Church–Rosser congruential deterministic context-free languages

Construct a natural structurally defined subclass of deterministic context-free languages that strictly extends the regular languages and whose members are Church–Rosser congruential, and determine whether membership in this subclass is decidable.

Background

Every regular language is known to be Church–Rosser congruential. The unresolved extension asks whether an analogous useful subclass exists among deterministic context-free languages and whether its finite confluent length-reducing presentations can be recognized algorithmically.

References

This problem consists of two interdependent parts: Is there a natural, structurally defined subclass of the deterministic context-free languages (DCFL) that strictly extends the class of all regular languages, such that all members of this subclass are Church-Rosser congruential? Is there an algorithm to decide whether an arbitrary DCFL belongs to this subclass --- or, equivalently, is it decidable whether an arbitrary DCFL can be presented by a finite, confluent, and length-reducing string-rewriting system with finitely many congruence classes?

— String Rewriting Systems: Brief Introduction and Sample of Open Problems  (2608.19397 - Kfoury, 19 Aug 2026) in Section 6, “Formal Languages that Are Church-Rosser Congruential”