Detectability of labeled weighted automata over monoids
Kuize Zhang · Discrete Event Dynamic Systems · 2022
Abstract 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 $\mathcal {A}^{\mathfrak {M}}$ A M for short), where the four notions are strong (periodic) detectability (SD and SPD) and weak (periodic) detectability (WD and WPD). The contributions of the current paper are as follows. Firstly, we formulate the notions of concurrent composition, observer, and detector for $\mathcal {A}^{\mathfrak {M}}$ A M . Secondly, we use the concurrent composition to give a necessary and sufficient condition for SD, use the detector to give a necessary and sufficient condition for SPD, and use the observer to give necessary and sufficient conditions for WD and WPD, all for general $\mathcal {A}^{\mathfrak {M}}$ A M without any assumption. Thirdly, we prove that for a labeled weighted automaton over monoid $(\mathbb {Q}^{k},+)$ ( ℚ k , + ) (denoted by $\mathcal {A}^{\mathbb {Q}^{k}}$ A ℚ k ), its concurrent composition, observer, and detector can be computed in N P , 2- E X P T I M E , and 2- E X P T I M E , respectively, by developing novel connections between $\mathcal {A}^{\mathbb {Q}^{k}}$ A ℚ k and the N P -complete exact path length problem (proven by [Nykänen and Ukkonen, 2002]) and a subclass of Presburger arithmetic. As a result, we prove that for $\mathcal {A}^{\mathbb {Q}^{k}}$ A ℚ k , SD can be verified in c o N P , while SPD, WD, and WPD can be verified in 2- E X P T I M E . Particularly, for $\mathcal {A}^{\mathbb {Q}^{k}}$ A ℚ k