Confluence after adding commutation

Determine whether confluence modulo a single commutation rule remains decidable for a finite confluent strictly length-reducing string-rewriting system, and characterize when that commutation preserves uniqueness of normal forms modulo the commutation theory.

Background

Confluence is undecidable for arbitrary strictly length-reducing systems supplemented by one commutation rule. The paper identifies the unresolved restricted case in which the original length-reducing system is already known to be confluent.

References

However, this classic result leaves a structural question regarding system perturbation. For example, if we narrow our scope from an arbitrary system to one that is already confluent, the behavioral threshold shifts. Specifically, if we start with a finite, strictly length-reducing system that is explicitly given to be confluent, the operational consequences of injecting a single commutative rule remain unknown.

— String Rewriting Systems: Brief Introduction and Sample of Open Problems  (2608.19397 - Kfoury, 19 Aug 2026) in Section 6, “Confluence of Length-Reducing Rules with Commutation”