  We consider the verification of distributed systems composed of an arbitrary
number of asynchronous processes. Processes are identical finite-state machines
that communicate by reading from and writing to a shared memory. Beyond the
standard model with finitely many registers, we tackle round-based
shared-memory systems with fresh registers at each round. In the latter model,
both the number of processes and the number of registers are unbounded, making
verification particularly challenging. The properties studied are generic
presence reachability objectives, which subsume classical questions such as
safety or synchronization by expressing the presence or absence of processes in
some states. In the more general round-based setting, we establish that the
parameterized verification of presence reachability properties is
PSPACE-complete. Moreover, for the roundless model with finitely many
registers, we prove that the complexity drops down to NP-complete and we
provide several natural restrictions that make the problem solvable in
polynomial time.
