139 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-polygraphs
This note presents the first known class of termination orders for
3-polygraphs, together with an application.Comment: 4 pages, 12 figure
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
Coherent presentations of Artin monoids
We compute coherent presentations of Artin monoids, that is presentations by
generators, relations, and relations between the relations. For that, we use
methods of higher-dimensional rewriting that extend Squier's and Knuth-Bendix's
completions into a homotopical completion-reduction, applied to Artin's and
Garside's presentations. The main result of the paper states that the so-called
Tits-Zamolodchikov 3-cells extend Artin's presentation into a coherent
presentation. As a byproduct, we give a new constructive proof of a theorem of
Deligne on the actions of an Artin monoid on a category
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
Coherence in monoidal track categories
We introduce homotopical methods based on rewriting on higher-dimensional
categories to prove coherence results in categories with an algebraic
structure. We express the coherence problem for (symmetric) monoidal categories
as an asphericity problem for a track category and we use rewriting methods on
polygraphs to solve it. The setting is extended to more general coherence
problems, seen as 3-dimensional word problems in a track category, including
the case of braided monoidal categories.Comment: 32 page
Two polygraphic presentations of Petri nets
This document gives an algebraic and two polygraphic translations of Petri
nets, all three providing an easier way to describe reductions and to identify
some of them. The first one sees places as generators of a commutative monoid
and transitions as rewriting rules on it: this setting is totally equivalent to
Petri nets, but lacks any graphical intuition. The second one considers places
as 1-dimensional cells and transitions as 2-dimensional ones: this translation
recovers a graphical meaning but raises many difficulties since it uses
explicit permutations. Finally, the third translation sees places as
degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is
a setting equivalent to Petri nets, equipped with a graphical interpretation.Comment: 28 pages, 24 figure
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
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