion. Sect ion I I introduces d e f i n i t i o n s which w i l l be used l a t e r . Sect ion I I I develops the E-Resolution system, and Sect ion IV i l l u s t r a t e s E-Resolution through examples. F i n a l l y , Sect ion V discusses search s t r a t e g i e s . I I . D e f i n i t i o n s The fo l l ow ing d e f i n i t i o n s are preparatory to the matter o f de f i n i ng E-Resolut ion. I t i s assumed tha t the reader Is f a m i l i a r w i t h the notat i o n used in [4] and [ 6 ] . Such terms as atom, l i t e r a l , c lause, wel l formed expression, s u b s t i t u t i o n , disagreement se t , Herbrand un iverse, e t c . are def ined there . I t is w e l l known tha t any set of sentences in the f i r s t o r d e r pred icate ca lcu lus w i t h equa l i t y can be converted to c lausa l form, i . e . , an equival en t set of c lauses, each of which is a d i s junc t o f l i t e r a l s . In the f o l l ow ing d e f i n i t i o n s , we borrow heav i l y from [ 8 ] . Equa l i t y atoms. An equa l i t y atom is an atom whose predicate symbol is the specia l symbol " = " . Rather than w r i t i n g an equa l i t y atom as " 2 a b " , we w i l l use the more readable "a b". Note that the pred icate symbol "=" is always of degree 2. Analogous to the above, " = a b " is known as an i nequa l i t y atom and w i l l be w r i t t e n as "a = b" . We w i l l sometimes speak of equa l i t y l i t e r a l s and i nequa l i t y l i t e r a l s . An equa l i t y l i t e r a l is an equa l i t y atom or a negated i n e q u a l i t y atom. An i nequa l i t y l i t e r a l is an i nequa l i t y atom or a negated equa l i t y atom. I n t e r p r e t a t i o n . An i n t e r p r e t a t i o n I of a set S of clauses is a set of l i t e r a l s such that f o r each atomic formula F that can be formed from a pred icate symbol of degree n occurr ing in S and n terms from the Herbrand universe Hg of S, exa c t l y one of the l i t e r a l s F or F is In I. An i n t e r p r e t a t i o n I is said to s a t i s f y a ground clause C i f a t leas t one l i t e r a l in C is also in I , and to f a l s i f y a ground clause C if the complement of every l i t e r a l in C is in I . An i n t e r p r e t a t i o n I is said to s a t i s f y a non-ground clause C i f i t s a t i s f i e s every ground instance of C over Hg, and is said to f a l s i f y a non-ground clause C i f I t f a l s i f i e s at leas t one ground Instance of C over Hg. An i n t e r p r e t a t i o n I is said to s a t i s f y a set S o f clauses i f i t s a t i s f i e s every clause in S, and is sa id to f a l s i f y a set o f clauses i f i t f a l s i f i e s at l eas t one clause in S. Q represents the empty c lause, and is not s a t i s f i e d by any