Decidability and Complexity of Decision Problems for Affine Continuous VASS

Abstract

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.