4 - String Rewriting and 2-Polygraphs

Summary

This chapter recasts the notion of string rewriting system into the language of polygraphs. This notion, which consists of a set of pairs of words called relations or rewriting rules over a fixed alphabet, is introduced along with a more general variant adapted to categories. It is shown that the rewriting paths form the morphisms of a sesquicategory, in which the traditional concepts for abstract rewriting systems can be instantiated. The word problem is then introduced, and it is shown that it can be efficiently solved for convergent, i.e., confluent and terminating rewriting systems. In practice, confluence can be checked by inspecting the critical branchings of the rewriting system, and termination by introducing a suitable reduction order. The convergence of a rewriting system is also useful to show that it forms a presentation of a given category. Finally, residuation techniques are introduced, which allow proving useful properties of categories (such as the existence of pushouts) by performing computations on their presentations.