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Register a new userOn detectability of labeled Petri nets and finite automata
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Added by
Kuize Zhang
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We study detectability properties for labeled Petri nets and finite automata. We first study weak approximate detectability (WAD) that implies that there exists an infinite observed output sequence of the system such that each prefix of the output sequence with length greater than a given value allows an observer to determine if the current state belongs to a given set. We also consider two new concepts called instant strong detectability (ISD) and eventual strong detectability (ESD). The former property implies that for each possible infinite observed output sequence each prefix of the output sequence allows reconstructing the current state. The latter implies that for each possible infinite observed output sequence, there exists a value such that each prefix of the output sequence with length greater than that value allows reconstructing the current state. Results: WAD: undecidable for labeled Petri nets, PSPACE-complete for finite automata ISD: decidable and EXPSPACE-hard for labeled Petri nets, belongs to P for finite automata ESD: decidable under promptness assumption and EXPSPACE-hard for labeled Petri nets, belongs to P for finite automata SD: belongs to P for finite automata, strengthens Shu and Lins 2011 results based on two assumptions of deadlock-freeness and promptness ISD<SD<ESD<WD<WAD for both labeled Petri nets and finite automata
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The categorical modeling of Petri nets has received much attention recently. The Dialectica construction has also had its fair share of attention. We revisit the use of the Dialectica construction as a categorical model for Petri nets generalizing the original application to suggest that Petri nets with different kinds of transitions can be modeled in the same categorical framework. Transitions representing truth-values, probabilities, rates or multiplicities, evaluated in different algebraic structures called lineales are useful and are modeled here in the same category. We investigate (categorical instances of) this generalized model and its connections to more recent models of categorical nets.
In this paper, by developing appropriate methods, 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. Particularly, for $mathcal{A}^{mathbb{Q}^k}$ in which from every state, a distinct state can be reached through some unobservable, instantaneous path, its detector can be computed in NP, and SPD can be verified in coNP. Finally, we prove that the problems of verifying SD and SPD of deterministic $mathcal{A}^{mathbb{N}}$ over monoid $(mathbb{N},+)$ are both NP-hard. The original methods developed in this paper will provide foundations for characterizing other fundamental properties (e.g., diagnosability and opacity) in $mathcal{A}^{mathfrak{M}}$.
Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and the currently best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound, i.e. that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi and other areas, that are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the currently best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack.
We consider approaches for causal semantics of Petri nets, explicitly representing dependencies between transition occurrences. For one-safe nets or condition/event-systems, the notion of process as defined by Carl Adam Petri provides a notion of a run of a system where causal dependencies are reflected in terms of a partial order. A well-known problem is how to generalise this notion for nets where places may carry several tokens. Goltz and Reisig have defined such a generalisation by distinguishing tokens according to their causal history. However, this so-called individual token interpretation is often considered too detailed. A number of approaches have tackled the problem of defining a more abstract notion of process, thereby obtaining a so-called collective token interpretation. Here we give a short overview on these attempts and then identify a subclass of Petri nets, called structural conflict nets, where the interplay between conflict and concurrency due to token multiplicity does not occur. For this subclass, we define abstract processes as equivalence classes of Goltz-Reisig processes. We justify this approach by showing that we obtain exactly one maximal abstract process if and only if the underlying net is conflict-free with respect to a canonical notion of conflict.
This paper proposes a semi-structural approach to verify the nonblockingness of a Petri net. We construct a structure, called minimax basis reachability graph (minimax-BRG): it provides an abstract description of the reachability set of a net while preserving all information needed to test if the net is blocking. We prove that a bounded deadlock-free Petri net is nonblocking if and only if its minimax-BRG is unobstructed, which can be verified by solving a set of integer constraints and then examining the minimax-BRG. For Petri nets that are not deadlock-free, one needs to determine the set of deadlock markings. This can be done with an approach based on the computation of maximal implicit firing sequences enabled by the markings in the minimax-BRG. The approach we developed does not require the construction of the reachability graph and has wide applicability.
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