  Vector addition system with states (VASS) is a popular model for the
verification of concurrent systems. VASS consists of finitely many control
states and a set of counters which can be incremented and decremented, but not
tested for zero. VASS is a relatively well-studied model of computation and
many results regarding the decidability of decision problems for VASS are
well-known. Given that the complexity of solving almost all problems for VASS
is very high, various tractable over-approximations of the reachability
relation of VASS have been proposed in the literature. One such tractable
over-approximation is the so-called continuous VASS, in which counters are
allowed to have non-negative rational values and whenever an update is
performed, the update is first scaled by an arbitrary non-zero fraction.
  In this paper, we consider affine continuous VASS, which extend continuous
VASS by allowing integer affine operations. Affine continuous VASS serve as an
over-approximation to the model of affine VASS, in the same way that continuous
VASS over-approximates the reachability relation of VASS. We investigate the
tractability of affine continuous VASS with respect to the reachability,
coverability and state-reachability problems for different classes of affine
operations and we prove an almost-complete classification of the decidability
of these problems. Namely, except for the coverability problem for a single
family of classes of affine operations, we completely determine the
decidability status of these problems for all classes. Furthermore, except for
this single family, we also complement all of our decidability results with
tight complexity-theoretic upper and lower bounds.
