On the Descriptional Complexity of Production Systems

In t h i s paper we f o r m a l i z e d i f f e r e n t methods f o r d e s c r i b i n g c o n t r o l knowledge in p roduc t i on systems by the concept of p r o d u c t i o n system schemes. In t h i s framework these methods are compared from the v iewpo in t o f d e s c r i p t i o n a l comp lex i t y g i v i n g us some i n s i g h t by which means problems can be descr ibed in an easy and succ inc t way. 1 I n t r o d u c t i o n In (Georgef f , 82) a p roduc t i on system a r c h i t e c t u r e i s descr ibed i n which p rocedura l knowledge i s s p e c i f i e d by c o n t r o l languages. I n t h i s framework he o u t l i n e d t r a n s f o r m a t i o n s on p roduc t i on systems to remove the amount of nondetermism to get a more e f f i c i e n t e x e c u t i o n . I n o rder t o c l e a r l y separate the means f o r d e s c r i b i n g c o n t r o l knowledge from the ( o b j e c t l e v e l ) semantics of a p r o d u c t i o n system and to compare d i f f e r e n t methods to desc r ibe cont ro lknowledge we w i l l a b s t r a c t t h i s concept by p r o d u c t i o n system schemes. D i f f e r e n t methods t o desc r ibe c o n t r o l i n f o r m a t i o n in p r o d u c t i o n systems can then be model led by d i f f e r e n t c lasses o f p r o d u c t i o n system schemes ( t h i s i s i n p a r a l l e l t o the theory o f program schemes (Gre ibach, 7 5 ) ) . The concept of p r o d u c t i o n system schemes might a l s o form the bas i s f o r a t heo ry o f t r a n s f o r m a t i o n s on p r o d u c t i o n systems. In t h i s paper we cons ider the f o l l o w i n g c lasses of p r o d u c t i o n system schemes: c o n t r o l l e d p r o d u c t i o n system schemes i n which p rocedura l knowledge i s desc r ibed by languages and p r o d u c t i o n system schemes w i t h markers , which use s p e c i a l database symbols to sequence the order o f p r o d u c t i o n s . These c lasses of p r o d u c t i o n system schemes w i l l be compared from the v i ewpo in t o f d e s c r i p t i o n a l c o m p l e x i t y . We show t h a t removing nondeterminism ( i f p o s s i b l e ) i n c o n t r o l l e d p roduc t i on systems might r e s u l t in a s i ze exp los ion o f the e q u i v a l e n t d e t e r m i n i s t i c p roduc t i on system. I n the same way we show t h a t in some cases the s p e c i f i c a t i o n o f c o n t r o l knowledge by markers can r e s u l t in d r a s t i c more succ inc t r ep resen ta t i ons than a separate d e s c r i p t i o n of cont ro lknowledge in even nond e t e r m i n i s t i c c o n t r o l l e d p roduc t i on systems• Def. t An i n t e r p r e t a t i o n of a PSS Q « A, f) is a p a i r I = (D, h) where D is a nonempty s e t , the database of Q, and h is a mapping ass ign ing to each pEP a p r e d i c a t e h (p ) from D i n t o (0,1) and to each a a r e l a t i o n over D. Def .: A c o n t r o l l e d p r o d u c t i o n system schema (CPSS) is a p a i r Q (L , P) where P is a PSS and L a language over the p r o d u c t i o n symbols of P.