Semilinearity of the reachability relation for symmetric 2-counter machines

Determine whether the reachability relation of every symmetric 2-counter machine is semilinear, and establish effective semilinearity if the property holds.

Background

Effective semilinearity is a central technical tool in the paper’s decidability proof, but the proof establishes semilinearity only for particular auxiliary relations and sets used by the decision procedure. It does not establish semilinearity of the full reachability relation.

The authors explicitly state that they do not know whether the full reachability relation is semilinear, although they believe that it is. This remains distinct from decidability: the reachability problem is decidable even though the stronger global semilinearity property is unresolved.