On the geometry and algebra of networks with state

In [1] an algebra of automata with interfaces, Span(Graph), was introduced with main operation being communicating-parallel composition – a system is represented by an expression in this algebra. A system so described has two aspects: an informal network geometry arising from the algebraic expression, and a space of states and transition given by its evaluation in Span(Graph). Note that Span(Graph) yields purely compositional descriptions of systems. In this paper we give an alternative globally (non-compositional) view of the same systems. In order to do this we make mathematically precise the network geometry in terms of monoidal graphs, and assignment of state in terms of morphisms of monoidal graphs. We call such globally described systems networks with state. To relate networks with state and Span(Graph) systems we use the algebra of cospans of monoidal graphs. As an illustration we give a new representation of a (non-compositionally described) class of Petri nets in the compositional setting of Span(Graph). We also present a tool for calculating with networks with state. Both algebras, of spans and of cospans, are symmetric monoidal categories with commutative separable algebra structures on the objects. We include a short section, following [2], in which we show that a simple modification of the algebra allows the description of networks in which the tangling of connectors may be represented, yielding a connection with the theory of knots.