In this paper, \emph{diagnosability} is characterized for a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ over a dioid $\mathcal{D}$ as a real-time system. In order to represent time elapsing, a special class of dioids called \emph{progressive} are considered, in which there is a total canonical order, there is at least one element greater than $\textbf{1}$, the product of sufficiently many elements greater than $\textbf{1}$ is arbitrarily large, and the cancellative law is satisfied. Then a notion of diagnosability is formulated for $\mathcal{A}^{\mathcal{D}}$ over a progressive dioid $\mathcal{D}$. By developing a notion of \emph{concurrent composition}, a sufficient and necessary condition is given for diagnosability of automaton $\mathcal{A}^{\mathcal{D}}$. It is also proven that the problem of verifying diagnosability of $\mathcal{A}^{\underline{\mathbb{Q}}}$ is coNP-complete, where coNP-hardness even holds for deterministic, deadlock-free, and divergence-free $\mathcal{A}^{\underline{\mathbb{N}}}$, where $\underline{\mathbb{Q}}$ and $\underline{\mathbb{N}}$ are the max-plus dioids having elements in $\mathbb{Q}\cup\{-\infty\}$ and $\mathbb{N}\cup\{-\infty\}$, respectively.

中文翻译：

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标记最大加自动机的可诊断性

在本文中，\emph{diagnosability} 被表征为一个标记的最大加自动机 $\mathcal{A}^{\mathcal{D}}$ 作为一个实时系统。为了表示时间的流逝，考虑了一类特殊的称为 \emph{progressive} 的二元类，其中有一个总规范顺序，至少有一个元素大于 $\textbf{1}$，充分大于 $\textbf{1}$ 的多个元素任意大，满足抵消律。然后在渐进二元组 $\mathcal{D}$ 上为 $\mathcal{A}^{\mathcal{D}}$ 制定了可诊断性的概念。通过发展\emph{并发组合}的概念，给出了自动机$\mathcal{A}^{\mathcal{D}}$的可诊断性的充分必要条件。