Detectability of labeled weighted weighted automata over monoids
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by
Kuize Zhang
2020
Abstract
Discrete-event systems (DESs) are generally composed of transitions between
discrete states caused by spontaneous occurrences of partially-observed (aka
labeled) events. Detectability is a fundamental property in labeled dynamical
systems, which describes whether one can use an observed label sequence to
reconstruct the current state. Labeled weighted automata (LWAs) can be regarded
as a timed model of DESs.
In this paper, by developing appropriate methods, we for the first time
obtain characterization of four fundamental notions of detectability for
general LWAs over monoids, 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 LWAs. 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 LWAs. Thirdly, we
prove that for an LWA over monoid (β,+,0) (denoted by
π^β), its concurrent composition, observer, and detector
can be computed in NP, 2-EXPTIME, and NP, respectively, by developing a novel
connection between π^β and the NP-complete exact path
length problem (proved by [NykΓ€nen and Ukkonen, 2002]). As a result, we
prove that for π^β, SD and SPD can be verified in coNP,
while WD and WPD can be verified in 2-EXPTIME. Finally, we prove that the
problems of verifying SD and SPD of deterministic π^β
are both coNP-hard.
The original methods developed in this paper will provide foundations for
characterizing other fundamental properties (e.g., diagnosability and opacity)
in LWAs over monoids.
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2006.14164v2
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