Compatibility of States in Input-Independent Machines

T h e pu rpose of th is no te is to find, for a cer ta in class of machines , t he smal l e s t in teger /c hav ing the p r o p e r t y t h a t , if two g iven s t a tes do n o n c o n t r a d i c t o r y work for all t apes of l eng th less t h a n or equa l to /~ , these two s ta tes do non -con t r a d i c to ry work for al l t apes of a n y length . Us ing the n o t a t i o n a n d t e rmino logy of [1], a mach ine S is said to be i n p u t i n d e p e n d e n t if for each s t a t e 1 p, ~ ( p , / 1 ) = X(p , /2) for each two inpu t s I1 and Is which are app l i cab le 2 to p. I n pa r t i cu la r , if S is comple te (i.e., ~ a n d X are def ined I for each s t a t e p and each i n p u t I ) , t hen S is a " m a c h i n e " as or ig ina l ly i n t roduced in [3]. Cal l a sequence of i npu t s a tape. T h e r e are m a n y p rope r t i e s a b o u t s t a t es whose def ini t ion involves the ph rase " for all t apes J ." F o r each of these p roper t i e s i t is of theore t i ca l concern to f ind the smal les t in teger 1¢ ( if i t exis ts) so t h a t t he p r o p e r t y holds if " for al l t a p e s J . " is r ep laced b y " fo r al l t a p e s J , of l eng th t ~ k, . . . ". F o r a long t ime no one knew of a n y nont r i v i a l p r o p e r t y for which the in teger k could be lowered when consider ing on ly s t a tes in i n p u t i n d e p e n d e n t machines . R e c e n t l y Lee [2] has shown t h a t t he p r o p e r t y of two s ta tes being compa t ib l e has th is phenomenon . ( T h e fac t t h a t c ompa t i b l e s t a t es p l a y a f u n d a m e n t a l role in the p r o b l e m of ob t a in ing m i n i m a l s t a t e machines he ightens the in te res t . ) I n [1, p. 272] t he fol lowing two resul t s on compa t ib l e s t a t es were n o t e d :