35 research outputs found
Polygraphs for termination of left-linear term rewriting systems
We present a methodology for proving termination of left-linear term
rewriting systems (TRSs) by using Albert Burroni's polygraphs, a kind of
rewriting systems on algebraic circuits. We translate the considered TRS into a
polygraph of minimal size whose termination is proven with a polygraphic
interpretation, then we get back the property on the TRS. We recall Yves
Lafont's general translation of TRSs into polygraphs and known links between
their termination properties. We give several conditions on the original TRS,
including being a first-order functional program, that ensure that we can
reduce the size of the polygraphic translation. We also prove sufficient
conditions on the polygraphic interpretations of a minimal translation to imply
termination of the original TRS. Examples are given to compare this method with
usual polynomial interpretations.Comment: 15 page
Termination orders for 3-dimensional rewriting
This paper studies 3-polygraphs as a framework for rewriting on
two-dimensional words. A translation of term rewriting systems into
3-polygraphs with explicit resource management is given, and the respective
computational properties of each system are studied. Finally, a convergent
3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In
order to prove these results, it is explained how to craft a class of
termination orders for 3-polygraphs.Comment: 30 pages, 35 figure
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
Intensional properties of polygraphs
We present polygraphic programs, a subclass of Albert Burroni's polygraphs,
as a computational model, showing how these objects can be seen as first-order
functional programs. We prove that the model is Turing complete. We use
polygraphic interpretations, a termination proof method introduced by the
second author, to characterize polygraphic programs that compute in polynomial
time. We conclude with a characterization of polynomial time functions and
non-deterministic polynomial time functions.Comment: Proceedings of TERMGRAPH 2007, Electronic Notes in Computer Science
(to appear), 12 pages, minor changes from previous versio
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
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
The three dimensions of proofs
In this document, we study a 3-polygraphic translation for the proofs of SKS,
a formal system for classical propositional logic. We prove that the free
3-category generated by this 3-polygraph describes the proofs of classical
propositional logic modulo structural bureaucracy. We give a 3-dimensional
generalization of Penrose diagrams and use it to provide several pictures of a
proof. We sketch how local transformations of proofs yield a non contrived
example of 4-dimensional rewriting.Comment: 38 pages, 50 figure
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half
Finite convergent presentations of plactic monoids for semisimple lie algebras
We study rewriting properties of the column presentation of plactic monoid
for any semisimple Lie algebra such as termination and confluence. Littelmann
described this presentation using L-S paths generators. Thanks to the shapes of
tableaux, we show that this presentation is finite and convergent. We obtain as
a corollary that plactic monoids for any semisimple Lie algebra satisfy
homological finiteness properties