Interacting Hopf Algebras

We introduce the theory IH of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IH are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid-comonoid pairs. This construction is instrumental in showing that IH is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R

Two interacting Hopf algebras of trees

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.Comment: Error in antipode formula (section 7) corrected. Erratum submitte

Refinement for Signal Flow Graphs

The symmetric monoidal theory of Interacting Hopf Algebras provides a sound and complete axiomatisation for linear relations over a given field. As is the case for ordinary relations, linear relations have a natural order that coincides with inclusion. In this paper, we give a presentation for this ordering by extending the theory of Interacting Hopf Algebras with a single additional inequation. We show that the extended theory gives rise to an abelian bicategory - a concept due to Carboni and Walters - and highlight similarities with the algebra of relations. Most importantly, the ordering leads to a well-behaved notion of refinement for signal flow graphs

Diagrammatic Polyhedral Algebra

We extend the theory of Interacting Hopf algebras with an order primitive, and give a sound and complete axiomatisation of the prop of polyhedral cones. Next, we axiomatise an affine extension and prove soundness and completeness for the prop of polyhedra

Universal Constructions for (Co)Relations: categories, monoidal categories, and props

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824

Interacting Frobenius Algebras are Hopf

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.Comment: 32 pages; submitte

Colored props for large scale graphical reasoning

The prop formalism allows representation of processes withstring diagrams and has been successfully applied in various areas such as quantum computing, electric circuits and control flow graphs. However, these graphical approaches suffer from scalability problems when it comes to writing large diagrams. A proposal to tackle this issue has been investigated for ZX-calculus using colored props. This paper extends the approach to any prop, making it a general tool for graphical languages manipulation

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