Composing Behaviors of Networks

This thesis aims to develop a compositional theory for the operational semantics of networks. The networks considered are described by either internal or enriched graphs. In the internal case we focus on \(\mathsf{Q}\)-nets, a generalization of Petri nets based on a Lawvere theory \(\mathsf{Q}\). \(\mathsf{Q}\)-nets include many known variants of Petri nets including pre-nets, integer nets, elementary net systems, and bounded nets. In the enriched case we focus on graphs enriched in a quantale \(R\) regarded as matrices with entries in \(R\). These \(R\)-matrices represent distance networks, Markov processes, capacity networks, non-deterministic finite automata, simple graphs, and more. The operational semantics of \(\mathsf{Q}\)-nets is constructed as an adjunction between \(\mathsf{Q}\)-nets and categories internal to the category of models of \(\mathsf{Q}\). Similarly, the operational semantics of \(R\)-matrices is constructed as an adjunction between \(R\)-matrices and categories enriched in \(R\). The left adjoint of this adjunction sends an \(R\)-matrix \(M\) to the \(R\)-category \(F_R(M)\) whose hom-objects are solutions of the algebraic path problem: a generalization of the shortest path problem to graphs weighted in \(R\). For both \(\mathsf{Q}\)-nets and \(R\)-matrices we use the theory of structured cospans to study the compositionality of the above operational semantics. For each type of network we construct a double category whose morphisms are "open networks", i.e. networks with certain vertices designated as input or output. We introduce the black-boxing of an open network, a profunctor describing the externally observable behavior of an open network. We introduce a class of open networks called "functional open networks" for which black-boxing preserves composition.