6 research outputs found
Presenting Finite Posets
We introduce a monoidal category whose morphisms are finite partial orders,
with chosen minimal and maximal elements as source and target respectively.
After recalling the notion of presentation of a monoidal category by the means
of generators and relations, we construct a presentation of our category, which
corresponds to a variant of the notion of bialgebra.Comment: In Proceedings TERMGRAPH 2014, arXiv:1505.0681
Posets with Interfaces for Concurrent Kleene Algebra
We introduce posets with interfaces (iposets) and generalise the serial
composition of posets to a new gluing composition of iposets. In partial order
semantics of concurrency, this amounts to designate events that continue their
execution across components. Alternatively, in terms of decomposing concurrent
systems, it allows cutting through some events, whereas serial composition may
cut through edges only.
We show that iposets under gluing composition form a category, extending the
monoid of posets under serial composition, and a 2-category when enriched with
a subsumption order and a suitable parallel composition as a lax tensor. This
generalises the interchange monoids used in concurrent Kleene algebra.
We also consider gp-iposets, which are generated from singletons by finitary
gluing and parallel compositions. We show that the class includes the
series-parallel posets as well as the interval orders, which are also well
studied in concurrency theory. Finally, we show that not all posets are
gp-iposets, exposing several posets that cannot occur as induced substructures
of gp-iposets
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half
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