Proof of a conjecture by A.W. Burks and H.Wang: Some relations between net cycles and states cycles

Abstract

In a foundational paper on the theory of automata A. W. Burks and H. Wang (1957) conjectured that a certain complexity measure involving the size of the strong components of a logical net formed a hierarchy for net behavior. This conjecture was established by Rhodes and Krohn. In this paper a strengthened version of the conjecture is proved by establishing that any logical net can be interpreted as a series-parallel composition of nets associated with its strong components. Some properties of the periodic behavior of machines, shown to be preserved under simulation and composition operations, are used to complete the proof. The relationship of this approach to algebraic proofs of series-parallel irreducibility is discussed.