论文标题
在单体上标记的加权自动机的可检测性
Detectability of labeled weighted automata over monoids
论文作者
论文摘要
在本文中，我们首次获得了四个基本概念的特征，可检测性的四个基本概念，用于一般标记的加权自动机（用$ \ mathcal {a}^{\ mathfrak {\ mathfrak {\ mathfrak {\ mathfrak {m}} $简短），其中四个概念是强（周期性的（周期性）可检测性（SECTECTIC）可检测性（sd）和spd和spd（Spect）和周期性（周期性）和w.首先，我们为$ \ mathcal {a}^{\ mathfrak {\ mathfrak {m}} $制定了并发构图，观察者和检测器的概念。其次，我们使用并发组合物为SD提供等效条件，使用检测器给出SPD的等效条件，并使用观察者为WD和WPD提供等效条件，全部用于一般的$ \ Mathcal {a} a}^{\ Mathfrak {\ Mathfrak {m}} $。第三，我们证明，对于在单型$（\ Mathbb {q}^k，+）上标记的加权自动机$（由$ \ Mathcal {a}^{\ mathbb {q}^k} $表示，与$ 2 $ 2 $ 2 $ 2 $ 2 $ -2 $ \ MATHCAL {a}^{\ MATHBB {Q}^K} $与NP完整的精确路径长度问题（由[Nykänen和Ukkonen，2002年证明）和一个Presburger arithmetic的子类之间的新颖连接。结果，我们证明，对于$ \ mathcal {a}^{\ mathbb {q}^k} $，可以在Conp中验证SD，而SPD，WD和WPD可以在$ 2 $ -EXPTIME中进行验证。最后，我们证明，通过monoid $（\ mathbb {n}，+）$验证确定性，无僵持和无差异$ \ Mathcal {a}^{\ mathbb {n}} $的问题的问题都是Conconp-Hard。 开发的原始方法将为$ \ Mathcal {a}^{\ Mathfrak {M Mathfrak {M}} $表征其他基础（例如，诊断性，不透明度）。我们最初还探索了标记的定时自动机中的可检测性，并证明SD验证问题是PSPACE-COMPLETE，而WD和WPD是不可决定的。
In this paper, we for the first time obtain characterization of four fundamental notions of detectability for general labeled weighted automata over monoids (denoted by $\mathcal{A}^{\mathfrak{M}}$ for short), where the four notions are strong (periodic) detectability (SD and SPD) and weak (periodic) detectability (WD and WPD). Firstly, we formulate the notions of concurrent composition, observer, and detector for $\mathcal{A}^{\mathfrak{M}}$. Secondly, we use the concurrent composition to give an equivalent condition for SD, use the detector to give an equivalent condition for SPD, and use the observer to give equivalent conditions for WD and WPD, all for general $\mathcal{A}^{\mathfrak{M}}$ without any assumption. Thirdly, we prove that for a labeled weighted automaton over monoid $(\mathbb{Q}^k,+)$ (denoted by $\mathcal{A}^{\mathbb{Q}^k}$), its concurrent composition, observer, and detector can be computed in NP, $2$-EXPTIME, and $2$-EXPTIME, respectively, by developing novel connections between $\mathcal{A}^{\mathbb{Q}^k}$ and the NP-complete exact path length problem (proved by [Nykänen and Ukkonen, 2002]) and a subclass of Presburger arithmetic. As a result, we prove that for $\mathcal{A}^{\mathbb{Q}^k}$, SD can be verified in coNP, while SPD, WD, and WPD can be verified in $2$-EXPTIME. Finally, we prove that the problems of verifying SD and SPD of deterministic, deadlock-free, and divergence-free $\mathcal{A}^{\mathbb{N}}$ over monoid $(\mathbb{N},+)$ are both coNP-hard. The developed original methods will provide foundations for characterizing other fundamental properties (e.g., diagnosability, opacity) for $\mathcal{A}^{\mathfrak{M}}$. We also initially explore detectability in labeled timed automata, and prove that the SD verification problem is PSPACE-complete, while WD and WPD are undecidable.
