On Petri Nets with Hierarchical Special Arcs

Relating SPO and DPO graph rewriting with Petri nets having read, inhibitor and reset arcs

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.

P/T nets with reset and transfer arcs can be seen as counter-machines with some restricted set of operations. Surprisingly, several problems related to boundedness are harder for Reset nets than for the more expressive Transfer nets. Our main result is that boundedness is undecidable for nets with three reset arcs, while it is decidable for nets with two resetable places.KeywordsReachability ProblemCounter MachinePath SchemeBoundedness ProblemReset Transition

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.

In this paper we present a survey on methods for the synthesis of Petri nets from behavioral descriptions given as languages. We consider place/transition Petri nets, elementary Petri nets and Petri nets with inhibitor arcs. For each net class we consider classical languages, step languages and partial languages as behavioral description. All methods are based on the notion of regions of languages. We identify two different types of regions and two different principles of computing from the set of regions of a language a finite Petri net generating this language. For finite or regular languages almost each combination of Petri net class, language type, region type and computation principle can be considered to compute such a net. Altogether, we present a framework for region-based synthesis of Petri nets from languages which integrates almost all known approaches and fills several remaining gaps in literature.

This article addresses expressiveness problems for Petri nets and their useful extension for modeling and control of a system that can be modeled with Petri nets. We construct some Petri net modules, namely, an enabling module, an inhibitor module and a Modulo-N counter, which are useful for system operations. We also present the simplifications for the enabling and inhibitor modules. Then, we propose two new types of arcs, namely, enabling and inhibitor arcs, with a weighted function set. The arcs with the weighted function set from places to transitions are the marked arcs. A function set is employed to denote the weight of an arc according to the system requirements. They are very useful to solve the resource reallocation problem in a resource allocation system, where resource sharing contributes to the occurrences of deadlocks. Finally, two examples are used to show the advantages of the presented new types of arcs.

As automated manufacturing systems become more complex, the need for an effective design tool to produce both high-level discrete event control systems (DECS) and low-level implementations becomes more important. Petri nets represent the most effective method for both the design and implementation of DECSs. In this paper, automation Petri nets (APN) are introduced to provide a new method for the design and implementation of DECSs. The APN is particularly well suited to multiproduct systems and provides a more effective solution than Grafcet in this context. Since ordinary Petri nets do not deal with sensors and actuators of DECSs, the Petri net concepts are extended, by including actions and sensor readings as formal structures within the APN. Moreover, enabling and inhibitor arcs, which can enable or disable transitions through the use of leading-edge, falling-edge and level of markings, are also introduced. In this paper, the methodology is explained by considering a fundamental APN structure. The conversion of APNs into the IEC1131-3 ladder diagrams (LD) for implementation on a PLC is also explained by using the token passing logic (TPL) concept. Finally, an illustrative example of how APNs can be applied to a discrete manufacturing problem is described in detail.

Information systems are developing and improving very fast. The requirements for synchronization for such systems also increase. In such systems linear recurrent sequences are widely used for synchronization. In this paper filtration algorithms of linear recurrent sequences sequences are synthesized. The synthesized algorithms can be classified as quasi-noncoherent (subincoherent) algorithms, as opposed to quasi-coherent algorithms, which use phase estimate of the received signal, or to noncoherent algorithms that use averaging over the entire region of phase uncertainty. The results of computer simulation of developed algorithms for various signal-to-noise ratios and accumulation time are presented.

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

In this paper, a state avoidance controller with inhibitor arcs, called IA-controller, is proposed for nonconvex forbidden state problems in Petri nets. The forbidden condition is transformed into non-convex marking constraints in controlled PNs. Simple formulas for IA-controllers are developed to avoid the non-convex marking condition. By extending enabling rule of controllable transitions, the IA-controller can realize OR-logics, which is impossible in ordinary PN structure-type controllers. The significance of IA-controller is that the controller can be easily implemented in the plant model. In addition, IA-controllers are synthesized in a modular way in order to solve multiple forbidden conditions.

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor /spl gamma/(n) > 1 in random exponential time 2/sup O(n)/, it is in AM for /spl gamma/(n) = 2, in coAM for /spl gamma/(n) = /spl radic/(n log n), and in NP /spl cap/ coNP for /spl gamma/(n) = /spl radic/n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor /spl gamma/(n) = log n, but cannot be solved in polynomial time for some /spl gamma/(n) = /spl Omega/(log log n) unless NP can be simulated in deterministic n/sup O(log log log n)/ time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is /spl Pi//sub 2/-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be /spl Pi//sub 2/-hard for approximation factors larger than 2. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor /spl gamma/(n) = /spl radic/(n/log n), solving an open problem of Blomer and Seifert (1999), and prove that the problem is also in coNP for /spl gamma/(n) = /spl radic/n. Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.

The best known linear recurrence sequence is the Fibonacci Sequence 1,1,2,3,5,8,..., where each term (after the first two terms) is a sum of the two preceding terms. We may extend it to become a “doubly infinite” sequence $$ ..., - 8,5, - 3,2, - 1,1,0,1,1,2,3,5,8,... . $$ .

Linear recurrent sequences and irrationality measures

On the ω-language Expressive Power of Extended Petri Nets

This paper introduces extended stochastic Petri nets to model wetlab experiments. The extentions include read and inhibitor arcs, stochastic transitions with freestyle rate functions as well as several deterministically timed transition types: immediate firing, deterministic firing delay, and scheduled firing. The extensions result into non-Markovian behaviour, which precludes analytical analysis approaches. But there are adapted stochastic simulation analysis (SSA) methods, ready to deal with the extended behaviour. Having the simulation traces, we apply simulative model checking of PLTL, a linear-time temporal logic (LTL) in a probabilistic setting.We present some typical model components, demonstrating the suitability of the introduced Petri net class for the envisaged application scenario. We conclude by looking briefly at a classical example of prokaryotic gene regulation, the lac operon case.KeywordsSystem Biology Markup LanguageReachability GraphStochastic Simulation AlgorithmStochastic TransitionSimulation TraceThese 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.