Consequences of restricting updates to a single counter

Determine how much weaker symmetric counter machines become when all update transitions are single-counter updates while zero-test transitions remain unrestricted.

Background

The variants section considers two restrictions: symmetry of zero-test transitions and the requirement that every update transition be nonzero on at most one coordinate. The paper shows that symmetric zero-tests alone do not reduce expressive power, and that imposing both restrictions yields a model essentially equivalent to a one-counter machine for zero source and target vectors.

The effect of imposing only the single-counter-update restriction is not determined. The authors explicitly state that they do not know its consequences, leaving open the precise loss of expressive power caused by this restriction.