We show that the regular separability problem of VASS reachability languages is decidable and $\mathbf{F}_{\omega}$-complete. At the heart of our decision procedure are doubly-marked graph transition sequences, a new proof object that tracks a suitable product of the VASS we wish to separate. We give a decomposition algorithm for DMGTS that not only achieves perfectness as known from MGTS, but also a new property called faithfulness. Faithfulness allows us to construct, from a regular separator for the $\mathbb{Z}$-versions of the VASS, a regular separator for the $\mathbb{N}$-versions. Behind faithfulness is the insight that, for separability, it is sufficient to track the counters of one VASS modulo a large number that is determined by the decomposition.

中文翻译：

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关于VASS可达性语言的可分性问题

我们证明了 VASS 可达性语言的正则可分性问题是可判定的且 $\mathbf{F}_{\omega}$ 完全的。我们决策过程的核心是双标记图转换序列，这是一个新的证明对象，用于跟踪我们希望分离的 VASS 的合适产品。我们给出了 DMGTS 的分解算法，它不仅实现了 MGTS 中已知的完美性，而且还实现了称为忠实度的新属性。忠实性允许我们从 VASS 的 $\mathbb{Z}$ 版本的常规分隔符构造 $\mathbb{N}$ 版本的常规分隔符。忠实性背后的洞察是，为了可分离性，跟踪一个 VASS 对分解确定的大数取模的计数器就足够了。