Proof of A Conjecture by A. W. Burks and H. Want: Some Relations between Net Cycles and States Cycles

In a foundat ional paper on the theory of au toma ta A. W. Burks and H. W a n g (1957) conjec tured tha t a certain complexi ty measu re involving the size of the s t rong c o m p o n e n t s of a logical ne t f o rmed a h ierarchy for ne t behavior . T h i s conjecture was es tabl ished by Rhodes and Krohn . I n this paper a s t r eng thened vers ion of the conjecture is p roved by es tabl ishing tha t any logical ne t can be in te rpre ted as a series-parallel compos i t ion of ne ts associated wi th its s t rong componen t s . Some proper t ies of the periodic behavior of machines , shown to be preserved u n d e r s imula t ion and composi t ion operat ions, are used to comple te the proof. T h e relat ionship of this approach to algebraic proofs of series-parallel i r reducibi l i ty is discussed.