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