  Automata over infinite alphabets have recently come to be studied extensively
as potentially useful tools for solving problems in verification and database
theory. One popular model of automata studied is the Class Memory Automata
(CMA), for which the emptiness problem is equivalent to Petri Net Reachability.
We identify a restriction - which we call weakness - of CMA, and show that
their emptiness problem is equivalent to Petri Net Coverability. Further, we
show that in the deterministic case they are closed under all Boolean
operations. We clarify the connections between weak CMA and existing automata
over data languages. We also extend CMA to operate over multiple levels of
nested data values, and show that while these have undecidable emptiness in
general, adding the weakness constraint recovers decidability of emptiness, via
reduction to coverability in well-structured transition systems. We also
examine connections with existing automata over nested data.
