Induction of Concepts in the Predicate Calculus

P o s i t i v e and n e g a t i v e i ns tances of a concept are assumed to be desc r i bed by a c o n j u n c t i o n o f l i t e r a l s i n the p r e d i c a t e c a l c u l u s , w i t h terms l i m i t e d t o cons tan ts and u n i v e r s a l l y q u a n t i f i e d v a r i a b l e s . A graph r e p r e s e n t a t i o n of a con junc t i o n o f l i t e r a l s , c a l l e d a "p roduc t g r a p h " , i s i n t r o d u c e d . I t i s d e s i r a b l e t o merge p o s i t i v e i ns tances by g e n e r a l i z a t i o n , w h i l e m a i n t a i n i n g d i s c r i m i n a t i o n aga ins t n e g a t i v e i n s t a n c e s . Th i s is accompl ished by an i n d u c t i o n procedure which opera tes on the p roduc t graph form of these posi t i v e and n e g a t i v e i n s t a n c e s . The c o r r e c t n e s s o f the procedure i s p r o v e n , t o g e t h e r w i t h seve ra l r e l a t e d r e s u l t s o f d i r e c t p r a c t i c a l s i g n i f i c a n c e . Th is work i s d i r e c t e d t o the goal o f p r o v i d i n g a fo rma l model f o r the i n d u c t i v e processes which are observed i n a r t i f i c i a l i n t e l l i g e n c e s t u d i e s i n s p e c i a l i z e d a reas . 1 . I n t r o d u c t i o n Lnduct ion may be b r o a d l y d e f i n e d as " r e a s o n ing f rom a p a r t to a who le , f rom p a r t i c u l a r s to g e n e r a l s , o r f rom the i n d i v i d u a l t o the u n i v e r s a l " [ 1 0 ] . C o n s i s t e n t w i t h t h i s , w e view " i n d u c t i o n " as a compu ta t i ona l process, and a " g e n e r a l i z a t i o n " as a s tatement computed by t h a t p rocess . The p resen t work is guided by the b e l i e f t h a t genera l purpose i n d u c t i o n procedures can be f o r mulated which would be independent o f the a r t i f i c i a l i n t e l l i g e n c e prob lem domain. I t i s s p e c i f i c a l l y concerned w i t h the i n d u c t i o n o f concepts from p o s i t i v e and nega t i ve i ns tances desc r i bed in the p r e d i c a t e c a l c u l u s , w i t h terms l i m i t e d t o cons t a n t s and u n i v e r s a l l y q u a n t i f i e d v a r i a b l e s . I n d u c t i v e processes i n a r t i f i c i a l i n t e l l i gence have been s t u d i e d from s e v e r a l aspec t s : v i s u a l ana log ies [ 3 ] , p r o p o s i t i o n a l l o g i c concept l e a r n i n g [ 4 , 9 ] , ana log ies i n p r e d i c a t e c a l c u l u s theorem p r o v i n g [ S ] , IQ t e s t comp le t i on problems [ 1 1 ] , t heo ry f o r m a t i o n f rom a data base [ 2 ] , v i s u a l concept l e a r n i n g f rom examples [ 1 2 ] , and i n d u c t i o n as a dual of deduc t i ve theorem p r o v i n g i n the p r e d i c a t e c a l c u l u s [ 1 , 6 , 8 ] , o r i n a gene r a l i z i t t i o n o f t he p r e d i c a t e c a l c u l u s [ 7 ] . The p resen t work may be regarded as a cont inuance of the concept l e a r n i n g research of Hunt and Tows te r , amongs o t h e r s , t o the p r e d i c a t e c a l c u l u s . I t s v i ewpo in t and many of the concepts are d e r i v e d from P l o t k i n . Throughou t , each p o s i t i v e and nega t i ve i n stance of a concept is assumed to be desc r i bed by a c o n j u n c t i o n o f l i t e r a l s i n the p r e d i c a t e c a l c u lus (no t n e c e s s a r i l y f i r s t o r d e r ) . Sec t i on 2 def i n e s t e r m i n o l o g y to be employed, o f which the most s i g n i f i c a n t i s t h e concept o f a "max ima l , c o n s i s t e n t , u n i f y i n g g e n e r a l i z a t i o n " , and con ta i ns some genera l o b s e r v a t i o n s on g e n e r a l i z a t i o n s . With t h i s background, a problem statement is then given in the def ined vocabulary. Section 3 i n t r o duces the "product graph", a graph representat ion of a conjunct ion of l i t e r a l s , which serves as a convenient medium fo r the d iscussion of the induct i o n process. This process is accomplished by s t ra igh t fo rward operat ions on these product graphs. Section 4 considers the application of product graphs to the quest ion of the "cons is tency" o f a genera l i za t ion in view of negat ive i n stances. Sect ion 5 contains concluding remarks. This work was p a r t i a l l y supported by DHEW Grant No. US-PHS-R01-MB-00114-01.