In this paper, we recall the basic features of the CospanSpan(Graph) algebra for the compositional description of reconfigurable hierarchical networks. In particular, we focus on compositionality and on the possibility of describing the interactions among physical/biological systems, using a parall el with communication operation not considered in the usual Kleene’s algebra. As a novel application, we give a complete compositional description in Span(Graph) of a simplified version of the heart system.

Span(Graph) was introduced by Katis, Sabadini and Walters as a categorical algebra of automata with interfaces, with main operation being communicating-parallel composition. Additional operations provide also a calculus of connectors or wires among components. 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). So, Span(Graph) yields purely compositional, hierarchical descriptions of networks with a fixed topology . The dual algebra Cospan(Graph) allows to describe also the sequential behaviour of systems. Both algebras, of spans and of cospans, are symmetrical monoidal categories with commutative separable algebra structures on the objects. Hence, the combined algebra CospanSpan(Graph) can be interpreted as a general algebra for reconfigurable/hierarchical networks, generalizing the usual Kleene's algebra for classical automata. We present some examples of systems described in this setting.

The context of this article is a program studying the bicategory of spans of graphs as an algebra of processes, with applications to concurrency theory. The objective here is to study functorial aspects of reachability, minimisation and minimal realisation. The compositionality of minimisation has application to model-checking.

In this paper, we propose a new and elegant definition of the class of recursive functions, which is analogous to Kleene's definition but differs in the primitives taken, thus demonstrating the computational power of the concurrent programming language introduced in Walters (1991), Walters (1992) and Khalil and Walters (1993). The definition can be immediately rephrased for any distributive graph in a countably extensive category with products, thus allowing a wide, natural generalization of computable functions.

Enumeration problems constitute a major part of combinatorial mathematics.