2020/06/25 by Kuize Zhang, Zhang, Kuize
Computer Science · Mathematics · #68Q45 #93B07 #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Optimization and Control (math.OC) #cs.FL #math.OC #msc:68Q45 #msc:93B07
paper · pdf · doi:10.48550/arxiv.2006.14164
64 pages, 25 figures
arxiv created 2022/01/23 · arxiv updated 2022/01/25
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 A^\mathfrakM 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 A^\mathfrakM. 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 A^\mathfrakM without any assumption. Thirdly, we prove that for a labeled weighted automaton over monoid (ℚk,+) (denoted by Aℚk), its concurrent composition, observer, and detector can be computed in NP, 2-EXPTIME, and 2-EXPTIME, respectively, by developing novel connections between Aℚk 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 Aℚk, 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 Aℕ over monoid (ℕ,+) are both coNP-hard. The developed original methods will provide foundations for characterizing other fundamental properties (e.g., diagnosability, opacity) for A^\mathfrakM. 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.