How to synthesize nets from languages: a survey

Synthesising systems from behavioural specifications is an attractive way of constructing implementations which are correct-by-design and thus requiring no costly validation efforts. In this paper, systems are modelled by Petri nets and the behavioural specifications are provided in the form of step transition systems, where arcs are labelled by multisets of executed actions. We focus on the problem of synthesising Petri nets with whole-place operations and localities (wpol-nets), which are a class of Petri nets powerful enough to express a wide range of system behaviours, including inhibition of actions, resetting of local states, and locally maximal executions.

This paper presents an algorithm for the synthesis of bounded Petri nets from transition systems. A bounded Petri net is always provided in case it exists. Otherwise, the events are split into several transitions to guarantee the synthesis of a Petri net with bisimilar behavior. The algorithm uses symbolic representations of multisets of states to efficiently generate all the minimal regions. The algorithm has been implemented in a tool. Experimental results show a significant net reduction when compared with approaches for the synthesis of safe Petri nets.

The elasticity principles have become one of the most important features of contemporary distributed systems. This novel paradigm has major consequences on Petri nets modeling. The problem is to decide whether an arbitrary Petri net satisfies the elasticity and how to make it elastic. The problem is solved by synthesis of Fractal Petri nets. Fractal Petri nets based on Iterated Algebraic System which makes possible elastic synthesis by incrementing or decrementing of original Petri net. One of the important tasks is recognition of a subnet which used as a pattern for elastic synthesis. For this purpose evolutionary classification of Petri nets based on subnets synthesis is introduced. And furthermore, algebraic operations replication and composition make elastic synthesis object-oriented. Control of the elasticity over the resources and system behavior is achieved with the use of patterns overlapping technique.

In this paper we present an algorithm to synthesize a finite unlabelled place/transition Petri net (p/t-net) from a possibly infinite partial language, which is given by a term over a finite set of labelled partial orders using operators for union, iteration, parallel composition and sequential composition. The synthesis algorithm is based on the theory of regions for partial languages presented in [17] and produces a p/t-net having minimal net behaviour including the given partial language. The algorithm uses linear programming techniques that were already successfully applied in [22] for the synthesis of p/t-nets from finite partial languages. Also, an equality test algorithm to check whether the behaviour of the synthesized p/t-net equals the given partial language is shown. Moreover, we present an implementation of the developed synthesis algorithm together with an example case study. Finally, a possible generalization of the presented term based representation of infinite partial languages is discussed.

Given a set of scenarios, we answer the question whether this set equals the set of all executions of a Petri net.Formally, scenarios are expressed by (isomorphism classes of) labelled partial orders (LPOs), also known as pomsets or partial words. An LPO is an execution of a Petri net if it is a sequentialization of an LPO generated by a process of the net. We propose a definition of regions for a set of LPOs, i.e for a partial language. Given a partial language of scenarios, we prove a necessary and sufficient condition (based on regions) for the partial language of scenarios to be the partial language of executions of a place/transition Petri net. Finally, we prove our notion of regions to be consistent with the notion of regions of trace languages.

With this contribution, we aim to draw a comprehensive classification of Petri nets with stopwatches w.r.t. expressiveness and decidability issues. This topic is too ambitious to be summarized in a single paper. That is why we present our results in two different parts. The scope of this first paper is to address the general results that apply for both dense-time and discrete-time semantics. We study the class of bounded Petri nets with stopwatches and reset arcs (rSwPNs), which is an extension of T-time Petri nets (TPNs) where time is associated with transitions. Stopwatches can be reset, stopped and started. We give the formal dense-time and discrete-time semantics of these models in terms of Transition Systems. We study the expressiveness of rSwPNs and its subclasses w.r.t. (weak) bisimilarity (behavioral semantics). The main results are following: 1) bounded rSw- PNs and 1-safe rSwPNs are equally expressive; 2) For all models, reset arcs add expressiveness. 3) The resulting partial classification of models is given by a set of relations explained in Fig. 7: in the forthcoming paper, we will complete these results by covering expressiveness and decidability issues when discrete-time nets are considered. For the sake of simplicity, our results are explained on a model such that the stopwatches behaviors are expressed using inhibitor arcs. Our conclusions can however be easily extended to the general class of Stopwatch Petri nets.

Pre-synthesis of Petri nets based on prime cycles and distance paths

1.1 Relationship between Petri net and linear logic Petri nets were first introduced by Petri in his seminal Ph.D. thesis, and both the theory and the applications of his model have flourished in concurrency theory (Reisig & Rozenberg, 1998a; Reisig & Rozenberg, 1998b). The relationships between Petri nets and linear logics have been studied by many researchers (Engberg & Winskel, 1997; Farwer, 1999; Hirai, 2000; Hirai 1999; Ishihara & Hiraish, 2001; Kamide, 2004, Kamide, 2006; Kanovich, 1995; Kanovich 1994; LarcheyWendling & Galmiche, 1998; Larchey-Wendling & Galmiche, 2000; Lilius, 1992; Mart -Oliet & Meseguer, 1991; Okada, 1998; Tanabe, 1997). A category theoretical investigation of such a relationship was given by Mart -Oliet and Meseguer (Mart -Oliet & Meseguer, 1991), purely syntactical approach using Horn linear logic was established by Kanovich (Kanovich, 1995; Kanovich 1994), a naive phase linear logic for a certain class of Petri nets was given by Okada (Okada, 1998), a linear logical view of object Petri nets were studied by Farwer (Farwer, 1999), and various Petri net interpretations of linear logic using quantale models were obtained by Ishihara and Hiraishi (Ishihara & Hiraish, 2001), Engberg and Winskel (Engberg & Winskel, 1997), Larchey-Wendling and Galmiche (Larchey-Wendling & Galmiche, 1998; Larchey-Wendling & Galmiche, 2000), and Lilius (Lilius, 1992). Petri net interpretations using Kripke semantics for various fragments and extensions of intuitionistic linear logic were studied by Kamide (Kamide, 2004; Kamide, 2006c). In (Kamide, 2004), Petri net interpretations of various fragments of a spatio-temporal soft linear logic were discussed. In (Kamide, 2006c), Petri nets with inhibitor arcs, which were first introduced by Kosaraju (Kosaraju, 1973) to show the limitation of the usual Petri nets, were described using Kripke semantics for intuitionistic linear logic with strong negation. The approarches using Kripke semantics can obtain a very simple correspondence between Petri net and linear logic.

Using elementary Petri nets, it is difficult to mixturely model the discrete, continuous and other complicated events, for example, DNA, RNA and amino acid sequence events. Therefore, Hybrid Functional Petri Net with extension (HFPNe) was introduced to overcome this difficulty and also to allow representation of pathway models without loss of biological details. First, this book chapter explains how modeling with Petri net is done. After which, a full definition of HFPNe is given, along with its relation to hybrid Petri net and other related Petri net extension. Finally, to demonstrate the elegance of HFPNe architecture in handling complex pathway modeling, we chose a biological pathway that involves discrete, continuous and sequence events (gene regulatory network of the cell fate determination in C. elegans). We provide the biological mechanisms of this network and show how intuitively it could be modeled on HFPNe architecture using a software tool, Cell Illustrator.KeywordsGene Regulatory NetworkCell Fate DeterminationInput PlaceOutput PlaceContinuous PlaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The synthesis problem of concurrent systems is the problem of synthesizing a concurrent system model from sequential observations. The paper studies the synthesis problem for elementary Petri nets and transition systems. A characterization of the class of transition systems which correspond to elementary Petri nets is proven. It is shown how to generate all elementary Petri nets corresponding to a given transition system. If there is any such elementary Petri net, it is proven that there always exists a small one which has only polynomially many elements in the size of the transition system.

In this paper, the synthesis of bounded Petri nets from deterministic modal transition systems is shown to be undecidable. The proof is built from three components. First, it is shown that the problem of synthesising bounded Petri nets satisfying a given formula of the conjunctive nu-calculus (a suitable fragment of the mu-calculus) is undecidable. Then, an equivalence between deterministic modal transition systems and a language-based formalism called modal specifications is developed. Finally, the claim follows from a known equivalence between the conjunctive nu-calculus and modal specifications.

Traffic lights are commonly used in the intersection in urban areas nowadays. In fact a network of the traffic lights regulating the urban vehicle can be viewed as a complex discrete-event system (DES). It is proved that Petri nets are one of the methodologies for modeling, analyzing and controlling on DES. A new approach via programmable logic controller (PLC) and Petri net synthesis is proposed in this paper. Firstly, we described the typical urban traffic lights control using Petri nets. Having compared with three methods for Petri nets realization, the best implementation method of the traffic lights control system was one which made with the PLC. The sequential function chart (SFC) of the PLC system is also given in the thesis. The result shown that the method has many advantages at improving flexibility and troubleshoots of system and reducing the difficulty in its realization.

When a Petri net is synthesised from a labelled transition system, it is frequently desirable that certain additional constraints are fulfilled. For example, in circuit design, one is often interested in constructing safe Petri nets. Targeting such subclasses of Petri nets is not necessarily computationally more efficient than targeting the whole class. For example, targeting safe nets is known to be NP-complete while targeting the full class of place/transition nets is polynomial, in the size of the transition system. In this paper, several classes of Petri nets are examined, and their suitability for being targeted through efficient synthesis from labelled transition systems is studied and assessed. The focus is on choice-free Petri nets and some of their subclasses. It is described how they can be synthesised efficiently from persistent transition systems, summarising and streamlining in tutorial style some of the authors’ and their groups’ work over the past few years.

We present in this paper a traffic model based on Petri nets and minplus algebra. A Petri net is represented by two matrices one in standard algebra and the other in minplus algebra. Then a system point of view is developed based on a matrix product combining these two algebras. Introducing inputs and outputs on transitions and places we can link two Petri nets by associating outputs with inputs of the two systems. This linking corresponds to a contraction operator. Combining elementary systems with this contraction operator we can build large systems. This point of view is used to define the traffic dynamics of a regular town from three elementary Petri nets.

In the area of Petri nets, many different developments have taken place within the last 30 years, in academia as well as in practice. For an adequate use in practice, a coherent and application oriented combination of various types and techniques for Petri nets is necessary. In order to attain a formal basis for different classes of Petri nets we introduce the concept of abstract Petri nets. The essential point of abstract Petri nets is to allow different kinds of net structures as well as the combination of various kinds of data types. This means that in abstract Petri nets the data type and the net structure part can be considered as abstract parameters which can be instantiated to different concrete net classes. We show that several net classes, like place/transition nets, elementary nets, S-graphs, algebraic high-level nets, and predicate/transitions nets are instantiations of abstract Petri nets. Moreover, we discuss coloured Petri nets in this context.KeywordsPredicate LogicForgetful FunctorUniform ApproachSpeci CationSignature Morphism