58 research outputs found
Relevant Categories and Partial Functions
A relevant category is a symmetric monoidal closed category with a diagonal
natural transformation that satisfies some coherence conditions. Every
cartesian closed category is a relevant category in this sense. The
denomination 'relevant' comes from the connection with relevant logic. It is
shown that the category of sets with partial functions, which is isomorphic to
the category of pointed sets, is a category that is relevant, but not cartesian
closed.Comment: 9 pages, one reference adde
Layer by layer - Combining Monads
We develop a method to incrementally construct programming languages. Our
approach is categorical: each layer of the language is described as a monad.
Our method either (i) concretely builds a distributive law between two monads,
i.e. layers of the language, which then provides a monad structure to the
composition of layers, or (ii) identifies precisely the algebraic obstacles to
the existence of a distributive law and gives a best approximant language. The
running example will involve three layers: a basic imperative language enriched
first by adding non-determinism and then probabilistic choice. The first
extension works seamlessly, but the second encounters an obstacle, which
results in a best approximant language structurally very similar to the
probabilistic network specification language ProbNetKAT
Bases as Coalgebras
The free algebra adjunction, between the category of algebras of a monad and
the underlying category, induces a comonad on the category of algebras. The
coalgebras of this comonad are the topic of study in this paper (following
earlier work). It is illustrated how such coalgebras-on-algebras can be
understood as bases, decomposing each element x into primitives elements from
which x can be reconstructed via the operations of the algebra. This holds in
particular for the free vector space monad, but also for other monads, like
powerset or distribution. For instance, continuous dcpos or stably continuous
frames, where each element is the join of the elements way below it, can be
described as such coalgebras. Further, it is shown how these
coalgebras-on-algebras give rise to a comonoid structure for copy and delete,
and thus to diagonalisation of endomaps like in linear algebra
Relating Operator Spaces via Adjunctions
This chapter uses categorical techniques to describe relations between
various sets of operators on a Hilbert space, such as self-adjoint, positive,
density, effect and projection operators. These relations, including various
Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual
adjunctions, and maps between them. Of particular interest is the connection
with quantum structures, via a dual adjunction between convex sets and effect
modules. The approach systematically uses categories of modules, via their
description as Eilenberg-Moore algebras of a monad
Quantum channels as a categorical completion
We propose a categorical foundation for the connection between pure and mixed
states in quantum information and quantum computation. The foundation is based
on distributive monoidal categories.
First, we prove that the category of all quantum channels is a canonical
completion of the category of pure quantum operations (with ancilla
preparations). More precisely, we prove that the category of completely
positive trace-preserving maps between finite-dimensional C*-algebras is a
canonical completion of the category of finite-dimensional vector spaces and
isometries.
Second, we extend our result to give a foundation to the topological
relationships between quantum channels. We do this by generalizing our
categorical foundation to the topologically-enriched setting. In particular, we
show that the operator norm topology on quantum channels is the canonical
topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
A Graph Model for Imperative Computation
Scott's graph model is a lambda-algebra based on the observation that
continuous endofunctions on the lattice of sets of natural numbers can be
represented via their graphs. A graph is a relation mapping finite sets of
input values to output values.
We consider a similar model based on relations whose input values are finite
sequences rather than sets. This alteration means that we are taking into
account the order in which observations are made. This new notion of graph
gives rise to a model of affine lambda-calculus that admits an interpretation
of imperative constructs including variable assignment, dereferencing and
allocation.
Extending this untyped model, we construct a category that provides a model
of typed higher-order imperative computation with an affine type system. An
appropriate language of this kind is Reynolds's Syntactic Control of
Interference. Our model turns out to be fully abstract for this language. At a
concrete level, it is the same as Reddy's object spaces model, which was the
first "state-free" model of a higher-order imperative programming language and
an important precursor of games models. The graph model can therefore be seen
as a universal domain for Reddy's model
Asynchronous games 4 : A fully complete model of propositional linear logic
International audienceWe construct a denotational model of propositional linear logic based on asynchronous games and winning uniform innocent strategies. Every formula A is interpreted as an asynchronous game [A] and every proof pi of A is interpreted as a winning uniform innocent strategy pi of the game A. We show that the resulting model is fully complete: every winning uniform innocent strategy sigma of the asynchronous game A is the denotation pi of a proof pi of the formula A
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