38 research outputs found
Encoding !-tensors as !-graphs with neighbourhood orders
Diagrammatic reasoning using string diagrams provides an intuitive language
for reasoning about morphisms in a symmetric monoidal category. To allow
working with infinite families of string diagrams, !-graphs were introduced as
a method to mark repeated structure inside a diagram. This led to !-graphs
being implemented in the diagrammatic proof assistant Quantomatic. Having a
partially automated program for rewriting diagrams has proven very useful, but
being based on !-graphs, only commutative theories are allowed. An enriched
abstract tensor notation, called !-tensors, has been used to formalise the
notion of !-boxes in non-commutative structures. This work-in-progress paper
presents a method to encode !-tensors as !-graphs with some additional
structure. This will allow us to leverage the existing code from Quantomatic
and quickly provide various tools for non-commutative diagrammatic reasoning.Comment: In Proceedings QPL 2015, arXiv:1511.0118
A first-order logic for string diagrams
Equational reasoning with string diagrams provides an intuitive means of
proving equations between morphisms in a symmetric monoidal category. This can
be extended to proofs of infinite families of equations using a simple
graphical syntax called !-box notation. While this does greatly increase the
proving power of string diagrams, previous attempts to go beyond equational
reasoning have been largely ad hoc, owing to the lack of a suitable logical
framework for diagrammatic proofs involving !-boxes. In this paper, we extend
equational reasoning with !-boxes to a fully-fledged first order logic called
with conjunction, implication, and universal quantification over !-boxes. This
logic, called !L, is then rich enough to properly formalise an induction
principle for !-boxes. We then build a standard model for !L and give an
example proof of a theorem for non-commutative bialgebras using !L, which is
unobtainable by equational reasoning alone.Comment: 15 pages + appendi
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
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