We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.

翻译：我们考虑基于一组接受状态（或者更一般地基于一些向上封闭条件）定义的带状态矢量加法系统（VASS）的语言包含和语言等价性问题。一般来说，即使对于一维 VASS 来说，语言等价性问题也是不可判定的。因此，为了获得可判定性，我们研究了一些受限制的子类。一方面，我们证明了在 k-有限模糊 VASS（对于任何自然数 k）中，VASS 语言包含问题是可判定的，甚至是 Ackermann 可判定的。另一方面，我们证明了即使对于确定性 VASS，语言等价性问题也是 Ackermann 硬的。这两个结果导致了在几个可能的限制条件下 VASS 语言包含和等价性问题都是 Ackermann 完备的。我们的一些技术也可以应用于更广泛的无限状态系统中，即某些严格结构化转移系统的子类。