Net Unfoldings Research Articles

Modular Construction of Finite and Complete Prefixes of Petri net Unfoldings

This paper considers distributed systems, defined as a collection of components interacting through interfaces. Components, interfaces and distributed systems are modeled as Petri nets. It is well known that the unfolding of such a distributed system factorises, in the sense that it can be expressed as the composition of unfoldings of its components. This factorised form of the unfolding generally provides a more compact representation of the system runs, because each component does not need to represent the possible choices (conflicts) appearing in the other components. Moreover, the unfolding factorisation makes it possible to analyse the system by parts. The paper focuses on the derivation of a finite and complete prefix (FCP) in the unfolding of a distributed system. Specifically, one would like to directly obtain such a in factorised form, without computing first a of the global distributed system and then factorising it. The construction of such a modular FCP is based on deriving summaries of component behaviours w.r.t. their interfaces, that are then communicated to the neighbouring components. The latter combine these summaries with their local behaviours, and prepare interface summaries for the next components. This globally takes the form of a message passing algorithm, where the global system is never considered.

Merged processes: a new condensed representation of Petri net behaviour

Model checking based on Petri net unfoldings is an approach widely applied to cope with the state space explosion problem. In this paper, we propose a new condensed representation of a Petri net’s behaviour called merged processes, which copes well not only with concurrency, but also with other sources of state space explosion, viz sequences of choices and non-safeness. Moreover, this representation is sufficiently similar to the traditional unfoldings, so that a large body of results developed for the latter can be re-used. Experimental results indicate that the proposed representation of a Petri net’s behaviour alleviates the state space explosion problem to a significant degree and is suitable for model checking.

Verification of bounded Petri nets using integer programming

Model checking based on the causal partial order semantics of Petri nets is an approach widely applied to cope with the state space explosion problem. One of the ways to exploit such a semantics is to consider (finite prefixes of) net unfoldings--themselves a class of acyclic Petri nets--which contain enough information, albeit implicit, to reason about the reachable markings of the original Petri nets. In [19], a verification technique for net unfoldings was proposed, in which deadlock detection was reduced to a mixed integer linear programming problem. In this paper, we present a further development of this approach. The essence of the proposed modifications is to transfer the information about causality and conflicts between the events involved in an unfolding, into a relationship between the corresponding integer variables in the system of linear constraints. Moreover, we present some problem-specific optimisation rules, reducing the search space. To solve other verification problems, such as mutual exclusion or marking reachability and coverability, we adopt Contejean and Devie's algorithm for solving systems of linear constraints over the natural numbers domain and refine it, by taking advantage of the specific properties of systems of linear constraints to be solved. Another contribution of this paper is a method of re-formulating some problems specified in terms of Petri nets as problems defined for their unfoldings. Using this method, we obtain a memory efficient translation of a deadlock detection problem for a safe Petri net into an LP problem. We also propose an on-the-fly deadlock detection method. Experimental results demonstrate that the resulting algorithms can achieve significant speedups.

Distributed Monitoring of Concurrent and Asynchronous Systems*

In this paper we study the diagnosis of distributed asynchronous systems with concurrency. Diagnosis is performed by a peer-to-peer distributed architecture of supervisors. Our approach relies on Petri net unfoldings and event structures, as means to manipulate trajectories of systems with concurrency. This article is an extended version of the paper with same title, which appeared as a plenary address in the Proceedings of CONCUR?2003.

FAULT DIAGNOSIS FOR DISTRIBUTED ASYNCHRONOUS DYNAMICALLY RECONFIGURED DISCRETE EVENT SYSTEMS

Diagnosis of concurrent and asynchronous systems, such as large telecommunication or information systems, requires powerful mathematical models. The use of Petri net unfoldings allows to formalize diagnosis using partial order semantics, a generalization from the global state model imposed by the use of automata. If, in addition to asynchronicity and distribution, the network topology itself is subject to dynamic changes, all static models, including Petri nets, reach their limits. Then, graph grammars can be used, encoding in the current local states not only the current values of state variables but also the current topology of the network connections; the fact that unfolding semantics is available allows to carry over the diagnosis algorithms to this setting.

State Feedback Control of Discrete Event Systems by using Petri net Unfoldings

A method for synthesizing feedback control logic for live and safe marked graphs has been given by Holloway and Krogh. The method strongly depends on the structure of marked graphs, and thus it has not been extended to wider classes. This paper considers the same problem of synthesizing the maximal permissive feedback control for a special condition by using unfoldings. Structural characteristics of unfoldings are similar to that of marked graphs. Transforming a Petri net to an unfolding allows us to apply the Holloway's approach to general bounded Petri nets.

SAT-Solving the Coverability Problem for Petri Nets

Net unfoldings have attracted great attention as a powerful technique for combating state space explosion in model checking, and have been applied to verification of finite state systems including 1-safe (finite) Petri nets and synchronous products of finite transition systems. Given that net unfoldings represent the state space in a distributed, implicit manner the verification algorithm is necessarily a two step process: generation of the unfolding and reasoning about it. In his seminal work McMillan (K.L. McMillan, Symbolic Model Checking. Kluwer Academic Publishers, 1993) showed that deadlock detection on unfoldings of 1-safe Petri nets is NP-complete. Since the deadlock problem on Petri nets is PSPACE-hard it is generally accepted that the two step process will yield savings (in time and space) provided the unfoldings are small. In this paper we show how unfoldings can be extended to the context of infinite-state systems. More precisely, we show how unfoldings can be constructed to represent sets of backward reachable states of unbounded Petri nets in a symbolic fashion. Furthermore, based on unfoldings, we show how to solve the coverability problem for unbounded Petri nets using a SAT-solver. Our experiments show that the use of unfoldings, in spite of the two-step process for solving coverability, has better time and space characteristics compared to a traditional reachability based implementation that considers all interleavings for solving the coverability problem.

Markov nets: probabilistic models for distributed and concurrent systems

For distributed systems, i.e., large complex networked systems, there is a drastic difference between a local view and knowledge of the system, and its global view. Distributed systems have local state and time, but do not possess global state and time in the usual sense. In this paper, motivated by the monitoring of distributed systems and in particular of telecommunications networks, we develop a generalization of Markov chains and hidden Markov models for distributed and concurrent systems. By a concurrent system, we mean a system in which components may evolve independently, with sparse synchronizations. We follow a so-called true concurrency approach, in which neither global state nor global time are available. Instead, we use only local states in combination with a partial order model of time. Our basic mathematical tool is that of Petri net unfoldings.

Canonical prefixes of Petri net unfoldings

In this paper, we develop a general technique for truncating Petri net unfoldings, parameterized according to the level of information about the original unfolding one wants to preserve. Moreover, we propose a new notion of completeness of a truncated unfolding. A key aspect of our approach is an algorithm-independent notion of cut-off events, used to truncate a Petri net unfolding. Such a notion is based on a cutting context and results in the unique canonical prefix of the unfolding. Canonical prefixes are complete in the new, stronger sense, and we provide necessary and sufficient conditions for its finiteness, as well as upper bounds on its size in certain cases. A surprising result is that after suitable generalization, the standard unfolding algorithm presented in [8], and the parallel unfolding algorithm proposed in [12], despite being non-deterministic, generate the canonical prefix. This gives an alternative correctness proof for the former algorithm, and a new (much simpler) proof for the latter one.

Diagnosis of asynchronous discrete-event systems: a net unfolding approach

In this paper, we consider the diagnosis of asynchronous discrete event systems. We follow a so-called true concurrency approach, in which no global state and no global time is available. Instead, we use only local states in combination with a partial order model of time. Our basic mathematical tool is that of net unfoldings originating from the Petri net research area. This study was motivated by the problem of event correlation in telecommunications network management.

Using Logic Programs with Stable Model Semantics to Solve Deadlock and Reachability Problems for 1-Safe Petri Nets

McMillan has presented a deadlock detection method for Petri nets based on finite complete prefixes (i.e. net unfoldings). The approach transforms the PSPACE-complete deadlock detection problem for a 1-safe Petri net into a potentially exponentially larger NP-complete problem of deadlock detection for a finite complete prefix. McMillan devised a branch-and-bound algorithm for deadlock detection in prefixes. Recently, Melzer and Romer have presented another approach, which is based on solving mixed integer programming problems. In this work it is shown that instead of using mixed integer programming, a constraint-based logic programming framework can be employed, and a linear-size translation from deadlock detection in prefixes into the problem of finding a stable model of a logic program is presented. As a side result also such a translation for solving the reachability problem is devised. Correctness proofs of both the translations are presented. Experimental results are given from an implementation combining the prefix generator of the PEP-tool, the translation, and an implementation of a constraint-based logic programming framework, the smodels system. The experiments show the proposed approach to be quite competitive, when compared to the approaches of McMillan and Melzer/Romer.

Deadlock prevention using Petri nets and their unfoldings

Unfoldings of Petri nets (PN) provide a method for the analysis of concurrent systems without restoring the state space of a system. This allows one to overcome the “state explosion” problem. Many properties of the initial PN (boundedness, safety, persistency and hazards) can be checked by constructing the unfolding. A deadlock prevention procedure first detects deadlocks using an unfolding. Then, the first method reduces the unfolding to a set of deadlock-free subunfoldings that cover all live behaviours. The second method uses a direct transformation at the level of the original PN. The methods are implemented as subroutines in the Berkeley program SIS. Although the deadlock detection problem is known to be NP-complete, experimental results show that for highly parallel specifications deadlock prevention by unfoldings is typically more efficient than deadlock prevention based on symbolic BDD (binary decision diagrams) traversal of the corresponding reachability graph.

A technique of state space search based on unfolding

Unfoldings of Petri nets provide a method of searching the state space of concurrent systems without considering all possible interleavings of concurrent events. A procedure is given for constructing the unfolding of a Petri net, terminating the construction when it is sufficient to represent all reachable markings. This procedure is applied to hazard and deadlock detection in asynchronous circuits. Examples are given of scalable systems with exponential size state spaces, but polynomial size unfoldings, including a distributed mutual exclusion ring circuit.

Model checking using net unfoldings

McMillan (1992) described a technique for deadlock detection based on net unfoldings. We extend its applicability to the properties of a temporal logic with a possibility operator. The new algorithm is shown to be polynomial in the size of the net for 1-safe conflict-free Petri nets, while the algorithms of the literature require exponential time.