Coherence of string rewriting systems by decreasingness

TL;DR

This paper extends the homotopical approach to string rewriting systems by weakening termination assumptions, using decreasingness methods to achieve coherence in presentations of monoids.

Contribution

It introduces decreasing two-dimensional polygraphs and provides conditions for their coherent extension, broadening the applicability of coherence results beyond terminating systems.

Findings

Weaker conditions for coherence in string rewriting systems.

Introduction of decreasing polygraphs for coherence.

Extension of confluent, quasi-terminating systems into coherent presentations.

Abstract

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.