1,340 research outputs 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
Towards 3-Dimensional Rewriting Theory
String rewriting systems have proved very useful to study monoids. In good
cases, they give finite presentations of monoids, allowing computations on
those and their manipulation by a computer. Even better, when the presentation
is confluent and terminating, they provide one with a notion of canonical
representative of the elements of the presented monoid. Polygraphs are a
higher-dimensional generalization of this notion of presentation, from the
setting of monoids to the much more general setting of n-categories. One of the
main purposes of this article is to give a progressive introduction to the
notion of higher-dimensional rewriting system provided by polygraphs, and
describe its links with classical rewriting theory, string and term rewriting
systems in particular. After introducing the general setting, we will be
interested in proving local confluence for polygraphs presenting 2-categories
and introduce a framework in which a finite 3-dimensional rewriting system
admits a finite number of critical pairs
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
Transformation of Attributed Structures with Cloning (Long Version)
Copying, or cloning, is a basic operation used in the specification of many
applications in computer science. However, when dealing with complex
structures, like graphs, cloning is not a straightforward operation since a
copy of a single vertex may involve (implicitly)copying many edges. Therefore,
most graph transformation approaches forbid the possibility of cloning. We
tackle this problem by providing a framework for graph transformations with
cloning. We use attributed graphs and allow rules to change attributes. These
two features (cloning/changing attributes) together give rise to a powerful
formal specification approach. In order to handle different kinds of graphs and
attributes, we first define the notion of attributed structures in an abstract
way. Then we generalise the sesqui-pushout approach of graph transformation in
the proposed general framework and give appropriate conditions under which
attributed structures can be transformed. Finally, we instantiate our general
framework with different examples, showing that many structures can be handled
and that the proposed framework allows one to specify complex operations in a
natural way
Rewriting Abstract Structures: Materialization Explained Categorically
The paper develops an abstract (over-approximating) semantics for
double-pushout rewriting of graphs and graph-like objects. The focus is on the
so-called materialization of left-hand sides from abstract graphs, a central
concept in previous work. The first contribution is an accessible, general
explanation of how materializations arise from universal properties and
categorical constructions, in particular partial map classifiers, in a topos.
Second, we introduce an extension by enriching objects with annotations and
give a precise characterization of strongest post-conditions, which are
effectively computable under certain assumptions
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
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