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The contributions of the current paper are as follows. Firstly, we formulate the notions of concurrent composition, observer, and detector for\n \n \n $\\mathcal {A}^{\\mathfrak {M}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . 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\n \n \n $\\mathcal {A}^{\\mathfrak {M}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n without any assumption. Thirdly, we prove that for a labeled weighted automaton over monoid\n \n \n $(\\mathbb {Q}^{k},+)$<\/jats:tex-math>\n \n (<\/mml:mo>\n \n \n \u211a<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n +<\/mml:mo>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n (denoted by\n \n \n $\\mathcal {A}^{\\mathbb {Q}^{k}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \u211a<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ), its concurrent composition, observer, and detector can be computed in\n N<\/jats:italic>\n P<\/jats:italic>\n , 2-\n E<\/jats:italic>\n X<\/jats:italic>\n P<\/jats:italic>\n T<\/jats:italic>\n I<\/jats:italic>\n M<\/jats:italic>\n E<\/jats:italic>\n , and 2-\n E<\/jats:italic>\n X<\/jats:italic>\n P<\/jats:italic>\n T<\/jats:italic>\n I<\/jats:italic>\n M<\/jats:italic>\n E<\/jats:italic>\n , respectively, by developing novel connections between\n \n \n $\\mathcal {A}^{\\mathbb {Q}^{k}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \u211a<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and the\n N<\/jats:italic>\n P<\/jats:italic>\n -complete exact path length problem (proven by [Nyk\u00e4nen and Ukkonen, 2002]) and a subclass of Presburger arithmetic. As a result, we prove that for\n \n \n $\\mathcal {A}^{\\mathbb {Q}^{k}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \u211a<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , SD can be verified in\n c<\/jats:italic>\n o<\/jats:italic>\n N<\/jats:italic>\n P<\/jats:italic>\n , while SPD, WD, and WPD can be verified in 2-\n E<\/jats:italic>\n X<\/jats:italic>\n P<\/jats:italic>\n T<\/jats:italic>\n I<\/jats:italic>\n M<\/jats:italic>\n E<\/jats:italic>\n . Particularly, for\n \n \n $\\mathcal {A}^{\\mathbb {Q}^{k}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \u211a<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in which from every state, a distinct state can be reached through some unobservable, instantaneous path, detector\n \n \n $\\mathcal {A}^{\\mathbb {Q}^{k}}_{det}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n d<\/mml:mi>\n e<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \u211a<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n can be computed in\n N<\/jats:italic>\n P<\/jats:italic>\n , and SPD can be verified in\n c<\/jats:italic>\n o<\/jats:italic>\n N<\/jats:italic>\n P<\/jats:italic>\n . Finally, we prove that the problems of verifying SD and SPD of deterministic, deadlock-free, and divergence-free\n \n \n $\\mathcal {A}^{\\mathbb {N}}$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \u2115<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n over monoid\n \n \n $(\\mathbb {N},+)$<\/jats:tex-math>\n \n (<\/mml:mo>\n \u2115<\/mml:mi>\n ,<\/mml:mo>\n +<\/mml:mo>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are both\n c<\/jats:italic>\n o<\/jats:italic>\n N<\/jats:italic>\n P<\/jats:italic>\n -hard. The original methods developed in this paper will provide foundations for characterizing other fundamental properties (e.g., diagnosability and opacity) in labeled weighted automata over monoids. In addition, in order to differentiate labeled weighted automata over monoids from labeled timed automata, we also initially explore detectability in labeled timed automata, and prove that the SD verification problem is\n P<\/jats:italic>\n S<\/jats:italic>\n P<\/jats:italic>\n A<\/jats:italic>\n C<\/jats:italic>\n E<\/jats:italic>\n -complete, while WD and WPD are undecidable.\n <\/jats:p>","DOI":"10.1007\/s10626-022-00362-8","type":"journal-article","created":{"date-parts":[[2022,5,23]],"date-time":"2022-05-23T05:04:58Z","timestamp":1653282298000},"page":"435-494","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Detectability of labeled weighted automata over monoids"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9547-103X","authenticated-orcid":false,"given":"Kuize","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,5,23]]},"reference":[{"key":"362_CR1","doi-asserted-by":"crossref","unstructured":"Adams S, Ouaknine J, Worrell J (2007) Undecidability of universality for timed automata with minimal resources. 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