  We study a dynamical system motivated by our earlier work on the statistical
physics of social balance on graphs that can be viewed as a generalization of
annihilating walks along two directions: first, the interaction topology is a
hypergraph; second, the ``number of particles`` at a vertex of the hypergraph
is an element of a finite field ${\bf Z}_{p}$ of integers modulo $p$, $p\geq
3$. Equivalently, particles move on a hypergraph, with a moving particle at a
vertex being replaced by one indistinguishable copy at each neighbor in a given
hyperedge; particles at a vertex collectively annihilate when their number
reaches $p$. The system we study can also be regarded as a natural
generalization of certain lights-out games to finite fields and hypergraph
topologies. Our result shows that under a liberal sufficient condition on the
nature of the interaction hypergraph there exists a polynomial time algorithm
(based on linear algebra over ${\bf Z}_{p}$) for deciding reachability and
recurrence of this dynamical system. Interestingly, we provide a counterexample
that shows that this connection does not extend to all graphs.
