Verification of Delayed Detectability for Unambiguous Weighted Automata

Detectability verification is an important problem that is tightly related to state estimation in the domain of discrete-event systems. Delayed detectability, referred to as (k1, k2)-detectability, is a property that necessitates the ability to determine the states of the system at the instant when the k1th event occurs, following an observation of k1 + k2 events. This differs from the conventional detectability problem due to the introduction of information delay. This paper delves into the examination of delayed detectability for unambiguous weighted automata. We employ a verification algorithm, strongly inspired by the work in [1], to verify this property. The fundamental concept involves verifying, after observing k1 events, whether each pair of current and subsequent state estimates can be ascertained after k2-step delays. Furthermore, we illustrate the pivotal steps of this algorithm using straightforward examples to validate the (k1, k2)-detectability of the examined unambiguous weighted automaton.