13 research outputs found
Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories
Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists astrongtracedfunctorintoFinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk.
Finite dimensional Hilbert spaces are complete for dagger compact closed categories
We show that an equation follows from the axioms of dagger compact closed
categories if and only if it holds in finite dimensional Hilbert spaces
Abstract Tensor Systems as Monoidal Categories
The primary contribution of this paper is to give a formal, categorical
treatment to Penrose's abstract tensor notation, in the context of traced
symmetric monoidal categories. To do so, we introduce a typed, sum-free version
of an abstract tensor system and demonstrate the construction of its associated
category. We then show that the associated category of the free abstract tensor
system is in fact the free traced symmetric monoidal category on a monoidal
signature. A notable consequence of this result is a simple proof for the
soundness and completeness of the diagrammatic language for traced symmetric
monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda
Inversion, Iteration, and the Art of Dual Wielding
The humble ("dagger") is used to denote two different operations in
category theory: Taking the adjoint of a morphism (in dagger categories) and
finding the least fixed point of a functional (in categories enriched in
domains). While these two operations are usually considered separately from one
another, the emergence of reversible notions of computation shows the need to
consider how the two ought to interact. In the present paper, we wield both of
these daggers at once and consider dagger categories enriched in domains. We
develop a notion of a monotone dagger structure as a dagger structure that is
well behaved with respect to the enrichment, and show that such a structure
leads to pleasant inversion properties of the fixed points that arise as a
result. Notably, such a structure guarantees the existence of fixed point
adjoints, which we show are intimately related to the conjugates arising from a
canonical involutive monoidal structure in the enrichment. Finally, we relate
the results to applications in the design and semantics of reversible
programming languages.Comment: Accepted for RC 201
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
On the completeness of the traced monoidal category axioms in (Rel,+)
It is shown that the traced monoidal category of finite sets and relations with coproduct as tensor is complete for the extension of the traced symmetric monoidal axioms by two simple axioms, which capture the additive nature of trace in this category. The result is derived from a theorem saying that already the structure of finite partial injections as a traced monoidal category is complete for the given axioms. In practical terms this means that if two biaccessible flowchart schemes are not isomorphic, then there exists an interpretation of the schemes by partial injections which distinguishes them
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
FInCo 2007 AGAPIA v0.1: A programming language for interactive systems and its typing system
Abstract A model (consisting of rv-systems), a core programming language (for developing rv-programs), several specification and analysis techniques appropriate for modeling, programming and reasoning about interactive computing systems have been recently introduced by Stefanescu using register machines and space-time duality, se