Diagnosability of labeled Dp-automata

In this paper, we formulate a notion of diagnosability for labeled weighted automata over a class of dioids which admit both positive and negative numbers as well as vectors. The weights can represent diverse physical meanings such as time elapsing and position deviations. We also develop an original tool called concurrent composition to verify diagnosability for such automata. These results are fundamentally new compared with the existing ones in the literature.

In a little more detail, diagnosability is characterized for a labeled weighted automaton $A^{D_p}$ over a special dioid $D_p$ called progressive, which can represent diverse physical meanings such as time elapsing and position deviations. In a progressive dioid, the canonical order is total, there is at least one eventually dominant element, there is no zero divisor, and the cancellative law is satisfied, where the functionality of an eventually dominant element t is to make every nonzero element a arbitrarily large by multiplying a by t for sufficiently many times. A notion of diagnosability is formulated for $A^{D_p}$. By developing a notion of concurrent composition, a necessary and sufficient condition is given for diagnosability of automaton $A^{D_p}$. It is proven that the problem of computing the concurrent composition for an automaton $A^{\underset{\_}{Q}}$ is $\mathrm{NP}$-complete, then the problem of verifying diagnosability of $A^{\underset{\_}{Q}}$ is proven to be $\mathrm{coNP}$-complete, where the $\mathrm{NP}$-hardness and $\mathrm{coNP}$-hardness results even hold for deterministic, deadlock-free, and divergence-free automaton $A^{\underset{\_}{N}}$, where $\underset{\_}{Q}$ and $\underset{\_}{N}$ are the max-plus dioids having elements in $Q\cup \{-\infty \}$ and $N\cup \{-\infty \}$, respectively. Several extensions of the main results have also been obtained.