  In a recent paper by Hellerstein [15], a tight relationship was conjectured
between the number of strata of a Datalog${}^\neg$ program and the number of
"coordination stages" required for its distributed computation. Indeed, Ameloot
et al. [9] showed that a query can be computed by a coordination-free
relational transducer network iff it is monotone, thus answering in the
affirmative a variant of Hellerstein's CALM conjecture, based on a particular
definition of coordination-free computation. In this paper, we present three
additional models for declarative networking. In these variants, relational
transducers have limited access to the way data is distributed. This variation
allows transducer networks to compute more queries in a coordination-free
manner: e.g., a transducer can check whether a ground atom $A$ over the input
schema is in the "scope" of the local node, and then send either $A$ or $\neg
A$ to other nodes.
  We show the surprising result that the query given by the well-founded
semantics of the unstratifiable win-move program is coordination-free in some
of the models we consider. We also show that the original transducer network
model [9] and our variants form a strict hierarchy of classes of
coordination-free queries. Finally, we identify different syntactic fragments
of Datalog${}^{\neg\neg}_{\forall}$, called semi-monotone programs, which can
be used as declarative network programming languages, whose distributed
computation is guaranteed to be eventually consistent and coordination-free.
