On the Linear Analysis of Synchronous Switching Networks

Abstract. An exposition of an earlier seminal paper on the linear analysisof synchronous switching networks is presented. This analysis, based onthe use of the finite or Galois field GF(2), resembles the linear analysis ofcontinuous systems and has important applications in genetics andbiochemistry. A synchronous switching network is represented by afunction matrix or by a transition matrix, which are related by a similaritytransformation in terms of a state matrix. Our use of a novel recursiveordering for the keys or indices of these matrices reveals several new andinteresting features and properties. The state matrix is observed to dependnot on the particular network but merely on its number of nodes, and isfurther given a novel interpretation via the modern concept ofsubsumption of a logical product by another. This reveals a recursivestructure of the state matrix and leads to a proof that it is involutory (selfinverse).The autonomous behavior of synchronous switching networks isstudied via the characteristic equations and eigenvectors of theaforementioned matrices. In general, the classical ideas are enriched withmodern concepts and terminology, supported with correct proofs, andclarified with detailed tutorial examples.