The Downward-Closure of Petri Net Languages (original) (raw)

On the Upward/Downward Closures of Petri Nets

ArXiv, 2017

We study the size and the complexity of computing finite state automata (FSA) representing and approximating the downward and the upward closure of Petri net languages with coverability as the acceptance condition. We show how to construct an FSA recognizing the upward closure of a Petri net language in doubly-exponential time, and therefore the size is at most doubly exponential. For downward closures, we prove that the size of the minimal automata can be non-primitive recursive. In the case of BPP nets, a well-known subclass of Petri nets, we show that an FSA accepting the downward/upward closure can be constructed in exponential time. Furthermore, we consider the problem of checking whether a simple regular language is included in the downward/upward closure of a Petri net/BPP net language. We show that this problem is EXPSPACE-complete (resp. NP-complete) in the case of Petri nets (resp. BPP nets). Finally, we show that it is decidable whether a Petri net language is upward/down...

A Well-Structured Framework for Analysing Petri Nets Extensions

1999

Transition systems defined from recursive functions INp! INpare introduced and named WSN,or well-structured nets. Such "minimally Petri net-like" transition systems sit conveniently betweenPetri net extensions and general transition systems. In a first part, we study decidability propertiesof WSN classes obtained by imposing natural restrictions to their defining functions, with respectto termination, to coverability, and to four variants of the boundedness

Symbolic analysis of bounded Petri nets

IEEE Transactions on Computers, 2001

AbstractÐThis work presents a symbolic approach for the analysis of bounded Petri nets. The structure and behavior of the Petri net is symbolically modeled by using Boolean functions, thus reducing reasoning about Petri nets to Boolean calculation. The set of reachable markings is calculated by symbolically firing the transitions in the Petri net. Highly concurrent systems suffer from the state explosion problem produced by an exponential increase of the number of reachable states. This state explosion is handled by using Binary Decision Diagrams (BDDs) which are capable of representing large sets of markings with small data structures. Petri nets have the ability to model a large variety of systems and the flexibility to describe causality, concurrency, and conditional relations. The manipulation of vast state spaces generated by Petri nets enables the efficient analysis of a wide range of problems, e.g., deadlock freeness, liveness, and concurrency. A number of examples are presented in order to show how large reachability sets can be generated, represented, and analyzed with moderate BDD sizes. By using this symbolic framework, properties requiring an exhaustive analysis of the reachability graph can be efficiently verified.

The Possibilities of Modeling Petri Nets and Their Extensions

Computer Simulation [Working Title], 2019

This chapter is dedicated to several structure features of Petri nets. There is detailed description of appropriate access in Petri nets and reachable tree mechanism construction. There is an algorithm that describes the minimum sequence of possible transitions. The algorithm developed by us finds the shortest possible sequence for the network promotion state, which transfers the mentioned network state to the coverage state. The corresponding theorem is proven, which states that due to the describing algorithm, the number of transitions in the covering state is minimal. This chapter studies the interrelation of languages of colored Petri nets and traditional formal languages. The Venn diagram, modified by the author, is presented, which shows the relationship between the languages of the colored Petri nets and some traditional languages. As a result, it is shown that the language class of colored Petri nets includes an entire class of context-free languages and some other classes. The results obtained show that it is not possible to model the Patil problem using the well-known semaphores P and V or classical Petri nets, so the mentioned systems have limited properties.

Petri Net Reachability Checking Is Polynomial with Optimal Abstraction Hierarchies

Lecture Notes in Computer Science, 2005

The Petri net model is a powerful state transition oriented model to analyse, model and evaluate asynchronous and concurrent systems. However, like other state transition models, it encounters the state explosion problem. The size of the state space increases exponentially with the system complexity. This paper is concerned with a method of abstracting automatically Petri nets to simpler representations, which are ordered with respect to their size. Thus it becomes possible to check Petri net reachability incrementally. With incremental approach we can overcome the exponential nature of Petri net reachability checking. We show that by using the incremental approach, the upper computational complexity bound for Petri net reachability checking with optimal abstraction hierarchies is polynomial. The method we propose considers structural properties of a Petri net as well an initial and a final marking. In addition to Petri net abstraction irrelevant transitions for a given reachability problem are determined. By removing these transitions from a net, impact of the state explosion problem is reduced even more.

An algebraic structure of petri nets

Lecture Notes in Computer Science, 1980

A relational model for non-deterministic programs is presented. Several predicate transformers are introduced and it is shown that one of them satisfies all the healthiness criteria indicated by Dijkstra for a useful total correctness predicate transformer.

A well-structured framework for analysing petri net extensions

Information and Computation, 2004

Transition systems de ned from recursive functions IN p ! IN p are introduced and named WSN, or well-structured nets. Such \minimally Petri net-like" transition systems sit conveniently between Petri net extensions and general transition systems. In a rst part, we study decidability properties of WSN classes obtained by imposing natural restrictions to their de ning functions, with respect to termination, to coverability, and to four variants of the boundedness problem. We are able to answer optimally almost all the questions which arise, thus gaining much insight into old and new generalized Petri net decidability results. In a second part, we specialize our analysis to WSN de ned from a ne functions. Such a ne WSN elegantly encompass most Petri net extensions studied in the litterature. Again, we study decidability properties of natural classes of a ne WSN with respect to six computational problems. In particular, we develop a nontrivial algorithm computing limits of iterated nonnegative a ne functions, in order to decide the path-place variant of the boundedness problem for the relevant a ne WSN. Undecidability results are scattered throughout the paper, concerning, for example, extending the domain of a recursive function from IN p to (IN f!g) p .

Accelerations for the Coverability Set of Petri Nets with Names

Fundamenta Informaticae, 2011

Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining ν-PNs and proved that they are strictly well structured (WSTS), so that coverability and boundedness are decidable. Here we use the framework recently developed by Finkel and Goubault-Larrecq for forward analysis for WSTS, in the case of ν-PNs, to compute the cover, that gives a good over approximation of the set of reachable markings. We prove that the least complete domain containing the set of markings is effectively representable. Moreover, we prove that in the completion we can compute least upper bounds of simple loops. Therefore, a forward Karp-Miller procedure that computes the cover is applicable. However, we prove that in general the cover is not computable, so that the procedure is non-terminating in general. As a corollary, we obtain the analogous result for Transfer Data nets and Data Nets. Finally, we show that a slight modification of the forward analysis yields decidability of a weak form of boundedness called width-boundedness, and identify a subclass of ν-PN that we call dw-bounded ν-PN, for which the cover is computable.

Extensions to the CEGAR Approach on Petri Nets

Acta Cybernetica, 2014

Formal verification is becoming more prevalent and often compulsory in the safety-critical system and software development processes. Reachability analysis can provide information about safety and invariant properties of the developed system. However, checking the reachability is a computationally hard problem, especially in the case of asynchronous or infinite state systems. Petri nets are widely used for the modeling and verification of such systems. In this paper we examine a recently published approach for the reachability checking of Petri net markings. We give proofs concerning the completeness and the correctness properties of the algorithm, and we introduce algorithmic improvements. We also extend the algorithm to handle new classes of problems: submarking coverability and reachability of Petri nets with inhibitor arcs.