Verification of D-detectability for Unambiguous Weighted Automata

The detectability problem has been widely studied in recent years, which is a property that characterizes the capability of discerning the current and future states of a discrete event system after a finite number of observations. However, the requirement to accurately determine a single state may be too stringent for some specific applications. Considering this, Shu proposes a relaxed version of detectability, called D-detectability, which focuses on distinguishing state pairs. It is worth noting that these works are all based on the model of logical automata. In this work, the notion of D-detectability is extended from logical automata [1] to unambiguous weighted automata (UWAs), and four types of D-detectability on UWAs are defined. Additionally, we establish that the observer structure presented in [2] for checking the detectability of UWAs can also be applied to verify D-detectabilities. We derive necessary and sufficient conditions from the observer to verify four types of D-detectability.