419 research outputs found
How to combine diagrammatic logics
This paper is a submission to the contest: How to combine logics? at the
World Congress and School on Universal Logic III, 2010. We claim that combining
"things", whatever these things are, is made easier if these things can be seen
as the objects of a category. We define the category of diagrammatic logics, so
that categorical constructions can be used for combining diagrammatic logics.
As an example, a combination of logics using an opfibration is presented, in
order to study computational side-effects due to the evolution of the state
during the execution of an imperative program
Deduction as Reduction
Deduction systems and graph rewriting systems are compared within a common
categorical framework. This leads to an improved deduction method in
diagrammatic logics
Scalability using effects
This note is about using computational effects for scalability. With this
method, the specification gets more and more complex while its semantics gets
more and more correct. We show, from two fundamental examples, that it is
possible to design a deduction system for a specification involving an effect
without expliciting this effect
Diagrammatic Inference
Diagrammatic logics were introduced in 2002, with emphasis on the notions of
specifications and models. In this paper we improve the description of the
inference process, which is seen as a Yoneda functor on a bicategory of
fractions. A diagrammatic logic is defined from a morphism of limit sketches
(called a propagator) which gives rise to an adjunction, which in turn
determines a bicategory of fractions. The propagator, the adjunction and the
bicategory provide respectively the syntax, the models and the inference
process for the logic. Then diagrammatic logics and their morphisms are applied
to the semantics of side effects in computer languages.Comment: 16 page
A parameterization process as a categorical construction
The parameterization process used in the symbolic computation systems Kenzo
and EAT is studied here as a general construction in a categorical framework.
This parameterization process starts from a given specification and builds a
parameterized specification by transforming some operations into parameterized
operations, which depend on one additional variable called the parameter. Given
a model of the parameterized specification, each interpretation of the
parameter, called an argument, provides a model of the given specification.
Moreover, under some relevant terminality assumption, this correspondence
between the arguments and the models of the given specification is a bijection.
It is proved in this paper that the parameterization process is provided by a
free functor and the subsequent parameter passing process by a natural
transformation. Various categorical notions are used, mainly adjoint functors,
pushouts and lax colimits
Data-Structure Rewriting
We tackle the problem of data-structure rewriting including pointer
redirections. We propose two basic rewrite steps: (i) Local Redirection and
Replacement steps the aim of which is redirecting specific pointers determined
by means of a pattern, as well as adding new information to an existing data ;
and (ii) Global Redirection steps which are aimed to redirect all pointers
targeting a node towards another one. We define these two rewriting steps
following the double pushout approach. We define first the category of graphs
we consider and then define rewrite rules as pairs of graph homomorphisms of
the form "L R". Unfortunately, inverse pushouts (complement pushouts)
are not unique in our setting and pushouts do not always exist. Therefore, we
define rewriting steps so that a rewrite rule can always be performed once a
matching is found
Categorical Abstract Rewriting Systems and Functoriality of Graph Transformation
Rewriting systems are often defined as binary relations over a given set of
objects. This simple definition is used to describe various properties of
rewriting such as termination, confluence, normal forms etc. In this paper, we
introduce a new notion of abstract rewriting in the framework of categories.
Then, we define the functoriality property of rewriting systems. This property
is sometimes called vertical composition. We show that most of graph
transformation systems are functorial and provide a counter-example of graph
transformation systems which is not functorial
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