Representation and Induction of Infinite Concepts and Recursive Action Sequences

A genera l method f o r the i n f e r e n c e of r e c u r s i v e f u n c t i o n s (programs) from examp les o f computa t ions i s d e s c r i b e d . I t i s based on the so c a l l e d a l g e b r a i c semant ics of r e c u r s i v e program schemes and in t h i s paper main ly a p p l i e d to the r e p r e s e n t a t i o n and i n d u c t i o n o f i n f i n i t e concepts and r e c u r s i v e a c t i o n sequences (which are of impor tance f o r problem s o l v i n g and h i e r a r c h i c a l p l a n n i n g ) . A d d i t i o n a l l y , the use of r e c u r s i v e program schemes leads to a new p r i n c i p l e of g e n e r a l i z a t i o n in the sense t h a t f a m i l i e s o f s t r u c t u r e s o r c lasses of programs cou ld be t r e a t e d s i m u l t a n e o u s l y and t h a t a l r e a d y e x i s t i n g s o l u t i o n s o f o l d problems cou ld be t r a n s f e r r e d to new problems which have to be so l ved . 1 . I n t r o d u c t i o n Program s y n t h e s i s by i n d u c t i o n as opposed to program s y n t h e s i s by deduc t i on from i n p u t o u t p u t s p e c i f i c a t i o n s becomes a mat te r o f i n c r e a s i n g i n t e r e s t i n A I and Computer Sc ience . In t h i s f i e l d much work has d e a l t w i t h the s y n t h e s i s o f LISP p r o grams from examples o f computa t ions / 3 / , / V , / 5 / . We are i n t e r e s t e d on the one hand in the development of a genera l f o rma l i sm not de pending on a s p e c i f i c programming l a n guage and on the o the r hand in the i n d u c t i o n o f c e r t a i n i n f i n i t e concepts and r e c u r s i v e a c t i o n sequences ex tend ing i n t h i s way our e a r l i e r work on concept l e a r n i n g / 6 / , / 7 / . B y i n f i n i t e concepts we mean c l asses of s t r i n g s and o t h e r s t u c t u r e s hav ing some i n t r i n s i c p e r i o d i c i t y ( I n P a t t e r n R e c o g n i t i o n on l y f e a t u r e v e c t o r s o f f i x e d d i m e n s i o n a l i t y have been cons ide red so f a r ) . Our method is based on the so c a l l e d a l g e b r a i c semant ics and r e c u r s i v e program schemes /1/, / 2 / and i t i s i n t ended to p rov ide a g e n e r a l framework f o r the abovementioned r e p r e s e n t a t i o n and i n d u c t i o n problems* The i n d u c t i o n p r i n c i p l e we i n t r o d u c e i n t h i s paper i s a more g e n e r a l v e r s i o n of SUMMERS' main s y n t h e s i s theorem / 5 / . Our main aim i s t o a p p l y i t t o the r e p r e s e n t a t i o n and i n d u c t i o n o f r e c u r s i v e a c t i o n sequencee f rom