Concurrency Theorems for Non-linear Rewriting Theories

TL;DR

This paper develops concurrency theorems for non-linear rewriting systems in categories like quasi-topoi, enabling compositional reasoning and static analysis techniques for complex graph transformations.

Contribution

It introduces the first concurrency theorem for non-linear SqPO rewriting in quasi-topoi and extends results to non-linear DPO rewriting in rm-adhesive categories.

Findings

Established a concurrency theorem for non-linear SqPO rewriting.

Revealed a 'backpropagation effect' in rule composition.

Extended concurrency results to non-linear DPO rewriting.

Abstract

Sesqui-pushout (SqPO) rewriting along non-linear rules and for monic matches is well-known to permit the modeling of fusing and cloning of vertices and edges, yet to date, no construction of a suitable concurrency theorem was available. The lack of such a theorem, in turn, rendered compositional reasoning for such rewriting systems largely infeasible. We develop in this paper a suitable concurrency theorem for non-linear SqPO-rewriting in categories that are quasi-topoi (subsuming the example of adhesive categories) and with matches required to be regular monomorphisms of the given category. Our construction reveals an interesting "backpropagation effect" in computing rule compositions. We derive in addition a concurrency theorem for non-linear double pushout (DPO) rewriting in rm-adhesive categories. Our results open non-linear SqPO and DPO semantics to the rich static analysis techniques available from concurrency, rule algebra and tracelet theory.