  We establish the equivalence between a class of asynchronous distributed
automata and a small fragment of least fixpoint logic, when restricted to
finite directed graphs. More specifically, the logic we consider is (a variant
of) the fragment of the modal $\mu$-calculus that allows least fixpoints but
forbids greatest fixpoints. The corresponding automaton model uses a network of
identical finite-state machines that communicate in an asynchronous manner and
whose state diagram must be acyclic except for self-loops. Exploiting the
connection with logic, we also prove that the expressive power of those
machines is independent of whether or not messages can be lost.
