# 20.7 Equivalence of Cost Register Automata: the quest for decidability

Consider the model introduced in this problem. It is mentioned therein that equivalence is undecidable for that model. We ask what further restrictions we can impose on that model to make equivalence decidable.

If the $$\min$$ operation is removed, then this is known to lead to decidability [Alur et al.'13]. But as soon as we have $$\min$$, we don't know of a model with decidable equivalence.

We conjecture that it is decidable is the CRA is ordered; this means that the registers can be ordered, say $$x_1, x_2, \ldots, x_n$$, such that for all updates, $$x_i$$ is only updated with registers $$x_j$$ for $$j \geq i$$.