A Non-Standard Semantics for Kahn Networks in Continuous Time

In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginning, process networks have been drawn as particular graphs, but this syntax is never formalized. We take the opportunity to clarify it by giving a precise definition of these graphs, that we call nets. The resulting category is shown to be a fixpoint category, i.e. a cartesian category which is traced wrt the monoidal structure given by the product, and interestingly this structure characterizes the category: we show that it is the free fixpoint category containing a given set of morphisms, thus providing a complete axiomatics that models of process networks should satisfy. We then use these tools to build a model of networks in which data vary over a continuous time, in order to elaborate on the idea that process networks should also be able to encompass computational models such as hybrid systems or electric circuits. We relate this model to Kahn's semantics by introducing a third model of networks based on non-standard analysis, whose elements form an internal complete partial order for which many properties of standard domains can be reformulated. The use of hyperreals in this model allows it to formally consider the notion of infinitesimal, and thus to make a bridge between discrete and continuous time: time is "discrete", but the duration between two instants is infinitesimal. Finally, we give some examples of uses of the model by describing some networks implementing common constructions in analysis.Comment: 201

Interacting Hopf Algebras: the theory of linear systems

Scientists in diverse fields use diagrammatic formalisms to reason about various kinds of networks, or compound systems. Examples include electrical circuits, signal flow graphs, Penrose and Feynman diagrams, Bayesian networks, Petri nets, Kahn process networks, proof nets, UML specifications, amongst many others. Graphical languages provide a convenient abstraction of some underlying mathematical formalism, which gives meaning to diagrams. For instance, signal flow graphs, foundational structures in control theory, are traditionally translated into systems of linear equations. This is typical: diagrammatic languages are used as an interface for more traditional mathematics, but rarely studied per se. Recent trends in computer science analyse diagrams as first-class objects using formal methods from programming language semantics. In many such approaches, diagrams are generated as the arrows of a PROP — a special kind of monoidal category — by a two-dimensional syntax and equations. The domain of interpretation of diagrams is also formalised as a PROP and the (compositional) semantics is expressed as a functor preserving the PROP structure. The first main contribution of this thesis is the characterisation of SVk, the PROP of linear subspaces over a field k. This is an important domain of interpretation for diagrams appearing in diverse research areas, like the signal flow graphs mentioned above. We present by generators and equations the PROP IH of string diagrams whose free model is SVk. The name IH stands for interacting Hopf algebras: indeed, the equations of IH arise by distributive laws between Hopf algebras, which we obtain using Lack’s technique for composing PROPs. The significance of the result is two-fold. On the one hand, it offers a canonical string diagrammatic syntax for linear algebra: linear maps, kernels, subspaces and the standard linear algebraic transformations are all faithfully represented in the graphical language. On the other hand, the equations of IH describe familiar algebraic structures — Hopf algebras and Frobenius algebras — which are at the heart of graphical formalisms as seemingly diverse as quantum circuits, signal flow graphs, simple electrical circuits and Petri nets. Our characterisation enlightens the provenance of these axioms and reveals their linear algebraic nature. Our second main contribution is an application of IH to the semantics of signal processing circuits. We develop a formal theory of signal flow graphs, featuring a string diagrammatic syntax for circuits, a structural operational semantics and a denotational semantics. We prove soundness and completeness of the equations of IH for denotational equivalence. Also, we study the full abstraction question: it turns out that the purely operational picture is too concrete — two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised — rewritten, using the equations of IH, into an executable form where the operational behaviour and the denotation coincide. This realisability theorem — which is the culmination of our developments — suggests a reflection about the role of causality in the semantics of signal flow graphs and, more generally, of computing devices