Weak Cost Register Automata are Still Powerful

We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over N with updates using min and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, namely equivalence, upperboundedness, and semilinearity. In particular, the undecidability of equivalence disproves a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.