2016/12/29 by Clément Alleaume, Alleaume, Clément, Philippe Malbos +1
Computer Science · #03D05 #18D05 #68Q42 #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Logic, programming, and type systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1612.09193
openalex publication_date 2016/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Squier introduced a homotopical method in order to describe all the relations amongst rewriting reductions of a confluent and terminating string rewriting system. From a string rewriting system he constructed a 2-dimensional combinatorial complex whose 2-cells are generated by relations induced by the rewriting rules. When the rewriting system is confluent and terminating, the homotopy of this complex can be characterized in term of confluence diagrams induced by the critical branchings of the rewriting system. Such a construction is now used to solve coherence problems for monoids using string rewriting systems. In this article, we show how to weaken the termination hypothesis in the description of all the relations amongst rewriting reductions. Our construction uses the decreasingness method introduced by van Oostrom. We introduce the notion of decreasing two-dimensional polygraph and we give sufficient conditions for a decreasing polygraph to be extended in a coherent way. In particular, we show how a confluent and quasi-terminating polygraph can be extended into a coherent presentation.