Decidability of reachability for symmetric counter machines with at least three counters

Determine whether the reachability problem is decidable for symmetric counter machines with three or more counters, in particular for symmetric 3-counter machines whose update transitions are invariant under all permutations of the counters, and establish or refute decidability for every number of counters.

Background

The paper proves that reachability is decidable for symmetric 2-counter machines, where update transitions are invariant under swapping the two counters. It also proves that reachability is undecidable for 3-counter machines invariant under the cyclic permutation group of order three, thereby locating a boundary between full permutation symmetry and weaker symmetry conditions.

The authors explain that their decidability proof relies on low-dimensional properties of reachable sets and reachability relations and does not extend to three counters. They conjecture that decidability holds for symmetric counter machines with any number of counters, but leave the three-counter and higher-dimensional cases unresolved.

References

We conjecture that the decidability extends to machines with 3 or more counters that are symmetric, meaning the set of update transitions is invariant under any permutation of counters. We leave this problem open for machines with 3 and more counters, while identifying a frontier of the impact of symmetry: we prove undecidability for 3-counter machines invariant under the cyclic permutation group of order~3.

— Symmetric two-counter machines  (2610.23495 - Jaros et al., 16 Oct 2026) in Abstract; also Section 1, subsection “Open questions”

It remains to be established the expressive power of the fragment, and the complexity of its reachability problem.

— Symmetric two-counter machines  (2610.23495 - Jaros et al., 16 Oct 2026) in Section 1, subsection “Open questions”