The Generalized Loneliness Detector and Weak System Models for k-Set Agreement

This paper presents two weak partially synchronous system models
${\cal M}^{{\rm anti}(n-k)}$
and
${\cal M}^{{\rm sink}(n-k)}$
, which are just strong enough for solving
$k$
-set agreement: We introduce the generalized
$(n-k)$
-loneliness failure detector
${\cal L}(k)$
, which we first prove to be sufficient for solving
$k$
-set agreement, and show that
${\cal L}(k)$
but not
${\cal L}(k-1)$
can be implemented in both models.
${\cal M}^{{\rm anti}(n-k)}$
and
${\cal M}^{{\rm sink}(n-k)}$
are hence the first message passing models that lie between models where
$\Omega$
(and therefore consensus) can be implemented and the purely asynchronous model. We also address
$k$
-set agreement in anonymous systems, that is, in systems where (unique) process identifiers are not available. Since our novel
$k$
-set agreement algorithm using
${\cal L}(k)$
also works in anonymous systems, it turns out that the loneliness failure detector
${\cal L}={\cal L}(n-1)$
introduced by Delporte et al. is also the weakest failure detector for set agreement in anonymous systems. Finally, we analyze the relationship between
${\cal L}(k)$
and other failure detectors suitable for solving
$k$
-set agreement.