On the Linear Analysis of Synchronous Switching Networks

Abstract

An exposition of an earlier seminal paper on the linear analysis of synchronous switching networks is presented. This analysis, based on the use of the finite or Galois field GF(2), resembles the linear analysis of continuous systems and has important applications in genetics and biochemistry. A synchronous switching network is represented by a function matrix or by a transition matrix, which are related by a similarity transformation in terms of a state matrix. Our use of a novel recursive ordering for the keys or indices of these matrices reveals several new and interesting features and properties. The state matrix is observed to depend not on the particular network but merely on its number of nodes, and is further given a novel interpretation via the modern concept of subsumption of a logical product by another. This reveals a recursive structure of the state matrix and leads to a proof that it is involutory (self- inverse). The autonomous behavior of synchronous switching networks is studied via the characteristic equations and eigenvectors of the aforementioned matrices. In general, the classical ideas are enriched with modern concepts and terminology, supported with correct proofs, and clarified with detailed tutorial examples.