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Be the first to write a review About this product. All listings for this product Buy it now Buy it now. Any condition Any condition. About this product Product Information Distributed algorithms, essential to networks of all kinds, are presented using the highly intuitive and technically elegant Petri net technique in this text for students and practitioners. A variety of distributed algorithms are presented, and proven correct using temporal logic. Additional Product Features Author s.

Show more Show less. No ratings or reviews yet. Be the first to write a review. Best-selling in Non Fiction See all. FSPNs allow us to consider stochastic systems which—due to incomplete or vague knowledge—have uncertain parameter values, and to analyze the uncertainties of outputs with output referring to a moment of the stochastic process resulting from the uncertainties associated with the input parameters. Our efficient analysis approach builds on simulation, which works equally for bounded and unbounded models, i.

We mainly consider the following scenario where our approach may be of help to cope with the challenges on hand. For a given biological system phenomenon , there may exist different qualitative and quantitative experimental findings data or models , each of which may describe complementary aspects of the system.

The appropriate combination of all these findings to form a more detailed model is of great significance to gain a deeper understanding of the system [ 7 , 8 ]. A simplistic approach might be to convert all of these qualitative and quantitative findings into qualitative descriptions and to build a model from a merely qualitative point of view. But it becomes more challenging the other way around.

In this paper, we aim to use our approach for integrating qualitative experimental findings into a quantitative model and to explore the system to be studied from the quantitative point of view. This may be achieved by describing qualitative or semi-quantitative findings using fuzzy sets. We apply a new modeling formalism, FSPNs, to cope simultaneously with both randomness and fuzziness of biological systems by combining SPNs and fuzzy sets.

Thus, a biological system can be analyzed even if it has some parameters that are unknown or not precisely estimated. We present a simulation-based analysis method for FSPNs, and thus we can analyze the uncertainties of outputs caused by the uncertainties associated with the input parameters for both bounded or unbounded models independently of the size of the state space.

Our approach permits to explore both the transient and steady state behavior, and can be easily varied to cope with different types of fuzzy numbers. This paper is structured as follows. After a brief review of related work in Section 2, we describe fuzzy stochastic Petri nets in Section 3, followed by a simulation-based analysis method for FSPNs in Section 4. After that, we use an example of the yeast polarization model to illustrate our method in Section 5, followed by the discussions and conclusions in Section 6.

Stochastic modeling is able to capture the randomness and fine grain behavior of biological systems which are not appropriately described by deterministic methods confining themselves to the averaged behavior. SPNs have been widely used for modeling stochastic biological systems since Gross et al. For example, Srivastava et al. Marwan et al. Mura et al.

See [ 6 , 18 , 19 ] and references therein for more applications. Like other quantitative modeling methods, SPNs rely on accurate kinetic data; in any case, the results shall be very carefully cross-validated. Parameter estimation plays an important role in the construction of quantitative models. A lot of work has been done to address this challenging problem, some of them within a Petri net framework.

For example, Shaw et al. Koh et al. It exploits the structure of a large pathway model to decompose it into smaller components and then estimates the parameters independently. Donaldson and Gilbert [ 22 ] characterize the desired behavior of SPN models by properties expressed in a Probabilistic Linear-time Temporal Logic with numerical constraints and combine Monte Carlo Model Checking with genetic algorithms to alter the parameters in the SPN model.

Li et al. In contrast, our approach in this paper differs from parameter estimation in the following way. Fuzzy logic is able to appropriately capture systems with only qualitative or incomplete quantitative information by describing the dynamic behavior of a system with a set of fuzzy rules.

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It has been applied for the modeling of a number of biological systems, see e. Fuzzy Petri nets, combining fuzzy logic and qualitative Petri nets, facilitate the use of fuzzy logic by help of the graphical representation coming with Petri nets. They have occasionally been used for biological modeling, see, e. The advantage of fuzzy logic or fuzzy Petri nets is that they can be deployed to model biological systems if the kinetic data are incomplete or even unavailable; the disadvantage is that they are usually used for constructing qualitative models, rather than quantitative ones.

To overcome this, researchers started to combine quantitative Petri nets with fuzzy logic. For example, Valette et al. Ding et al. But to the best of our knowledge, there are so far no applications of these fuzzy quantitative Petri nets to the modeling of biological systems. In this paper, we combine SPNs and fuzzy logic in order to quantitatively model biological systems with uncertain kinetic parameters. We will use a similar way to associate the rate of a stochastic transition with a fuzzy number as in [ 29 ]. However, in [ 29 ], they analytically determine the steady state distribution of an SPN model by generating the corresponding continuous time Markov chain CTMC.

As this approach requires a finite state space, they only can deal with bounded Petri nets. But, there are many biological models that are unbounded.

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Consequently, all numerical approaches requiring a closed representation of the CTMC do not work. Even if the model is bounded, the size of the state space often exceeds the computational resources required for numerical analyses; see [ 30 ] for details. Therefore, in this paper we will adopt a simulation approach to tackle both bounded and unbounded models in order to overcome the limitations of the method developed in [ 29 ].

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Moreover, our approach immediately reveals both the transient and steady state behavior, and can be easily varied to cope with different types of fuzzy numbers. Petri nets [ 31 , 32 ] see Fig 1 for an example are weighted, directed, bipartite multigraphs, which consist of places, transitions and arcs. In biological models, places may represent chemical species or any kind of compounds, e.

For a chemical reaction, its precursors correspond to the preplaces of a transition, while products correspond to its postplaces. The preplaces postplaces of a transition are also called the input output places of the transition. The arcs connect places with transitions, or transitions with places. The weight of an arc indicates its multiplicity, reflecting e. The weight 1 is usually not explicitly labeled.

A place may contain an arbitrary natural number of tokens, represented as black dots if the number of tokens is smaller, or a natural number if it is larger. A specific distribution of tokens over all places of a Petri net is called a marking, which represents a state of the Petri net.


The formal definition of a qualitative Petri net is given in the following. Cells grow or die. The virus may enter a healthy cell UCell and infect it ICell.

Then the virus starts the replication of itself and more viruses are released note the arc weight of Besides, infected cells may die and viruses may degrade. A transition is called enabled if each of its preplaces contains tokens whose number is greater than or equal to the weight of the corresponding arcs. If a transition is enabled, it may fire. The firing of a transition moves tokens from its preplaces to its postplaces according to the weights of the corresponding arcs, updating the current marking to a new reachable one.

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  • The repeated firing of transitions establishes the behavior of a net. The set of markings reachable from the initial marking form the state space of the net. These reachable markings and transitions between them constitute the reachability graph of the net. Stochastic Petri nets are a popular extension of Petri nets [ 6 , 35 ], and more specifically can be considered as an extension of Time Petri nets TPN [ 36 ].