Fixed-bounded rule sizes

Determine whether confluence is undecidable for strictly length-reducing cycle-rewriting systems in which both the left-hand and right-hand sides of every rule are bounded by fixed constants, thereby extending undecidability beyond the bounded-shape construction with a growing alphabet and rule count.

Background

The paper proves that confluence is undecidable, indeed Π⁰₁-complete, for finite nonerasing strictly length-reducing cycle-rewriting systems over a fixed binary alphabet. It also gives a bounded-shape variant in which left-hand sides have length at most five, right-hand sides have length at most four, and symbol weights belong to a fixed finite set, but the alphabet and number of rules are allowed to grow with the instance.

The unresolved issue is whether both sides of every rule can be bounded by fixed constants while retaining undecidability. The authors note that any such family would require a growing alphabet, because fixing both the alphabet and rule-length bounds would leave only finitely many possible systems.

References

It remains open whether confluence is undecidable for strictly length-reducing cycle systems with a fixed bound on both sides of every rule. Such a family would necessarily use a growing alphabet, since a fixed alphabet with bounded rule lengths admits only finitely many systems.

— Undecidability of Confluence for Binary Length-Reducing Cycle Rewriting  (2608.18859 - Campbell, 19 Aug 2026) in Section 6, “Conclusion and Future Work”

It is open whether undecidability can be obtained with a fixed number of binary length-reducing rules.

— Undecidability of Confluence for Binary Length-Reducing Cycle Rewriting  (2608.18859 - Campbell, 19 Aug 2026) in Section 6, “Conclusion and Future Work”