Refinement Strategies for Inductive Leaming of Simple Prolog Programs

T h i s paper ex tends Shap i ro ' s M o d e l Inference System for syn thes iz ing logic p rog rams f r o m examples of i n p u t / o u t p u t behav io r . A new ref inement ope ra to r for clause genera t i on , based u p o n the d e c o m p o s i t i o n o f P ro l og p rog rams i n t o skeletons, basic P ro l og p rog rams w i t h a we l l unde rs tood c o n t r o l f l ow, a n d techniques, s t a n d a r d P r o l o g p r o g r a m m i n g pract ices is descr ibed. Shap i ro ' s o r i g i na l sys tem is i n t r o duced , skeletons and techniques are discussed, a n d s imp le examples are p r o v i d e d , to f am i l i a r ize the reader w i t h t he necessary t e rm ino logy . T h e M o d e l Inference Sys tem equ ipped w i t h th is new re f inement ope ra to r is compa red a n d cont ras ted w i t h the o r i g i na l vers ion presented by Shap i ro . T h e s t rengths a n d weaknesses o f app l y i n g skeletons a n d techniques to syn thes iz ing P r o l o g p rog rams is discussed. 1 I n t r o d u c t i o n I n d u c t i v e l ea rn ing of concepts, g iven a set of examples a n d coun te rexamples , has been g iven a l o t of a t t e n t i o n i n the A r t i f i c i a l In te l l igence c o m m u n i t y . T h i s paper concerns a special case of i n d u c t i v e l e a r n i n g , syn thes iz ing P ro log p rog rams f r o m examples o f t he i r i n p u t / o u t p u t behav io r . A n i n c r e m e n t a l i n d u c t i v e inference a l g o r i t h m was developed in [Shap i ro , 1983] fo r syn thes iz ing logic p rog rams . Shap i ro n a m e d his i m p l e m e n t a t i o n the M o d e l Inference Sys tem ( M I S ) . M I S has several componen ts i n c l u d i n g : de tec t ion and remova l o f a false clause, de tec t i on of the i n a b i l i t y to prove a goa l k n o w n to be t r u e , and the a b i l i t y to f i nd a new clause to j u s t i f y t he k n o w n t r u t h o f a goa l . Anyone e x p e r i m e n t i n g w i t h M I S q u i c k l y discovers t h a t some p ro g rams are easy to l ea rn , o thers can be synthesized w i t h d i f f i cu l t y , and o thers are beyond the scope of the syst e m . T h e reason fo r t he v a r i a t i o n in pe r fo rmance can be t raced to the re f inement opera to r used to p roduce new clauses, a n d the search s t ra tegy emp loyed to de te rm ine i f t he new clause co r rec t l y imp l ies the examples . Hence the re f inement ope ra to r , due to i t s in f luence u p o n the scope o f t he sys tem, is t he foca l p o i n t fo r th i s paper . T h i s paper w i l l descr ibe M I S w i t h one o f i t s re f inement L e o n S. S te r l i ng Computer Engineering & Science Dept. Case Western Reserve University Cleveland, Ohio 44106 U.S.A. opera tors and w i t h a new re f inement opera to r based on w o r k on decompos ing P ro log p rog rams i n t o skeletons , basic P ro l og p rog rams w i t h a we l l unde rs tood cont r o l f l ow , and techniques, s t a n d a r d P ro log p r o g r a m m i n g pract ices. In cont ras t to Shap i ro 's re f inement opera to r , wh i ch checks and adds new clauses one at a t i m e , the new opera to r produces a l l re f inements and then checks the clauses generated. In fac t , every t i m e M I S t r ies to learn a new clause the re f inement opera to r goes t h r o u g h the same order o f clause genera t ion . By check ing p rev ious ly re fu ted clauses, the p r o g r a m ref ra ins f r o m repea t ing i ts mis takes. We w i l l denote M I S equ ipped w i t h our new opera to r as the M o d e l Inference System w i t h Skeletons and Techniques ( M I S S T ) . I n M I S S T the re f inement opera to r consists o f two phases. T h e f i rs t phase matches the necessary d a t a s t ruc tu res w i t h a skeleton an a p p r o p r i a t e con t ro l f low for the p r o g r a m . T h i s genera t ion of a skeleton is accomp l i shed by c rea t i ng a t e m p l a t e us ing on l y the i n p u t a rgumen ts f r o m the p r o g r a m to be synthes ized. Once the skeleton is c reated, the second phase enhances the skeleton by a p p l y i n g a techn ique to i t . Each technique w i l l generate a p r o g r a m to be checked for correctness. Examp les o f the types o f p r o g r a m s w h i c h are ha rd o r imposs ib le to l ea rn and those easy to learn w i l l be given for each sys tem. We w i l l examine the i m p l i c a t i o n s o f the resul ts and c i te areas for f u t u r e research. 2 T h e M o d e l In fe rence Sys tem T h e M o d e l Inference Sys tem is an i m p l e m e n t a t i o n o f an i n c r e m e n t a l i n d u c t i v e inference a l g o r i t h m . G i v e n a set o f examples and counte rexamples o f a new concept , M I S produces a set of H o r n clauses to represent the concept . Whenever the cu r ren t set of clauses prove a counterexa m p l e t rue , the p r o o f tree is used to de te rm ine the f au l t y clause in the set w h i c h is then removed . I f there exists an examp le no t exp la ined by the cur ren t set of clauses, a new clause is generated us ing a re f inement opera to r . A m a j o r assump t i on o f the system is t h a t an oracle exists w h i c h knows the t r u t h o r fa ls i t y o f any pa r t i cu l a r g r o u n d instance of the concept to be learned. T h e re f inement opera to r de termines the t ype o f P ro log p rog rams w h i c h can e a s i l y / n o t easi ly be learned t h r o u g h M I S . One re f inement opera to r g iven in [Shap i ro , 1983] was special ized fo r genera t ing clauses for de f in i te clause Kirschenbaum and Sterling 757 Ref inement Strategies for Induc t i ve Learn ing Of Simple Prolog Programs g r a m m a r s whereas a second opera to r was presented as a more general opera to r . Each opera to r was designed for syn thes iz ing a d i f ferent t ype o f p r o g r a m . In th i s paper, we use the general ised re f inement ope ra to r fo r a l l compar isons w i t h M I S S T . M I S needs to have access to ce r t a i n knowledge to successful ly synthesize a logic p r o g r a m . T h e f o l l o w i n g t w o types o f knowledge are specif ic to the i n tended ta rge t p r o g r a m a n d are supp l ied by the user. Dec la ra t ions a b o u t the t ype a n d m o d e o f each var iab le are used to de te rm ine how the var iab les w i l l be i n s t a n t i a t e d . I n f o r m a t i o n a b o u t ' a l l owab le ' predicates guides clause c rea t ion . An added goa l i s re la ted to the o ther goals in the clause t h r o u g h the i n s t a n t i a t i o n o f var iab les as specif ied by the t ype a n d m o d e dec la ra t ions . O the r knowledge is i nc l uded as p a r t o f the M I S database. For examp le , there is a l i s t of i n s t a n t i a t i o n s for each t y p e . A l i s t va r iab le can be i n s t a n t i a t e d to [ ] or [X|X.]. T h e general re f inement opera to r pe r fo rms i n the f o l -