147 research outputs found
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
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