We propose a reachability verification technique that combines the Petri net state equation (a linear algebraic overapproximation of the set of reachable states) with the concept of counterexample guided abstraction refinement. In essence, we replace the search through the set of reachable states by a search through the space of solutions of the state equation. We demonstrate the excellent performance of the technique on several real-world examples. The technique is particularly useful in those cases where the reachability query yields a negative result: While state space based techniques need to fully expand the state space in this case, our technique often terminates promptly. In addition, we can derive some diagnostic information in case of unreachability while state space methods can only provide witness paths in the case of reachability.
For every finite Petri net, we construct a commutative polynomial in two variables and with coefficients from the semiring of natural numbers. We also present an inverse construction and show that multiplication of polynomials correspondence to the product of the corresponding Petri nets in the category of Petri nets with Winskel's morphisms. We endow the set of all Petri nets with Zariski topology.
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.
Timed-arc Petri nets (TAPN) are a well-known time extension of the Petri net model and several translations to networks of timed automata have been proposed for this model. We present a direct, DBM-based algorithm for forward reachability analysis of bounded TAPNs extended with transport arcs, inhibitor arcs and age invariants. We also give a complete proof of its correctness, including reduction techniques based on symmetries and extrapolation. Finally, we augment the algorithm with a novel state-space reduction technique introducing a monotonic ordering on markings and prove its soundness ev…
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 …
We introduce a novel technique for checking reachability in Petri nets that relies on a recently introduced compositional algebra of nets. We prove that the technique is correct, and discuss our implementation. We report promising experimental results on some well-known examples.
A Petri net is structurally cyclic if every configuration is reachable from itself in one or more steps. We show that structural cyclicity is decidable in deterministic polynomial time. For this, we adapt the Kosaraju's approach for the general reachability problem for Petri nets.
Bipolar synchronization systems (BP-systems) constitute a class of coloured Petri nets, well suited for modeling the control flow of discrete, dynamical systems. Every BP-system has an underlying ordinary Petri net, which is a T-system. Moreover, it has a second ordinary net attached, which is a free-choice system. We prove that a BP-system is live and safe if the T-system and the free-choice system are live and safe and if the free-choice system has no frozen tokens. This result is the converse of a theorem of Genrich and Thiagarajan and proves an elder conjecture. The proof compares the diff…
The reachability semantics for Petri nets can be studied using open Petri nets. For us an "open" Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri})$, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\math…
In this paper,the authors show the versatility of the Signed Petri Net (SPN) introduced by them by showing the equivalence between a Logic Signed Petri Net (LSPN) and Logic Petri Net (LPN).The capacity of each place in all these nets is at most one, i.e.,a place has either zero or one token in it.
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.
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.
The theory of Petri Nets provides a general framework to specify the behaviors of real-time reactive systems and Time Petri Nets were introduced to take also temporal specifications into account. We present in this paper a forward zone-based algorithm to compute the state space of a bounded Time Petri Net: the method is different and more efficient than the classical State Class Graph. We prove the algorithm to be exact with respect to the reachability problem. Furthermore, we propose a translation of the computed state space into a Timed Automaton, proved to be timed bisimilar to the original…
The verification of safety properties for concurrent systems often reduces to the coverability problem for Petri nets. This problem was shown to be ExpSpace-complete forty years ago. Driven by the concurrency revolution, it has regained a lot of interest over the last decade. In this paper, we propose a generic and simple approach to solve this problem. Our method is inspired from the recent approach of Blondin, Finkel, Haase and Haddad. Basically, we combine forward invariant generation techniques for Petri nets with backward reachability for well- structured transition systems. An experiment…
Model checking is an important aim of the theoretical computer science. It enables the verification of a model with a set of properties such as liveness, deadlock or safety. One of the typical modelling techniques are Petri nets they are well understood and can be used for a model checking. Reconfigurable Petri nets are based on a Petri nets with a set of rules. These rules can be used dynamically to change the net. Missing is the possibility to verify a reconfigurable net and properties such as deadlocks or liveness. This paper introduces a conversion from reconfigurable Petri net to Maude, t…
We present a concurrent operational Petri net semantics for the join-calculus, a process calculus for specifying concurrent and distributed systems. There often is a gap between system specifications and the actual implementations caused by synchrony assumptions on the specification side and asynchronously interacting components in implementations. The join-calculus is promising to reduce this gap by providing an abstract specification language which is asynchronously distributable. Classical process semantics establish an implicit order of actually independent actions, by means of an interlea…
In recent work, the second and third authors introduced a technique for reachability checking in 1-bounded Petri nets, based on wiring decompositions, which are expressions in a fragment of the compositional algebra of nets with boundaries. Here we extend the technique to the full algebra and introduce the related structural property of decomposition width on directed hypergraphs. Small decomposition width is necessary for the applicability of the reachability checking algorithm. We give examples of families of nets with constant decomposition width and develop the underlying theory of decompo…
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.
Algebraic Petri nets are a formalism for modeling distributed systems and algorithms, describing control and data flow by combining Petri nets and algebraic specification. One way to specify correctness of an algebraic Petri net model $N$ is to specify a linear equation $E$ over the places of $N$ based on term substitution, and coefficients from an abelian group $G$. Then, $E$ is valid in $N$ iff $E$ is valid in each reachable marking of $N$ . Due to the expressive power of Algebraic Petri nets, validity is generally undecidable. Stable linear equations form a class of linear equations for whi…
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 cl…
We investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order and modal languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.
Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is the verification of safety and liveness properties in this model; despite the extensive research effort, some basic problems remain open, which is exemplified by the open complexity status of the reachability problem. The liveness problems are known to be closely related to the reachability problem, and many structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the…
We propose a method to count the number of reachable markings of a Petri net without having to enumerate these rst. The method relies on a structural reduction system that reduces the number of places and transitions of the net in such a way that we can faithfully compute the number of reachable markings of the original net from the reduced net and the reduction history. The method has been implemented and computing experiments show that reductions are eective on a large benchmark of models.
We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ω-labeled input and output arcs, that is well-suited to analyse parametric concurrent systems with dynamic thread creation. Most techniques (such as the Karp and Miller tree or the Rackoff technique) that have been proposed in the setting of plain Petri nets do not apply directly to ωPN because ωPN define transition systems that have infinite branching. This motivates a thorough analysis of the computational aspects of ωPN. We show that an ωPN can be turned into an plain Petri net that allows to recover the reachability set…
For a given labelled transition system (LTS), synthesis is the task to find an unlabelled Petri net with an isomorphic reachability graph. Even when just demanding an embedding into a reachability graph instead of an isomorphism, a solution is not guaranteed. In such a case, label splitting is an option, i.e. relabelling edges of the LTS such that differently labelled edges remain different. With an appropriate label splitting, we can always obtain a solution for the synthesis or embedding problem. Using the label splitting, we can construct a labelled Petri net with the intended bahaviour (e.…
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…
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…
Two formal stochastic models are said to be bisimilar if their solutions as a stochastic process are probabilistically equivalent. Bisimilarity between two stochastic model formalisms means that the strengths of one stochastic model formalism can be used by the other stochastic model formalism. The aim of this paper is to explain bisimilarity relations between stochastic hybrid automata, stochastic differential equations on hybrid space and stochastic hybrid Petri nets. These bisimilarity relations make it possible to combine the formal verification power of automata with the analysis power of…
Unfoldings provide an efficient way to avoid the state-space explosion due to interleavings of concurrent transitions when exploring the runs of a Petri net. The theory of adequate orders allows one to define finite prefixes of unfoldings which contain all the reachable markings. In this paper we are interested in reachability of a single given marking, called the goal. We propose an algorithm for computing a finite prefix of the unfolding of a 1-safe Petri net that preserves all minimal configurations reaching this goal. Our algorithm combines the unfolding technique with on-the-fly model red…
Petri-nets are a simple formalism for modeling concurrent computation. Recently, they have emerged as a powerful tool for the modeling and analysis of biochemical reaction networks, bridging the gap between purely qualitative and quantitative models. These networks can be large and complex, which makes their study difficult and computationally challenging. In this paper, we focus on two structural properties of Petri-nets, siphons and traps, that bring us information about the persistence of some molecular species. We present two methods for enumerating all minimal siphons and traps of a Petri…
Automated synthesis from behavioural specifications is an attractive and powerful way of constructing concurrent systems. Here we focus on the problem of synthesising a membrane system from a behavioural specification given in the form of a transition system which specifies the desired state space of the system to be constructed. We demonstrate how a Petri net solution to this problem, based on the notion of region of a transition system, yields a method of automated synthesis of membrane systems from state spaces.
HSE University（俄罗斯国立高等经济大学）；Free University of Bozen-Bolzano（博尔扎诺自由大学）
Recently introduced Petri net-based formalisms advocate the importance of proper representation and management of case objects as well as their co-evolution. In this work we build on top of one of such formalisms and introduce the notion of soundness for it. We demonstrate that for nets with non-deterministic synchronization between case objects, the soundness problem is decidable.
We investigate the problem asking whether the intersection of a context-free language (CFL) and a Petri net language (PNL) is empty. Our contribution to solve this long-standing problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data domain, is to identify a class of CFLs called the finite-index CFLs for which the problem is decidable. The k-index approximation of a CFL can be obtained by discarding all the words that cannot be derived within a budget k on the number of occurrences of non-terminals. A finite-index CFL is thus a CFL which coin…
We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a net using comonads, and externally, using lax-monoidal-lax functors. Our external semantics is non-local, meaning that tokens are endowed with properties that say something about the global state of the net. We then prove, in both cases, that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construct…
