Rob van Glabbeek, Ursula Goltz, Jens-Wolfhard Schicke
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This paper continues and completes the work of arXiv:1103.5916. Together, these papers show that a structural conflict net is conflict-free iff it has exactly one maximal run---equivalently formalised as maximal BD-run, maximal BD-process or maximal GR-process up to swapping equivalence. The "only if" direction, together with a taxonomy of suitable notions of maximality, is contributed here
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.
A new formalism of Petri nets, based on the adoption of the "position-arc-transition" triad and "transition-arc-position" triad as structure-forming units is introduced. In accordance with the Fusion principle, an analytical representation of Petri nets is developed. We propose the concept of Creative Petri Nets, which allows to implement structural changes in Petri net by procreation/deletion of structural units or complexes.
New subclasses of Petri nets - Petri nets receptors and Petri nets effectors are introduced. The introduction/exclusion of such substructures in the main Petri net may be fulfilled in accordance with the Fusion/Defusion principles. We propose two pairs of entities: position marking receptor (effector) and transition marking receptor (effector), which allow to observe parameters of the main Petri net and, if necessary, to carry out their regulation.
Brown and Gurr have introduced a model of Petri Nets that is based on de~Paiva's Dialectica categories. This model was refined in an unpublished technical report, where Petri nets with multiplicities, instead of {\em elementary} nets (i.e., nets with multiplicities zero and one only) were considered. In this note we expand this modelling to deal with {\em fuzzy} petri nets. The basic idea is to use as the dualizing object in the Dialectica categories construction, the unit interval that has all the properties of a {\em lineale} structure.
In a previous paper we introduced immediate observation Petri nets, a subclass of Petri nets with application domains in distributed protocols (population protocols) and theoretical chemistry (chemical reaction networks). IO nets enjoy many useful properties, but like the general case of conservative Petri nets they have a PSPACE-complete reachability problem. In this paper we explore two restrictions of the reachability problem for IO nets which lower the complexity of the problem drastically. The complexity is NP-complete for the first restriction with applications in distributed protocols, …
Developing algorithms for distributed systems is an error-prone task. Formal models like Petri nets with transits and Petri games can prevent errors when developing such algorithms. Petri nets with transits allow us to follow the data flow between components in a distributed system. They can be model checked against specifications in LTL on both the local data flow and the global behavior. Petri games allow the synthesis of local controllers for distributed systems from safety specifications. Modeling problems in these formalisms requires defining extended Petri nets which can be cumbersome wh…
Petri nets are a formalism for modelling and reasoning about the behaviour of distributed systems. Recently, a reversible approach to Petri nets, Reversing Petri Nets (RPN), has been proposed, allowing transitions to be reversed spontaneously in or out of causal order. In this work we propose an approach for controlling the reversal of actions of an RPN, by associating transitions with conditions whose satisfaction/violation allows the execution of transitions in the forward/reversed direction, respectively. We illustrate the framework with a model of a novel, distributed algorithm for antenna…
We provide an Ackermannian complexity lower bound for the reachability problem for checking programs, a model equivalent to Petri nets. Moreover in fixed dimension $2d+4$, we show that the problem is $\mathbb{F}_d$-hard. As a direct corollary, the reachability problem in dimension 10 is not elementary.
We introduce stochastic decision Petri nets (SDPNs), which are a form of stochastic Petri nets equipped with rewards and a control mechanism via the deactivation of controllable transitions. Such nets can be translated into Markov decision processes (MDPs), potentially leading to a combinatorial explosion in the number of states due to concurrency. Hence we restrict ourselves to instances where nets are either safe, free-choice and acyclic nets (SAFC nets) or even occurrence nets and policies are defined by a constant deactivation pattern. We obtain complexity-theoretic results for such cases …
Solutions proposed for the longstanding problem of automatic decomposition of Petri nets into concurrent processes, as well as methods developed in Grenoble for the automatic conversion of safe Petri nets to NUPNs (Nested-Unit Petri Nets), require certain properties to be computed on Petri nets. We notice that, although these properties are theoretically interesting and practically useful, they are not currently implemented in mainstream Petri net tools. Taking into account such properties would open fruitful research directions for tool developers, and new perspectives for the Model Checking …
In a previous paper we introduced immediate observation (IO) Petri nets, a class of interest in the study of population protocols and enzymatic chemical networks. In the first part of this paper we show that IO nets are globally flat, and so their safety properties can be checked by efficient symbolic model checking tools using acceleration techniques, like FAST. In the second part we study Branching IO nets (BIO nets), whose transitions can create tokens. BIO nets extend both IO nets and communication-free nets, also called BPP nets, a widely studied class. We show that, while BIO nets are no…
Synthesis for a type $τ$ of Petri nets is the following search problem: For a transition system $A$, find a Petri net $N$ of type $τ$ whose state graph is isomorphic to $A$, if there is one. To determine the computational complexity of synthesis for types of bounded Petri nets we investigate their corresponding decision version, called feasibility. We show that feasibility is NP-complete for (pure) $b$-bounded P/T-nets if $b\in \mathbb{N}^+$. We extend (pure) $b$-bounded P/T-nets by the additive group $\mathbb{Z}_{b+1}$ of integers modulo $(b+1)$ and show feasibility to be NP-complete for the …
To model check concurrent systems, it is convenient to distinguish between the data flow and the control. Correctness is specified on the level of data flow whereas the system is configured on the level of control. Petri nets with transits and Flow-LTL are a corresponding formalism. In Flow-LTL, both the correctness of the data flow and assumptions on fairness and maximality for the control are expressed in linear time. So far, branching behavior cannot be specified for Petri nets with transits. In this paper, we introduce Flow-CTL* to express the intended branching behavior of the data flow w…
We introduce immediate observation Petri nets, a class of interest in the study of population protocols (a model of distributed computation), and enzymatic chemical networks. In these areas, relevant analysis questions translate into parameterized Petri net problems: whether an infinite set of Petri nets with the same underlying net, but different initial markings, satisfy a given property. We study the parameterized reachability, coverability, and liveness problems for immediate observation Petri nets. We show that all three problems are in PSPACE for infinite sets of initial markings defined…
SMPT (for Satisfiability Modulo Petri Net) is a model checker for reachability problems in Petri nets. It started as a portfolio of methods to experiment with symbolic model checking, and was designed to be easily extended. Some distinctive features are its ability to benefit from structural reductions and to generate verdict certificates. Our tool is quite mature and performed well compared to other state-of-the-art tools in the Model Checking Contest.
Petri nets are a mathematical language for modeling and reasoning about distributed systems. In this paper we propose an approach to Petri nets for embedding reversibility, i.e., the ability of reversing an executed sequence of operations at any point during operation. Specifically, we introduce machinery and associated semantics to support the three main forms of reversibility namely, backtracking, causal reversing, and out-of-causal-order reversing in a variation of cyclic Petri nets where tokens are persistent and are distinguished from each other by an identity. Our formalism is influenced…
The correctness of networks is often described in terms of the individual data flow of components instead of their global behavior. In software-defined networks, it is far more convenient to specify the correct behavior of packets than the global behavior of the entire network. Petri nets with transits extend Petri nets and Flow-LTL extends LTL such that the data flows of tokens can be tracked. We present the tool AdamMC as the first model checker for Petri nets with transits against Flow-LTL. We describe how AdamMC can automatically encode concurrent updates of software-defined networks as Pe…
State Space Computation and Analysis of Time Petri Nets
Petri网理论提供了一个通用的框架来描述实时反应式系统的行为，时间Petri网的引入也考虑了时间规范。本文提出了一种基于前向区域的算法来计算有界时间Petri网的状态空间：该方法与经典的状态类图不同，并且更有效。我们证明该算法是精确的可达性问题。此外，我们提出了一个翻译的计算状态空间到一个时间自动机，证明了原来的时间Petri网的时间双相似。当该方法产生单个时间自动机时，可以应用语法时钟减少方法（例如Daws和Yovine）来产生具有更少时钟的自动机。然后，我们的方法允许通过使用高效的时间自动机工具进行模型检查TTPN。逻辑程序设计理论与实践（Theory and Practice of Logic Programming，TPLP）
Finite 1-safe Petri nets, also called \emph{net systems}, are natural models of asynchronous concurrency. The event structure of a net system describes all its possible executions and their concurrent nature: two events may be causally ordered, occur in parallel or be conflicting. Monadic second order logic (MSO) can be used to specify behavioural properties of net systems. Thiagarajan's conjecture states that MSO is decidable if and only if the net system is grid-free. The present paper gives a positive answer to this conjecture.
Reversing Petri nets (RPNs) have recently been proposed as a net-basedapproach to model causal and out-of-causal order reversibility. They are based on the notion of individual tokens that can be connected together via bonds. In this paper we extend RPNs by allowing multiple tokens of the same type to exist within a net based on the individual token interpretation of Petri nets. According to this interpretation, tokens of the same type are distinguished via their causal path. We develop a causal semantics of the model and we prove that the expressive power of RPNs with multiple tokens is equiv…
We propose a semi-decision procedure for checking generalized reachability properties, on generalized Petri nets, that is based on the Property Directed Reachability (PDR) method. We actually define three different versions, that vary depending on the method used for abstracting possible witnesses, and that are able to handle problems of increasing difficulty. We have implemented our methods in a model-checker called SMPT and give empirical evidences that our approach can handle problems that are difficult or impossible to check with current state of the art tools.
