Picturing resources in concurrency

Inspired by the pioneering work of Petri and the rise of diagrammatic formalisms to reason about networks of open systems, we introduce the resource calculus---a graphical language for distributed systems. Like process algebras, the resource calculus is modular, with primitive connectors from which all diagrams can be built. We characterise its equational theory by proving a full completeness result for an interpretation in the symmetric monoidal category of additive relations---a result that constitutes the main contribution of this thesis. Additive relations are frequently exploited by model-checking algorithms for Petri nets. In this thesis, we recognise them as a fundamental algebraic structure of concurrency and use them as an axiomatic framework. Surprisingly, the resource calculus has the same syntax as that of interacting Hopf algebras, a diagrammatic formalism for linear (time-invariant dynamical) systems. Indeed, the approach stems from the simple but fruitful realisation that, by replacing values in a field with values in the semiring of non-negative integers, concurrent behaviour patterns emerge. This change of model reflects the interpretation of diagrams as systems manipulating limited and discrete resources instead of continuous signals. We also extend the resource calculus in two orthogonal directions. First, by adding an affine primitive to express access to a constant quantity of resources. The extended calculus is remarkably expressive and allows the formulation of non-additive patterns of behaviour, like mutual exclusion. Once more, we characterise it---this time as the equational theory of the symmetric monoidal category of polyhedral relations, discrete analogues of polyhedra in convex geometry. Secondly, we add a synchronous register to model stateful systems. The stateful resource calculus is expressive enough to faithfully capture the behaviour of Petri nets while being strictly more expressive. It is also shown to axiomatise a category of open Petri nets, in the style of the connector algebras of nets with boundaries first studied by Bruni, Melgratti, Montanari and Sobociński.