We consider a variant of VASS extended with integer counters, denoted VASS+ℤ. These are automata equipped with ℕ- and ℤ-counters; the ℕ-counters are required to remain nonnegative and the ℤ-counters do not have this restriction. We study the complexity of the reachability problem for VASS+ℤ when the number of ℕ-counters is fixed. We show that reachability is NP-complete in 1-VASS+ℤ (i.e. when there is only one ℕ-counter) regardless of unary or binary encoding. For d ≥ 2, using a KLMST-based algorithm, we prove that reachability in d-VASS+ℤ lies in the complexity class ℱ_{d+2}. Our upper bound improves on the naively obtained Ackermannian complexity by simulating the ℤ-counters with ℕ-counters. To complement our upper bounds, we show that extending VASS with integer counters significantly lowers the number of ℕ-counters needed to exhibit hardness. We prove that reachability in unary 2-VASS+ℤ is PSpace-hard; without ℤ-counters this lower bound is only known in dimension 5. We also prove that reachability in unary 3-VASS+ℤ is Tower-hard. Without ℤ-counters, reachability in 3-VASS has elementary complexity and Tower-hardness is only known in dimension 8.

The reachability problem for Vector Addition Systems (VAS) is a central decision problem in the theory of infinite-state systems, first solved by Kosaraju and Mayr in the 1980s. An alternative, conceptually simpler approach introduced by Leroux shows that non-reachability is always witnessed by semilinear inductive invariants, yielding a decision procedure by combining an enumeration of runs with a search for such invariants. However, the construction of these invariants relies on a back-and-forth scheme that depends symmetrically on the source and the target. As a result, the invariants are not guaranteed to reflect the structural properties of the VAS, and the construction is difficult to extend to asymmetric models such as Branching VAS. We introduce a new forward-only construction of semilinear inductive invariants for VAS. Our method builds invariants from the source configuration alone and avoids the need for backward reasoning. This yields invariants that are more canonical and better aligned with the structure of the system. In particular, our method produces periodic inductive invariants for periodic VAS. Beyond its intrinsic interest, our approach provides a step toward extending invariant-based techniques to Branching VAS.

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