Exact complexity of reachability for symmetric 2-counter machines

Determine the exact computational complexity of the reachability problem for symmetric 2-counter machines, narrowing the gap between the elementary upper bound established by the decision procedure and the currently known lower bound inherited from symmetric 2-vector addition systems with states.

Background

The main theorem establishes decidability of reachability for symmetric 2-counter machines, and the complexity analysis provides an elementary upper bound for the resulting decision procedure. However, the paper reports only a lower bound inherited from the corresponding symmetric 2-vector addition system model.

The authors explicitly leave the exact complexity unresolved and note that it may be complete for the relevant complexity class in every dimension at least two. A later discussion reiterates that narrowing this apparent complexity gap remains future research.