  We characterize the complexity of the safety verification problem for
parameterized systems consisting of a leader process and arbitrarily many
anonymous and identical contributors. Processes communicate through a shared,
bounded-value register. While each operation on the register is atomic, there
is no synchronization primitive to execute a sequence of operations atomically.
We analyze the complexity of the safety verification problem when processes are
modeled by finite-state machines, pushdown machines, and Turing machines. The
problem is coNP-complete when all processes are finite-state machines, and is
PSPACE-complete when they are pushdown machines. The complexity remains
coNP-complete when each Turing machine is allowed boundedly many interactions
with the register. Our proofs use combinatorial characterizations of
computations in the model, and in case of pushdown-systems, some
language-theoretic constructions of independent interest.
