Complexity of Deciding Detectability in Discrete Event Systems

Detectability of discrete event systems (DESs) is a question whether the current and subsequent states can be determined based on observations. Shu and Lin designed a polynomial-time algorithm to check strong (periodic) detectability and an exponential-time (polynomial-space) algorithm to check weak (periodic) detectability. Zhang showed that checking weak (periodic) detectability is PSpace-complete. This intractable complexity opens a question whether there are structurally simpler DESs for which the problem is tractable. In this paper, we show that it is not the case by considering DESs represented as deterministic finite automata without non-trivial cycles, which are structurally the simplest deadlock-free DESs. We show that even for such very simple DESs, checking weak (periodic) detectability remains intractable. On the contrary, we show that strong (periodic) detectability of DESs can be efficiently verified on a parallel computer

Complexity of Detectability, Opacity and A-Diagnosability for Modular Discrete Event Systems

We study the complexity of deciding whether a modular discrete event system is detectable (resp. opaque, A-diagnosable). Detectability arises in the state estimation of discrete event systems, opacity is related to the privacy and security analysis, and A-diagnosability appears in the fault diagnosis of stochastic discrete event systems. Previously, deciding weak detectability (opacity, A-diagnosability) for monolithic systems was shown to be PSPACE-complete. In this paper, we study the complexity of deciding weak detectability (opacity, A-diagnosability) for modular systems. We show that the complexities of these problems are significantly worse than in the monolithic case. Namely, we show that deciding modular weak detectability (opacity, A-diagnosability) is EXPSPACE-complete. We further discuss a special case where all unobservable events are private, and show that in this case the problems are PSPACE-complete. Consequently, if the systems are all fully observable, then deciding weak detectability (opacity) for modular systems is PSPACE-complete

On Verification of D-Detectability for Discrete Event Systems

Detectability has been introduced as a generalization of state-estimation properties of discrete event systems studied in the literature. It asks whether the current and subsequent states of a system can be determined based on observations. Since, in some applications, to exactly determine the current and subsequent states may be too strict, a relaxed notion of D-detectability has been introduced, distinguishing only certain pairs of states rather than all states. Four variants of D-detectability have been defined: strong (periodic) D-detectability and weak (periodic) D-detectability. Deciding weak (periodic) D-detectability is PSpace-complete, while deciding strong (periodic) detectability or strong D-detectability is polynomial (and we show that it is actually NL-complete). However, to the best of our knowledge, it is an open problem whether there exists a polynomial-time algorithm deciding strong periodic D-detectability. We solve this problem by showing that deciding strong periodic D-detectability is a PSpace-complete problem, and hence there is no polynomial-time algorithm unless PSpace = P. We further show that there is no polynomial-time algorithm deciding strong periodic D-detectability even for systems with a single observable event, unless P = NP. Finally, we propose a class of systems for which the problem is tractable.Comment: Extended version of a paper accepted for WODES 202

Deciding Detectability for Labeled Petri Nets

Detectability of discrete event systems (DESs) is a property to determine a priori whether the current and subsequent states can be determined based on observations. In this paper, we investigate the verification of two detectability properties -- strong detectability and weak detectability -- for DESs modeled by labeled Petri nets. Strong detectability requires that we can always determine, after a finite number of observations, the current and subsequent markings of the system, while weak detectability requires that we can determine, after a finite number of observations, the current and subsequent markings for some trajectories of the system. We show that for DESs modeled by labeled Petri nets, checking strong detectability is decidable whereas checking weak detectability is undecidable. Our results extend the existing studies on the verification of detectability from finite-state automata to labeled Petri nets. As a consequence, we strengthen a result on checking current-state opacity for labeled Petri nets

Critical Observability for Automata and Petri Nets

Critical observability is a property of cyber-physical systems to detect whether the current state belongs to a set of critical states. In safety-critical applications, critical states model operations that may be unsafe or of a particular interest. De Santis et al. introduced critical observability for linear switching systems, and Pola et al. adapted it for discrete-event systems, focusing on algorithmic complexity. We study the computational complexity of deciding critical observability for systems modeled as (networks of) finite-state automata and Petri nets. We show that deciding critical observability is (i) NL-complete for finite automata, that is, it is efficiently verifiable on parallel computers, (ii) PSPACE-complete for networks of finite automata, that is, it is very unlikely solvable in polynomial time, and (iii) undecidable for labeled Petri nets, but becoming decidable if the set of critical states (markings) is finite or co-finite, in which case the problem is as hard as the non-reachability problem for Petri nets.Comment: Accepted for publication in IEEE TA

Detectability of labeled weighted automata over monoids

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 timed models 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 without any assumption. Thirdly, we prove that for an LWA over monoid $(\mathbb{Q},+,0)$ (denoted by $\mathcal{A}^{\mathbb{Q}}$), its concurrent composition, observer, and detector can be computed in NP, NEXPTIME, and NP, respectively, by developing a novel connection between $\mathcal{A}^{\mathbb{Q}}$ and the NP-complete exact path length problem (proved by [Nyk\"{a}nen and Ukkonen, 2002]). As a result, we prove that for $\mathcal{A}^{\mathbb{Q}}$, SD and SPD can be verified in coNP, while WD and WPD can be verified in NEXPTIME. Finally, we prove that the problems of verifying SD and SPD of deterministic $\mathcal{A}^{\mathbb{N}}$ over monoid $(\mathbb{N},+,0)$ are both coNP-hard.Comment: 41 pages, 19 figur

Verification of Detectability Using Petri Nets and Detector

Detectability describes the property of a system to uniquely determine, after a finite number of observations, the current and subsequent states. In this paper, to reduce the complexity of checking the detectability properties in the framework of bounded labeled Petri nets, we use a new tool, which is called detector, to verifying the strong detectability and periodically strong detectability. First, an approach, which is based on the reachable graph and its detector, is proposed. Then, we develop a novel approach which is based on the analysis of the detector of the basis reachability graph. Without computing the whole reachability space, and without building the observer, the proposed approaches are more efficient.Comment: 8 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:1903.09298, arXiv:1903.0782

Revisiting delayed strong detectability of discrete-event systems

Among notions of detectability for a discrete-event system (DES), strong detectability implies that after a finite number of observations to every output/label sequence generated by the DES, the current state can be uniquely determined. This notion is strong so that by using it the current state can be easily determined. In order to keep the advantage of strong detectability and weaken its disadvantage, we can additionally take some "subsequent outputs" into account in order to determine the current state. Such a modified observation will make some DES that is not strongly detectable become "strongly detectable in a weaker sense", which we call "{\it $K$-delayed strong detectability}" if we observe at least $K$ outputs after the time at which the state need to be determined. In this paper, we study $K$-delayed strong detectability for DESs modeled by finite-state automata (FSAs), and give a polynomial-time verification algorithm by using a novel concurrent-composition method. Note that the algorithm applies to all FSAs. Also by the method, an upper bound for $K$ has been found, and we also obtain polynomial-time verification algorithms for $(k_1,k_2)$-detectability and $(k_1,k_2)$-D-detectability of FSAs firstly studied by [Shu and Lin, 2013]. Our algorithms run in quartic polynomial time and apply to all FSAs, are more effective than the sextic polynomial-time verification algorithms given by [Shu and Lin 2013] based on the usual assumptions of deadlock-freeness and having no unobservable reachable cycle. Finally, we obtain polynomial-time synthesis algorithms for enforcing delayed strong detectability, which are more effective than the exponential-time synthesis algorithms in the supervisory control framework in the literature.Comment: 25 pages, 11 figures, partially submitted to the 58th IEEE Conference on Decision and Control (2019

Verification of Detectability in Petri Nets Using Verifier Nets

Detectability describes the property of a system whose current and the subsequent states can be uniquely determined after a finite number of observations. In this paper, we developed a novel approach to verifying strong detectability and periodically strong detectability of bounded labeled Petri nets. Our approach is based on the analysis of the basis reachability graph of a special Petri net, called Verifier Net, that is built from the Petri net model of the given system. Without computing the whole reachability space and without enumerating all the markings, the proposed approaches are more efficient.Comment: arXiv admin note: text overlap with arXiv:1903.0782

A unified method to decentralized state inference and fault diagnosis/prediction of discrete-event systems

The state inference problem and fault diagnosis/prediction problem are fundamental topics in many areas. In this paper, we consider discrete-event systems (DESs) modeled by finite-state automata (FSAs). There exist results for decentralized versions of the latter problem but there is almost no result for a decentralized version of the former problem. We propose a decentralized version of strong detectability called co-detectability which implies that once a system satisfies this property, for each generated infinite-length event sequence, at least one local observer can determine the current and subsequent states after a common observation time delay. We prove that the problem of verifying co-detectability of FSAs is coNP-hard. Moreover, we use a unified concurrent-composition method to give PSPACE verification algorithms for co-detectability, co-diagnosability, and co-predictability of FSAs, without any assumption or modifying the FSAs under consideration, where co-diagnosability is firstly studied by [Debouk & Lafortune & Teneketzis 2000], while co-predictability is firstly studied by [Kumar \& Takai 2010]. By our proposed unified method, one can see that in order to verify co-detectability, more technical difficulties will be met compared to verifying the other two properties, because in co-detectability, generated outputs are counted, but in the latter two properties, only occurrences of events are counted. For example, when one output was generated, any number of unobservable events could have occurred. The PSPACE-hardness of verifying co-diagnosability is already known in the literature. In this paper, we prove the PSPACE-hardness of verifying co-predictability.Comment: 30 pages,12 figures, 2 table