Network Algebra

This book is devoted to a general, algebraic study of networks and their behavior. The term "network" is used in a broad sense here as consisting in a collection of interconnecting cells. Two radically different particular interpretations of this enlarged notion of networks are studied in more details. Virtual networks are obtained using the Cantorian interpretation in which at most one cell is active at a given time. With this interpretation, Network Algebra covers the classical models of control, including infinite automata or flowchart schemes. In a second Cartesian interpretation, each cell is always active, hence models for reactive and concurrent systems as Petri nets or dataflow networks may be covered as well. Points to a more advanced research setting which mixes the above interpretations are included. The results are presented in the unified framework of the calculus of flownomials (an abstract calculus very similar to the classical calculus of polynomials). After their introduction in the context of control-flow charts setting (Stefanescu, 1986), the Basic Network Algebra axioms were rediscovered in various fields ranging from circuit theory to action calculi, from dataflow networks to knot theory (traced monoidal categories), from process graphs to functional progamming. The book is suited for use as teaching material for graduate students as well as for more advanced material for researchers.