1,462 research outputs found
Higher-dimensional normalisation strategies for acyclicity
We introduce acyclic polygraphs, a notion of complete categorical cellular
model for (small) categories, containing generators, relations and
higher-dimensional globular syzygies. We give a rewriting method to construct
explicit acyclic polygraphs from convergent presentations. For that, we
introduce higher-dimensional normalisation strategies, defined as homotopically
coherent ways to relate each cell of a polygraph to its normal form, then we
prove that acyclicity is equivalent to the existence of a normalisation
strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical
finiteness condition for higher categories which extends Squier's finite
derivation type for monoids. We relate this homotopical property to a new
homological finiteness condition that we introduce here.Comment: Final versio
Extensional and Intensional Strategies
This paper is a contribution to the theoretical foundations of strategies. We
first present a general definition of abstract strategies which is extensional
in the sense that a strategy is defined explicitly as a set of derivations of
an abstract reduction system. We then move to a more intensional definition
supporting the abstract view but more operational in the sense that it
describes a means for determining such a set. We characterize the class of
extensional strategies that can be defined intensionally. We also give some
hints towards a logical characterization of intensional strategies and propose
a few challenging perspectives
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
A Purely Functional Computer Algebra System Embedded in Haskell
We demonstrate how methods in Functional Programming can be used to implement
a computer algebra system. As a proof-of-concept, we present the
computational-algebra package. It is a computer algebra system implemented as
an embedded domain-specific language in Haskell, a purely functional
programming language. Utilising methods in functional programming and prominent
features of Haskell, this library achieves safety, composability, and
correctness at the same time. To demonstrate the advantages of our approach, we
have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's
and , in a composable way.Comment: 16 pages, Accepted to CASC 201
Higher-dimensional categories with finite derivation type
We study convergent (terminating and confluent) presentations of
n-categories. Using the notion of polygraph (or computad), we introduce the
homotopical property of finite derivation type for n-categories, generalizing
the one introduced by Squier for word rewriting systems. We characterize this
property by using the notion of critical branching. In particular, we define
sufficient conditions for an n-category to have finite derivation type. Through
examples, we present several techniques based on derivations of 2-categories to
study convergent presentations by 3-polygraphs
Rewriting in higher dimensional linear categories and application to the affine oriented Brauer category
In this paper, we introduce a rewriting theory of linear monoidal categories.
Those categories are a particular case of what we will define as linear (n,
p)-categories. We will also define linear (n, p)-polygraphs, a linear adapation
of n-polygraphs, to present linear (n -- 1, p)-categories. We focus then on
linear (3, 2)-polygraphs to give presentations of linear monoidal categories.
We finally give an application of this theory in linear (3, 2)-polygraphs to
prove a basis theorem on the category AOB with a new method using a rewriting
property defined by van Ostroom: decreasingness
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