Detectability of labeled weighted automata over monoids

TL;DR

This paper characterizes four types of detectability in labeled weighted automata over monoids, providing complexity results and methods for verification, and extends some results to timed automata.

Contribution

It introduces a comprehensive characterization of detectability notions for automata over monoids, with complexity bounds and novel verification methods.

Findings

SD verification in coNP for automata over $ ext{Q}^k$

SPD, WD, WPD verification in 2-EXPTIME for automata over $ ext{Q}^k$

SD verification in PSPACE and WDs/WPDs undecidable for timed automata

Abstract

In this paper, we for the first time obtain characterization of four fundamental notions of detectability for general labeled weighted automata over monoids (denoted by $\mathcal{A}^{\mathfrak{M}}$ for short), where the four notions are strong (periodic) detectability (SD and SPD) and weak (periodic) detectability (WD and WPD). Firstly, we formulate the notions of concurrent composition, observer, and detector for $\mathcal{A}^{\mathfrak{M}}$. 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 $\mathcal{A}^{\mathfrak{M}}$ without any assumption. Thirdly, we prove that for a labeled weighted automaton over monoid $(\mathbb{Q}^k,+)$ (denoted by $\mathcal{A}^{\mathbb{Q}^k}$), its concurrent composition, observer, and detector can be computed in NP, $2$-EXPTIME, and $2$-EXPTIME, respectively, by developing novel connections between $\mathcal{A}^{\mathbb{Q}^k}$ and the NP-complete exact path length problem (proved by [Nyk\"{a}nen and Ukkonen, 2002]) and a subclass of Presburger arithmetic. As a result, we prove that for $\mathcal{A}^{\mathbb{Q}^k}$, SD can be verified in coNP, while SPD, WD, and WPD can be verified in $2$-EXPTIME. Finally, we prove that the problems of verifying SD and SPD of deterministic, deadlock-free, and divergence-free $\mathcal{A}^{\mathbb{N}}$ over monoid $(\mathbb{N},+)$ are both coNP-hard. The developed original methods will provide foundations for characterizing other fundamental properties (e.g., diagnosability, opacity) for $\mathcal{A}^{\mathfrak{M}}$. We also initially explore detectability in labeled timed automata, and prove that the SD verification problem is PSPACE-complete, while WD and WPD are undecidable.