Detectability of labeled weighted automata over monoids

In this paper, by developing appropriate methods, 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 [Nyk{a}nen 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. Particularly, for $mathcal{A}^{mathbb{Q}^k}$ in which from every state, a distinct state can be reached through some unobservable, instantaneous path, its detector can be computed in NP, and SPD can be verified in coNP. Finally, we prove that the problems of verifying SD and SPD of deterministic $mathcal{A}^{mathbb{N}}$ over monoid $(mathbb{N},+)$ are both NP-hard. The original methods developed in this paper will provide foundations for characterizing other fundamental properties (e.g., diagnosability and opacity) in $mathcal{A}^{mathfrak{M}}$.