Petri net - Knowledge

3398: 4837:, i.e., when the transformation is "working" it is marked. This allows for the transformation to fire (or be marked) multiple times representing the real-world behavior of process throughput. Marking of the transformation assumes that transformation time must be greater than zero. A zero transformation time used in many typical Petri Nets may be mathematically appealing but impractical in representing real-world processes. dP-Nets also exploit the power of Petri Nets' hierarchical abstraction to depict 4710:) with the original Petri net, some add properties that cannot be modelled in the original Petri net formalism (e.g. timed Petri nets). Although backwards-compatible models do not extend the computational power of Petri nets, they may have more succinct representations and may be more convenient for modeling. Extensions that cannot be transformed into Petri nets are sometimes very powerful, but usually lack the full range of mathematical tools available to analyse ordinary Petri nets. 2981: 3477: 100: 4855: 3393:{\displaystyle W^{-}={\begin{bmatrix}*&t1&t2\\p1&1&0\\p2&0&1\\p3&0&1\\p4&0&0\end{bmatrix}},\ W^{+}={\begin{bmatrix}*&t1&t2\\p1&0&1\\p2&1&0\\p3&1&0\\p4&0&1\end{bmatrix}},\ W^{T}={\begin{bmatrix}*&t1&t2\\p1&-1&1\\p2&1&-1\\p3&1&-1\\p4&0&1\end{bmatrix}}} 392: 384: 702:, seen in the bottom figure. In the diagram, the firing rule for a transition can be characterised by subtracting a number of tokens from its input places equal to the multiplicity of the respective input arcs and accumulating a new number of tokens at the output places equal to the multiplicity of the respective output arcs. 4833:(dP-Nets) is a Petri Net extension developed by E. Dawis, et al. to better represent real-world process. dP-Nets balance the duality of change/no-change, action/passivity, (transformation) time/space, etc., between the bipartite Petri Net constructs of transformation and place resulting in the unique characteristic of 4778:

Another popular extension of Petri nets is hierarchy; this in the form of different views supporting levels of refinement and abstraction was studied by Fehling. Another form of hierarchy is found in so-called object Petri nets or object systems where a Petri net can contain Petri nets as its tokens

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One thing that makes Petri nets interesting is that they provide a balance between modeling power and analyzability: many things one would like to know about concurrent systems can be automatically determined for Petri nets, although some of those things are very expensive to determine in the general

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Extended WF-net is a Petri net that is composed of a WF-net with additional transition t (feedback transition). The sink place is connected as the input place of transition t and the source place as its output place. Firing of the transition causes iteration of the process (Note, the extended WF-net

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WRI (Well-handled with Regular Iteration) WF-net, is an extended acyclic well-handled WF-net. WRI-WF-net can be built as composition of nets, i.e., replacing a transition within a WRI-WF-net with a subnet which is a WRI-WF-net. The result is also WRI-WF-net. WRI-WF-nets are G-sound, therefore by

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of process activities. The WF-net transitions are assigned to tasks or activities, and places are assigned to the pre/post conditions. The WF-nets have additional structural and operational requirements, mainly the addition of a single input (source) place with no previous transitions, and output

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Instead of extending the Petri net formalism, we can also look at restricting it, and look at particular types of Petri nets, obtained by restricting the syntax in a particular way. Ordinary Petri nets are the nets where all arc weights are 1. Restricting further, the following types of ordinary

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that has two types of elements: places and transitions. Place elements are depicted as white circles and transition elements are depicted as rectangles. A place can contain any number of tokens, depicted as black circles. A transition is enabled if all places connected to it as inputs contain at

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There are many more extensions to Petri nets, however, it is important to keep in mind, that as the complexity of the net increases in terms of extended properties, the harder it is to use standard tools to evaluate certain properties of the net. For this reason, it is a good idea to use the most

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where each transition is labelled by a transition from the Petri net. The Petri net is then synchronised with the finite state automaton, i.e., a transition in the automaton is taken at the same time as the corresponding transition in the Petri net. It is only possible to take a transition in the

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In the diagram of a Petri net (see top figure right), places are conventionally depicted with circles, transitions with long narrow rectangles and arcs as one-way arrows that show connections of places to transitions or transitions to places. If the diagram were of an elementary net, then those

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add priorities to transitions, whereby a transition cannot fire, if a higher-priority transition is enabled (i.e. can fire). Thus, transitions are in priority groups, and e.g. priority group 3 can only fire if all transitions are disabled in groups 1 and 2. Within a priority group, firing is

4841:. Complex process systems are modeled as a series of simpler nets interconnected through various levels of hierarchical abstraction. The process architecture of a packet switch is demonstrated in, where development requirements are organized around the structure of the designed system. 3596:

It is a matter of walking the reachability-graph defined above, until either the requested-marking is reached or it can no longer be found. This is harder than it may seem at first: the reachability graph is generally infinite, and it isn't easy to determine when it is safe to stop.

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automaton if the corresponding transition in the Petri net is enabled, and it is only possible to fire a transition in the Petri net if there is a transition from the current state in the automaton labelled by it. (The definition of VASS is usually formulated slightly differently.)

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The non-deterministic property has been a very valuable one, as it lets the user abstract a large number of properties (depending on what the net is used for). In certain cases, however, the need arises to also model the timing, not only the structure of a model. For these cases,

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Petri net is a net in which there are no fully distinct elementary paths between a place and a transition (or transition and a place), i.e., if there are two paths between the pair of nodes then these paths share a node. An acyclic well-handled WF-net is sound (G-sound).

2966: 602:. In the given diagram of a Petri net (see right), the place circles may encompass more than one token to show the number of times a place appears in a configuration. The configuration of tokens distributed over an entire Petri net diagram is called a 3636: 5421: 1944: 5189: 4810:

have evolved, where there are transitions that are timed, and possibly transitions which are not timed (if there are, transitions that are not timed have a higher priority than timed ones). A subsidiary of timed Petri nets are the

171:, i.e. there are sufficient tokens in all of its input places; when the transition fires, it consumes the required input tokens, and creates tokens in its output places. A firing is atomic, i.e. a single non-interruptible step. 5458:-sound WF-net). Additionally, all the transitions in the process could fire (i.e., for each transition there is a reachable state in which the transition is enabled). A general sound (G-sound) WF-net is defined as being 3604:-hard years before it was shown to be decidable at all (Mayr, 1981). Papers continue to be published on how to do it efficiently. In 2018, Czerwiński et al. improved the lower bound and showed that the problem is not 5922:. Proceedings of the 6th International Symposium on Abstraction, Reformulation and Approximation—SARA 2005. Lecture notes in computer science. Vol. 3607. Airth Castle, Scotland, UK: Springer. pp. 149–164. 3463: 1659: 2113: 5069: 4968: 2118: 1691: 877: 4634: 1609: 1117: 1197: 2340: 333: 5506:

are realization of DSM-based plans into workflow processes by Petri nets, and are equivalent to WRI-WF-nets. The DSM-net construction process ensures the soundness property of the resulting net.

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does not impose a precondition on firing, and empties the place when the transition fires; this makes reachability undecidable, while some other properties, such as termination, remain decidable;

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an assignment of capacities to (some or all) places, and the transition relation is the usual one restricted to the markings in which each place with a capacity has at most that many tokens.

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While reachability seems to be a good tool to find erroneous states, for practical problems the constructed graph usually has far too many states to calculate. To alleviate this problem,

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imposes the precondition that the transition may only fire when the place is empty; this allows arbitrary computations on numbers of tokens to be expressed, which makes the formalism

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through the property that all input places have sufficient number of tokens (shown in the figures as dots) "equal to or greater" than the multiplicities on their respective arcs to

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Czerwiński, Wojciech; Lasota, Sławomir; Lazic, Ranko; Leroux, Jerome; Mazowiecki, Filip (2018). "The Reachability Problem for Petri Nets is Not Elementary (Extended Abstract)".

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ter Hofstede, Arthur H. M.; van der Aalst, Wil M. P.; Adams, Michael; Russell, Nick (2010). Hofstede, Arthur H. M; Aalst, Wil M. P; Adams, Michael; Russell, Nick (eds.).

3790: 1020: 4253: 4088: 4010: 3936: 96:. Unlike these standards, Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis. 4779:

inducing a hierarchy of nested Petri nets that communicate by synchronisation of transitions on different levels. See for an informal introduction to object Petri nets.

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to find if indeed a state can be reached, by finding a set of necessary conditions for the state to be reached then proving that those conditions cannot be satisfied.

4289: 3693: 1345: 774: 1473: 1434: 144:. Arcs run from a place to a transition or vice versa, never between places or between transitions. The places from which an arc runs to a transition are called the 4390: 4316: 4211: 4180: 4147: 4116: 4038: 3964: 3887: 3856: 3746: 3719: 3664: 2658: 2552: 2052: 1809: 1374: 5084: 3527:

case. Several subclasses of Petri nets have been studied that can still model interesting classes of concurrent systems, while these determinations become easier.

2837: 2264: 2010: 2519: 2489: 2448: 1818: 6847:. Formal Methods for Industrial Critical Systems - 15th International Workshop, FMICS 2010. Lecture Notes in Computer Science. Vol. 6371. pp. 215–230. 4690:

As well as for discrete events, there are Petri nets for continuous and hybrid discrete-continuous processes that are useful in discrete, continuous and hybrid

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Fernandez, J. L.; Sanz, R.; Paz, E.; Alonso, C. (19–23 May 2008). "Using hierarchical binary Petri nets to build robust mobile robot applications: RoboGraph".

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Since firing is nondeterministic, and multiple tokens may be present anywhere in the net (even in the same place), Petri nets are well suited for modeling the

7120:. Advanced Information Systems Engineering - 25th International Conference, CAiSE 2013. Lecture Notes in Computer Science. Vol. 7908. pp. 400–416. 3404: 4408: 6577: 712:

in the firing rule, where subtle variations on the algebraic properties can lead to other classes of Petri nets; for example, algebraic Petri nets.

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is used for many Petri net formalisms that extend the basic P/T net formalism; this includes coloured Petri nets, hierarchical Petri nets such as

6067: 6927:"PetriNet Editor + PetriNet Engine: New Software Tool For Modelling and Control of Discrete Event Systems Using Petri Nets and Code Generation" 5772: 3612:, independently by Jerome Leroux and by Wojciech Czerwiński and Łukasz Orlikowski. These results thus close the long-standing complexity gap. 5077:

net (FC), every arc from a place to a transition is either the only arc from that place or the only arc to that transition, i.e. there can be

4118:-live if it can fire infinitely often, i.e. if there is some fixed (necessarily infinite) firing sequence in which for every positive integer 7916: 7894: 7872: 7851: 7832: 7813: 7750: 7690: 7671: 7610: 7582: 7504: 7479: 7439: 7379: 7285: 7239: 7201: 7135: 7061: 7020: 6977: 6860: 6823: 6726: 6189: 6146: 4663: 2454: 5439:

place (sink) with no following transitions. Accordingly, start and termination markings can be defined that represent the process status.

4640: 6606: 2230:{\displaystyle M_{0}{\underset {G,t_{1}}{\longrightarrow }}M_{1}\wedge \cdots \wedge M_{n-1}{\underset {G,t_{n}}{\longrightarrow }}M_{n}} 7987: 5588: 5519: 4871:(SM), every transition has one incoming arc, and one outgoing arc, and all markings have exactly one token. As a consequence, there can 2343: 2057: 1621: 77: 6534:. Proc of the 3rd Workshop on Modelling of Objects, Components, and Agents. Vol. 571. Aarhus, Denmark: DAIMI PB. pp. 21–36. 4991: 4890: 1664: 7647:

Grobelna, Iwona (2011). "Formal verification of embedded logic controller specification with computer deduction in temporal logic".

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Fahland, D.; Gierds, C. (2013). "Analyzing and Completing Middleware Designs for Enterprise Integration Using Coloured Petri Nets".

6689: 6511: 6422: 6400: 6367: 6249: 5935: 5820: 5754: 5917: 4721:, and all other extensions sketched in this section. The term is also used specifically for the type of coloured nets supported by 818: 6162: 4786:

is an equivalent formalism to Petri nets. However, it can be superficially viewed as a generalisation of Petri nets. Consider a

4678:-bounded by adding a "counter-place" with flow opposite to that of the place, and adding tokens to make the total in both places 4599: 6300: 4768: 1048: 2806:

can be used to describe the reachable markings in terms of matrix multiplication, as follows. For any sequence of transitions

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It can be useful to explicitly impose a bound on places in a given net. This can be used to model limited system resources.

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places in a configuration would be conventionally depicted as circles, where each circle encompasses a single dot called a

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The precise meaning of "equal to or greater" will depend on the precise algebraic properties of addition being applied on

54: 2961:{\displaystyle R(N)=\{M\mid \exists w:\ w{\text{ is a firing sequence of }}N\ {\text{ and }}\ M=M_{0}+W^{T}\cdot o(w)\}} 1694: 1959: 1700: 7992: 5838: 3620: 7862: 6387:. 2001 IEEE Pacific Rim Conference on Communications, Computers and signal Processing. Vol. 1. pp. 323–6. 6101:"New Software Tool for Modeling and Control of Discrete-Event and Hybrid Systems Using Timed Interpreted Petri Nets" 5633: 179: 81: 6004:

Czerwiński, Wojciech; Orlikowski, Łukasz (2021). "Reachability in vector addition systems is Ackermann-complete".

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is usually used to 'time' these nets. In this case, the nets' reachability graph can be used as a continuous time

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An approach to relating some of these models of concurrency is proposed in the chapter by Winskel and Nielsen.

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in the Petri net is defined as the sequence of nodes (places and transitions) linked by the directed arcs. An

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Kučera, Erik; Haffner, Oto; Drahoš, Peter; Cigánek, Ján; Leskovský, Roman; Štefanovič, Juraj (January 2020).

3624: 6034: 5675: 5618: 5515: 5499: 198: 7928:"Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition in Considering Conflicting Opinions" 6611:

At – Automatisierungstechnik – Methoden und Anwendungen der Steuerungs-, Regelungs- und Informationstechnik

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Araki, T.; Kasami, T. (1977). "Some Decision Problems Related to the Reachability Problem for Petri Nets".

4321: 6618: 6437: 5573: 4787: 4783: 4714: 178:(e.g. a strict ordering of transitions, describing precedence) is defined, the execution of Petri nets is 7079:"High-level Petri nets for the process description and control in service-oriented manufacturing systems" 1556:

We are generally interested in what may happen when transitions may continually fire in arbitrary order.

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Dufourd, C.; Finkel, A.; Schnoebelen, Ph. (1998). "Reset Nets Between Decidability and Undecidability".

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Desel, Jörg; Juhás, Gabriel (2001). "What Is a Petri Net? Informal Answers for the Informed Reader". In

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if there are enough tokens in its input places for the consumptions to be possible, i.e. if and only if

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Modelling of Concurrent Systems: Structural and Semantical Methods in the High Level Petri Net Calculus

5416:{\displaystyle \forall s_{1},s_{2}\in S:(s_{1}{}^{\bullet }\cap s_{2}{}^{\bullet }\neq \emptyset )\to } 2976:

is a firing sequence; allowing arbitrary sequences of transitions will generally produce a larger set.

6075: 4807: 7782: 6889: 6755: 6354:. 2001 IEEE International Conference on Systems, Man, and Cybernetics. Vol. 3. pp. 1554–8. 5737:

Rozenburg, G.; Engelfriet, J. (1998). "Elementary Net Systems". In Reisig, W.; Rozenberg, G. (eds.).

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Mazurkiewicz, Antoni (1995). "Introduction to Trace Theory". In Diekert, V.; Rozenberg, G. (eds.).

6442: 5578: 5471: 4707: 4548: 3609: 2368: 1258: 970:. In many textbooks, arcs can only have multiplicity 1. These texts often define Petri nets using 50: 5431: 7729: 7661: 7545: 7389: 7340: 7299: 7098: 7026: 6983: 6907: 6645: 6623: 6609:[The Boolean differential calculus – A method for analysis and synthesis of Petri nets]. 6455: 6282: 6005: 5984: 5963: 5670: 4756: 4674:

Alternatively, places can be made bounded by extending the net. To be exact, a place can be made

3755: 987: 159:. Any distribution of tokens over the places will represent a configuration of the net called a 116:, after whom such structures are named, analyzed Petri nets extensively in his 1962 dissertation 4706:

There are many extensions to Petri nets. Some of them are completely backwards-compatible (e.g.

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Architecture of an SS7 Protocol Stack on a Broadband Switch Platform using Dualistic Petri Nets

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The Petri net that follows after the transition fires (Initial Petri net in the figure above).

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Leroux, Jérôme (2021). "The Reachability Problem for Petri Nets is Not Primitive Recursive".

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using only WRI-WF-net building blocks, one can get WF-nets that are G-sound by construction.

5184:{\displaystyle \forall s\in S:(|s^{\bullet }|\leq 1)\vee ({}^{\bullet }(s^{\bullet })=\{s\})} 4261: 3669: 1312: 741: 7939: 7790: 7719: 7711: 7633: 7570: 7537: 7467: 7427: 7367: 7332: 7323: 7273: 7263: 7227: 7217: 7189: 7121: 7090: 7049: 7008: 6965: 6938: 6897: 6848: 6811: 6773: 6763: 6714: 6629: 6499: 6447: 6388: 6355: 6274: 6237: 6214: 6177: 6112: 6043: 5923: 5847: 5808: 5781: 5742: 5717: 5650: 5613: 5539: 4718: 4471: 3531: 1939:{\displaystyle R(N)\ {\stackrel {D}{=}}\ \left\{M'{\Bigg |}M_{0}{\xrightarrow{*}}M'\right\}} 1443: 1404: 673:. Once and only once a transition is enabled will the transition fire. In this example, the 99: 73: 7909:

Modeling, Simulation, & Control of Flexible Manufacturing Systems: A Petri Net Approach

6607:"Der Boolesche Differentialkalkül – eine Methode zur Analyse und Synthese von Petri-Netzen" 4368: 4294: 4189: 4158: 4125: 4094: 4016: 3942: 3865: 3834: 3724: 3697: 3642: 2636: 2530: 2030: 1787: 1784:, we are interested in the firings that can be performed starting with the initial marking 1352: 590:

can be interpreted as representing a non-singleton set of configurations. In this respect,

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Kučera, Erik; Haffner, Oto; Drahoš, Peter; Leskovský, Roman; Cigánek, Ján (January 2020).

5628: 5527: 4775:, where the arc and guard expressions are restricted to make it easier to analyse the net. 4747: 4695: 2813: 2240: 1986: 979: 113: 63: 58: 6463: 4976:(MG), every place has one incoming arc, and one outgoing arc. This means, that there can 2494: 2464: 2423: 163:. In an abstract sense relating to a Petri net diagram, a transition of a Petri net may 7786: 6893: 6841:"Automatic Structure-Based Code Generation from Coloured Petri Nets: A Proof of Concept" 6759: 5941: 5713: 4655:, both places are assigned capacity 2, we obtain a Petri net with place capacities, say 3538:

results for Petri nets and some subclasses, can be found in Esparza and Nielsen (1995).

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Proceedings of the 25th International Colloquium on Automata, Languages and Programming

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Some definitions of Petri nets explicitly allow this as a syntactic feature. Formally,

3859: 982: 182:: when multiple transitions are enabled at the same time, they will fire in any order. 5898: 3476: 7956: 7864:

Clans of Petri Nets: Verification of protocols and performance evaluation of networks

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These definitions are in accordance with Murata's overview, which additionally uses

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least one token. Some sources state that Petri nets were invented in August 1939 by

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form a configuration. Similarly, if a Petri net is not an elementary net, then the

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Scheuring, Rainer; Wehlan, Herbert "Hans" (1991-12-01) . Bretthauer, Georg (ed.).

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Petri Net Reachability Checking Is Polynomial with Optimal Abstraction Hierarchies

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tokens in all reachable markings, including the initial marking; it is said to be

2453: 7622:"A new paradigm for uncertain knowledge representation by 'Plausible Petri nets'" 7459: 7126: 7094: 7077:

Mendes, J. Marco; Leitão, Paulo; Colombo, Armando W.; Restivo, Francisco (2012).

6852: 6768: 6706: 6633: 6498:. Lecture Notes in Computer Science. Vol. 2678. Springer. pp. 337–356. 6489:"Soundness and separability of workflow nets in the stepwise refinement approach" 6136: 155:

Graphically, places in a Petri net may contain a discrete number of marks called

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of the transition; the places to which arcs run from a transition are called the

7419: 7371: 6902: 6878:"A Petri Net Neural Network Robust Control for New Paste Backfill Process Model" 6877: 5741:. Lecture Notes in Computer Science. Vol. 1491. Springer. pp. 12–121. 5660: 5598: 5531: 4433: 2013: 7078: 7053: 6840: 6805: 6392: 6303:. Department of Computer Science, University of Aarhus, Denmark. Archived from 7795: 7770: 7724: 7638: 7621: 7574: 7530:"Modélisation des pannes d'une antenne active et modifications d'architecture" 7528:

Juan, Marion; Mailland, David; Fifis, Nicolas; Gregoris, Guy (December 2021).

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Models of Software Architecture – Design and Analysis with UML and Petri-Nets

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Other ways of modelling concurrent computation have been proposed, including

5502:(DSM) can model process relations, and be utilized for process planning. The 4759:, every token has a value. In popular tools for coloured Petri nets such as 7564: 7221: 7183: 7002: 6959: 5812: 4760: 4722: 3623:

to prove that such states cannot be reached. Linear temporal logic uses the

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Chiachio, Manuel; Chiachio, Juan; Presscott, Darren; Andrews, John (2018).

6787: 4040:-live if it can fire arbitrarily often, i.e. if for every positive integer 7715: 7529: 7277: 5866: 2839:

for the vector that maps every transition to its number of occurrences in

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van der Aalst, Wil M. P.; Stahl, Christian; Westergaard, Michael (2013).

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Meseguer, Jose; Montanari, Ugo (October 1990). "Petri nets are monoids".

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tokens in its source place, can reach the termination state marking with

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A Petri net is bounded if and only if its reachability graph is finite.

891:(or weight); note that no arc may connect two places or two transitions. 594:

extends the concept of configuration for elementary nets to Petri nets.

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Petri Net Synthesis for Discrete Event Control of Manufacturing Systems

7336: 7153:"Modeling shortest path games with Petri nets: a Lyapunov based theory" 6236:. Lecture Notes in Computer Science. Vol. 1443. pp. 103–115. 6181: 6176:. Lecture Notes in Computer Science. Vol. 1217. pp. 203–208. 3458:{\displaystyle M_{0}={\begin{bmatrix}1&0&2&1\end{bmatrix}}} 85: 2349:

Another common variation, e.g. in Desel and Juhás (2001), is to allow

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Modern Business Process Automation - YAWL and its Support Environment

7318: 7220:; Kishinevsky, M.; Kondratyev, A.; Lavagno, L.; Yakovlev, A. (2002). 3970:), if and only if it may fire, i.e. it is in some firing sequence in 2274:

A common variation is to disallow arc multiplicities and replace the

1891: 563:, so that the count (or weight) for each arc is a measure of the arc 7683:

Formális módszerek az informatikában (Formal methods in informatics)

7360:"Strategies for Modeling Complex Processes Using Colored Petri Nets" 6047: 5770:

Reisig, Wolfgang (1991). "Petri Nets and Algebraic Specifications".

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Petri nets can be described as having different degrees of liveness

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of a Petri net (graph) is a multiset of its places, i.e., a mapping

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and is commonly described with reference to Petri net diagrams as a

66:—at the age of 13—for the purpose of describing chemical processes. 6010: 5989: 5968: 5872: 4853: 4661: 4638: 4406: 3634: 2108:{\displaystyle {\vec {\sigma }}=\langle t_{1}\cdots t_{n}\rangle } 1654:{\displaystyle M{\overset {*}{\underset {G}{\longrightarrow }}}M'} 736: 390: 382: 212: 98: 7366:. Lecture Notes in Computer Science. Vol. 7. pp. 6–55. 7157:

International Journal of Applied Mathematics and Computer Science

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Architecture of Computer-based Systems using Dualistic Petri Nets

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Zaitsev, D. A. (2013). "Toward the Minimal Universal Petri Net".

3896:, if it can never fire, i.e. it is not in any firing sequence in 5064:{\displaystyle \forall s\in S:|s^{\bullet }|=|{}^{\bullet }s|=1} 4963:{\displaystyle \forall t\in T:|t^{\bullet }|=|{}^{\bullet }t|=1} 7001:

Carmona, J.; van Dongen, B.F.; Solti, A.; Weidlich, M. (2018).

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Lectures on Petri Nets I: Basic Models – Advances in Petri Nets

1686:{\displaystyle {\overset {*}{\underset {G}{\longrightarrow }}}} 7364:

Transactions on Petri Nets and Other Models of Concurrency VII

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IEEE International Conference on Robotics and Automation, 2008

5434:(WF-nets) are a subclass of Petri nets intending to model the 3471: 7226:. Springer Series in Advanced Microelectronics. Vol. 8. 4755:

In a standard Petri net, tokens are indistinguishable. In a

872:{\displaystyle W:(S\times T)\cup (T\times S)\to \mathbb {N} } 7182:

Yakovlev, Alex; Gomes, Luis; Lavagno, Luciano, eds. (2000).

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property, indicating that a process with a start marking of

1199:. Definitions of pre- and postsets of places are analogous. 7223:

Logic Synthesis for Asynchronous Controllers and Interfaces

6681:

Modeling Business Processes - A Petri Net-Oriented Approach

6267:

IEEE Transactions on Systems, Man, and Cybernetics: Systems

4763:, the values of tokens are typed, and can be tested (using 4629:{\displaystyle C:P\rightarrow \!\!\!\shortmid \mathbb {N} } 4460:

when all of its places are. A Petri net (graph) is called

6585:

Handbook of Logic and the Foundations of Computer Science

6530:

Ping, L.; Hao, H.; Jian, L. (2004). Moldt, Daniel (ed.).

5538:. Different models provide tradeoffs of concepts such as 4258:

Note that these are increasingly stringent requirements:

1112:{\displaystyle {}^{\bullet }t=\{s\in S\mid W(s,t)>0\}} 648:(bottom figure) be a Petri net with a marking configured 7317:

Bernardeschi, C.; De Francesco, N.; Vaglini, G. (1995).

1240:. We say the marking assigns to each place a number of 571:

If a Petri net is equivalent to an elementary net, then

6961:

Process Mining - Data Science in Action, Second Edition

5079:

both concurrency and conflict, but not at the same time

3488: 1192:{\displaystyle t^{\bullet }=\{s\in S\mid W(t,s)>0\}} 7453: 7451: 6423:"The application of Petri nets to workflow management" 4819:

through adjustable randomness of the transitions. The

4325: 4324: 3426: 3271: 3137: 3003: 2461:

Its transition relation can be described as a pair of

2335:{\displaystyle F\subseteq (S\times T)\cup (T\times S)} 978:. When using this convention, a Petri net graph is a 634:(top figure) be a Petri net with a marking configured 328:{\displaystyle F\subseteq (P\times T)\cup (T\times P)} 7413: 7411: 7272:. Lecture Notes in Computer Science. Vol. 2549. 5899:"The Reachability Problem Requires Exponential Space" 5218: 5193:

Extended free choice (EFC) – a Petri net that can be

5087: 4994: 4893: 4846:

simple net type possible for a given modelling task.

4602: 4551: 4494: 4464:

if it is bounded for every possible initial marking.

4371: 4297: 4264: 4219: 4192: 4161: 4128: 4097: 4054: 4019: 3976: 3945: 3902: 3868: 3837: 3798: 3758: 3727: 3700: 3672: 3645: 3564: 3407: 2984: 2852: 2816: 2753: 2666: 2639: 2560: 2533: 2497: 2467: 2426: 2371: 2292: 2243: 2121: 2060: 2033: 1989: 1962: 1821: 1790: 1733: 1703: 1667: 1624: 1579: 1499: 1446: 1407: 1355: 1315: 1261: 1212: 1133: 1051: 990: 905: 887:, i.e. it assigns to each arc a non-negative integer 821: 813:, i.e. no object can be both a place and a transition 744: 715:

The following formal definition is loosely based on (

575:

can be the countable set {0,1} and those elements in

285: 220: 7266:; Yakovlev, Alex; Rozenberg, Grzegorz, eds. (2002). 7004:

Conformance Checking - Relating Processes and Models

6807:

Modeling in Systems Biology - The Petri Net Approach

6525: 6523: 4659:; its reachability graph is displayed on the right. 1604:{\displaystyle M{\underset {G}{\longrightarrow }}M'} 201:

that extend a class of nets called elementary nets.

6027:"Petri Nets: Properties, Analysis and Applications" 6587:. Vol. 4. OUP. pp. 1–148. Archived from 5415: 5183: 5063: 4962: 4628: 4588: 4537: 4384: 4354: 4310: 4283: 4247: 4205: 4174: 4141: 4110: 4082: 4032: 4004: 3958: 3930: 3881: 3850: 3823: 3784: 3740: 3713: 3687: 3658: 3585: 3457: 3392: 2960: 2831: 2795: 2733: 2652: 2624: 2546: 2513: 2483: 2442: 2408: 2334: 2258: 2229: 2107: 2046: 2004: 1975: 1938: 1803: 1776: 1724:; that is, if it is reachable in 0 or more steps. 1716: 1685: 1653: 1603: 1544: 1467: 1428: 1368: 1339: 1298: 1232: 1191: 1111: 1014: 962: 871: 768: 327: 250: 6810:. Computational Biology. Vol. 16. Springer. 6673: 6671: 6487:van Hee, K.; Sidorova, N.; Voorhoeve, M. (2003). 6350:Dawis, E. P.; Dawis, J. F.; Koo, Wei-Pin (2001). 5807:. LNCS. Vol. 2128. Springer. pp. 1–25. 4694:, and related to discrete, continuous and hybrid 4617: 4616: 4615: 2353:to be defined on places. This is discussed under 1976:{\displaystyle {\underset {G}{\longrightarrow }}} 1871: 1717:{\displaystyle {\underset {G}{\longrightarrow }}} 6744:"esyN: Network Building, Sharing and Publishing" 6130: 6128: 681:generates a map that has the marking configured 7319:"A Petri nets semantics for data flow networks" 6678:van der Aalst, W.M.P.; Stahl, C. (2011-05-27). 6494:. In van der Aalst, W. M. P.; Best, E. (eds.). 5867:"Decidability issues for Petri nets – a survey" 5479:includes every node in the sequence only once. 4732:Additional types of arcs; two common types are 4666:A two-bounded Petri net, obtained by extending 3550:for Petri nets is to decide, given a Petri net 627:is an output place to the same transition. Let 6845:Formal Methods for Industrial Critical Systems 4771:. A subsidiary of coloured Petri nets are the 6532:On 1-soundness and soundness of workflow nets 6416: 6414: 6412: 6167:"A brief introduction to coloured Petri Nets" 4728:A short list of possible extensions follows: 2237:. The set of firing sequences is denoted as 8: 7743:Petri Net Theory and the Modeling of Systems 7083:International Journal of Production Research 6804:; Reisig, Wolfgang; Schreiber, Falk (2011). 5204:net (AC), concurrency and conflict (in sum, 5175: 5169: 3608:. In 2021, this problem was shown to be non- 2955: 2868: 2361:Formulation in terms of vectors and matrices 2102: 2076: 1186: 1147: 1106: 1067: 957: 912: 484:), which extends the elementary net so that 117: 88:for stepwise processes that include choice, 7769:Petri, Carl Adam; Reisig, Wolfgang (2008). 7700:Peterson, James Lyle (1977). "Petri Nets". 7601:Cardoso, Janette; Camargo, Heloisa (1999). 6839:Kristensen, L. M.; Westergaard, M. (2010). 6138:Discrete, continuous, and hybrid Petri Nets 5694:Petri, Carl Adam; Reisig, Wolfgang (2008). 4686:Discrete, continuous, and hybrid Petri nets 963:{\displaystyle F=\{(x,y)\mid W(x,y)>0\}} 6430:Journal of Circuits, Systems and Computers 6174:A brief introduction to colored Petri nets 5865:Esparza, Javier; Nielsen, Mogens (1995) . 4863:Petri nets are commonly used and studied: 7943: 7794: 7723: 7637: 7125: 6942: 6901: 6777: 6767: 6496:Application and Theory of Petri Nets 2003 6441: 6116: 6009: 5988: 5967: 5721: 5401: 5399: 5392: 5379: 5377: 5370: 5351: 5349: 5342: 5329: 5327: 5320: 5292: 5290: 5283: 5270: 5268: 5261: 5239: 5226: 5217: 5157: 5144: 5142: 5121: 5115: 5106: 5086: 5050: 5041: 5039: 5033: 5025: 5019: 5010: 4993: 4949: 4940: 4938: 4932: 4924: 4918: 4909: 4892: 4784:vector addition system with states (VASS) 4750:and implies existence of a universal net. 4622: 4621: 4601: 4577: 4550: 4526: 4493: 4376: 4370: 4333: 4326: 4323: 4302: 4296: 4269: 4263: 4236: 4218: 4197: 4191: 4166: 4160: 4133: 4127: 4102: 4096: 4071: 4053: 4024: 4018: 3993: 3975: 3950: 3944: 3919: 3901: 3873: 3867: 3842: 3836: 3812: 3797: 3776: 3763: 3757: 3732: 3726: 3705: 3699: 3671: 3650: 3644: 3563: 3421: 3412: 3406: 3266: 3257: 3132: 3123: 2998: 2989: 2983: 2934: 2921: 2903: 2892: 2851: 2815: 2787: 2774: 2758: 2752: 2734:{\displaystyle \forall s,t:W^{+}=W(t,s).} 2686: 2665: 2644: 2638: 2580: 2559: 2538: 2532: 2506: 2498: 2496: 2476: 2468: 2466: 2435: 2427: 2425: 2397: 2370: 2291: 2242: 2221: 2208: 2192: 2180: 2161: 2148: 2132: 2126: 2120: 2096: 2083: 2062: 2061: 2059: 2038: 2032: 1988: 1963: 1961: 1892: 1886: 1880: 1870: 1869: 1845: 1840: 1838: 1837: 1820: 1795: 1789: 1765: 1732: 1704: 1702: 1668: 1666: 1628: 1623: 1583: 1578: 1545:{\displaystyle \forall s:M(s)\geq W(s,t)} 1498: 1445: 1406: 1360: 1354: 1314: 1287: 1260: 1226: 1225: 1211: 1138: 1132: 1055: 1053: 1050: 989: 904: 865: 864: 820: 743: 609:In the top figure (see right), the place 284: 219: 3619:is usually used in conjunction with the 2625:{\displaystyle \forall s,t:W^{-}=W(s,t)} 2452: 719:). Many alternative definitions exist. 716: 7499:. Andre Kleyner (5th ed.). Wiley. 7118:Active Flow and Combustion Control 2018 5686: 4618: 4470:Boundedness is decidable by looking at 387:A Petri net with an enabled transition. 193:Formal definition and basic terminology 7397: 7387: 7089:(6). Taylor & Francis: 1650–1665. 7048:. Pasadena, CA, USA. pp. 1372–7. 4355:{\textstyle \textstyle {j\in {1,2,3}}} 3600:In fact, this problem was shown to be 7536:. Sécurité des Systèmes Industriels. 6094: 6092: 5454:tokens in its sink place (defined as 3522:Mathematical properties of Petri nets 1983:restricted to its reachable markings 1436:tokens from each of its input places 27:Model to describe distributed systems 7: 7764:(Ph. D. thesis). University of Bonn. 6301:"Very Brief Introduction to CP-nets" 4767:expressions) and manipulated with a 4213:-live in every reachable marking in 4186:) if it may always fire, i.e. it is 1475:tokens in each of its output places 7867:. LAP LAMBERT Academic Publishing. 6135:David, René; Alla, Hassane (2005). 5589:Diagnosis (artificial intelligence) 5520:communicating finite-state machines 3510:and Montanari considered a kind of 2894: is a firing sequence of 2420:of non-negative integers of length 1380:, a marking of the Petri net graph. 1233:{\displaystyle M:S\to \mathbb {N} } 78:Business Process Model and Notation 7823:Riemann, Robert-Christoph (1999). 6565:. World Scientific. pp. 3–67. 5301: 5219: 5088: 4995: 4894: 2877: 2796:{\displaystyle W^{T}=-W^{-}+W^{+}} 2667: 2561: 2346:as both can represent each other. 1500: 25: 7497:Practical reliability engineering 4423:if it does not contain more than 4419:A place in a Petri net is called 4048:times in some firing sequence in 1255:by some, see above) is a 4-tuple 189:behavior of distributed systems. 7495:O'Connor, Patrick D. T. (2012). 7466:. Springer. pp. 4716–4717. 7464:Encyclopedia of Database Systems 7426:. Springer. pp. 4717–4718. 7424:Encyclopedia of Database Systems 6711:Encyclopedia of Database Systems 6421:van der Aalst, W. M. P. (1998). 6326:"LLPN - Linear Logic Petri Nets" 4486:Petri nets with place capacities 3639:A Petri net in which transition 3475: 616:is an input place of transition 103:(a) Petri net trajectory example 69:Like industry standards such as 7911:. World Scientific Publishing. 7269:Concurrency and Hardware Design 6656:from the original on 2017-10-16 5916:Küngas, P. (July 26–29, 2005). 4821:exponential random distribution 4769:functional programming language 4538:{\displaystyle (S,T,W,C,M_{0})} 4448:A (marked) Petri net is called 1777:{\displaystyle N=(S,T,W,M_{0})} 735:by some, but see below) is a 3- 7973:Concurrency (computer science) 7963:Formal specification languages 7907:; Venkatesh, Kurapati (1998). 7889:. Kluwer Academic Publishers. 7472:10.1007/978-1-4614-8265-9_1476 7458:van der Aalst, W.M.P. (2018). 7418:van der Aalst, W.M.P. (2018). 7185:Hardware Design and Petri Nets 6958:van der Aalst, W.M.P. (2016). 6719:10.1007/978-1-4614-8265-9_1179 6713:. Springer. pp. 370–374. 6705:van der Aalst, W.M.P. (2018). 5410: 5407: 5363: 5357: 5313: 5310: 5307: 5304: 5254: 5178: 5163: 5150: 5138: 5132: 5122: 5107: 5103: 5051: 5034: 5026: 5011: 4950: 4933: 4925: 4910: 4612: 4583: 4552: 4532: 4495: 4242: 4223: 4077: 4058: 3999: 3980: 3925: 3906: 3818: 3799: 3580: 3574: 3468:Category-theoretic formulation 2952: 2946: 2862: 2856: 2826: 2820: 2725: 2713: 2704: 2692: 2619: 2607: 2598: 2586: 2507: 2499: 2477: 2469: 2436: 2428: 2403: 2372: 2329: 2317: 2311: 2299: 2282:with a simple set, called the 2253: 2247: 2194: 2134: 2067: 1999: 1993: 1965: 1911: 1893: 1831: 1825: 1771: 1740: 1706: 1671: 1631: 1585: 1539: 1527: 1518: 1512: 1462: 1450: 1423: 1411: 1334: 1316: 1293: 1262: 1222: 1177: 1165: 1097: 1085: 1009: 991: 948: 936: 927: 915: 861: 858: 846: 840: 828: 763: 745: 322: 310: 304: 292: 245: 227: 112:The German computer scientist 1: 7741:Peterson, James Lyle (1981). 7432:10.1007/978-1-4614-8265-9_826 6707:"Business Process Management" 6684:. MIT Press. pp. 1–400. 5559:Boolean differential calculus 4589:{\displaystyle (S,T,W,M_{0})} 3889:-live, where a transition is 3512:symmetric monoidal categories 2409:{\displaystyle (S,T,W,M_{0})} 2054:is a sequence of transitions 1299:{\displaystyle (S,T,W,M_{0})} 55:discrete event dynamic system 7806:A Primer in Petri Net Design 7127:10.1007/978-3-642-38709-8_26 7095:10.1080/00207543.2011.575892 6876:Gao, X.; Hu, Xinyan (2020). 6853:10.1007/978-3-642-15898-8_14 6769:10.1371/journal.pone.0106035 6634:10.1524/auto.1991.39.112.226 6219:10.1016/0304-3975(76)90067-0 6207:Theoretical Computer Science 6072:www.techfak.uni-bielefeld.de 6025:Murata, Tadao (April 1989). 5852:10.1016/0890-5401(90)90013-8 5786:10.1016/0304-3975(91)90203-e 5773:Theoretical Computer Science 2365:The markings of a Petri net 2270:Variations on the definition 1695:reflexive transitive closure 269:are disjoint finite sets of 57:. A Petri net is a directed 7762:Kommunikation mit Automaten 7372:10.1007/978-3-642-38143-0_2 6903:10.1109/ACCESS.2020.2968510 6742:Favrin, Bean (2014-09-02). 5905:. Yale University: 305–329. 5839:Information and Computation 5634:Workflow management systems 5510:Other models of concurrency 4858:Petri net types graphically 4651:For example, if in the net 3862:all of its transitions are 3785:{\displaystyle L_{1}-L_{4}} 2023:for a Petri net with graph 1956:is the transition relation 1015:{\displaystyle (S\cup T,F)} 119:Kommunikation mit Automaten 82:event-driven process chains 8009: 7988:Software modeling language 7885:; Dicesare, Frank (1993). 7681:Pataricza, András (2004). 7649:Przegląd Elektrotechniczny 7054:10.1109/ROBOT.2008.4543394 6393:10.1109/PACRIM.2001.953588 4431:if it is 1-bounded; it is 4411:The reachability graph of 4248:{\displaystyle R(N,M_{0})} 4083:{\displaystyle L(N,M_{0})} 4005:{\displaystyle L(N,M_{0})} 3931:{\displaystyle L(N,M_{0})} 7804:Reisig, Wolfgang (1992). 7796:10.4249/scholarpedia.6477 7760:Petri, Carl Adam (1962). 7639:10.1016/j.ins.2018.04.029 7575:10.1007/978-3-642-03121-2 7534:Techniques de l'Ingénieur 7460:"Workflow Model Analysis" 7232:10.1007/978-3-642-55989-1 7194:10.1007/978-1-4757-3143-9 7013:10.1007/978-3-319-99414-7 6970:10.1007/978-3-662-49851-4 6816:10.1007/978-1-84996-474-6 6617:(7). Stuttgart, Germany: 6576:Winskel, G.; Nielsen, M. 6452:10.1142/s0218126698000043 6360:10.1109/ICSMC.2001.973505 6279:10.1109/TSMC.2012.2237549 5723:10.4249/scholarpedia.6477 5656:Petri Net Markup Language 5564:Business process modeling 4488:can be defined as tuples 3824:{\displaystyle (N,M_{0})} 3586:{\displaystyle M\in R(N)} 2972:It must be required that 1727:For a (marked) Petri net 695:and results in Petri net 251:{\displaystyle N=(P,T,F)} 7861:Zaitsev, Dmitry (2013). 7842:Störrle, Harald (2000). 7151:Clempner, Julio (2006). 6578:"Models for Concurrency" 6504:10.1007/3-540-44919-1_22 5747:10.1007/3-540-65306-6_14 5594:Discrete process control 4643:An unbounded Petri net, 3534:, with decidability and 199:state-transition systems 130:A Petri net consists of 7603:Fuzziness in Petri Nets 6035:Proceedings of the IEEE 5813:10.1007/3-540-45541-8_1 5676:Vector addition systems 5619:Reliability engineering 5516:vector addition systems 5500:design structure matrix 4284:{\displaystyle L_{j+1}} 3688:{\displaystyle j>0,} 3666:is dead, while for all 3625:semi-decision technique 1340:{\displaystyle (S,T,W)} 769:{\displaystyle (S,T,W)} 655:. The configuration of 535:extends the concept of 335:is a set of (directed) 49:for the description of 7827:. Herbert Utz Verlag. 7632:(July 2018): 323–345. 5803:; et al. (eds.). 5574:Concurrent programming 5417: 5195:transformed into an FC 5185: 5065: 4964: 4859: 4835:transformation marking 4795:Prioritised Petri nets 4788:finite state automaton 4773:well-formed Petri nets 4671: 4670:with "counter-places". 4648: 4630: 4590: 4539: 4474:, by constructing the 4462:(structurally) bounded 4416: 4386: 4356: 4312: 4285: 4249: 4207: 4176: 4143: 4112: 4084: 4034: 4006: 3960: 3932: 3883: 3852: 3825: 3786: 3749: 3742: 3715: 3689: 3660: 3587: 3459: 3394: 2962: 2833: 2797: 2744:Then their difference 2735: 2654: 2626: 2548: 2515: 2485: 2458: 2444: 2410: 2342:. This does not limit 2336: 2260: 2231: 2109: 2048: 2006: 1977: 1940: 1921: 1805: 1778: 1718: 1687: 1655: 1605: 1559:We say that a marking 1546: 1469: 1468:{\displaystyle W(t,s)} 1430: 1429:{\displaystyle W(s,t)} 1370: 1341: 1300: 1234: 1193: 1113: 1016: 964: 873: 770: 396: 388: 329: 252: 118: 104: 7968:Models of computation 7716:10.1145/356698.356702 7703:ACM Computing Surveys 7660:Jensen, Kurt (1997). 7278:10.1007/3-540-36190-1 6383:Dawis, E. P. (2001). 5609:Kahn process networks 5569:Computational biology 5524:Kahn process networks 5418: 5186: 5066: 4965: 4857: 4817:nondeterministic time 4813:stochastic Petri nets 4665: 4642: 4631: 4591: 4540: 4410: 4387: 4385:{\displaystyle L_{0}} 4357: 4313: 4311:{\displaystyle L_{j}} 4286: 4250: 4208: 4206:{\displaystyle L_{1}} 4177: 4175:{\displaystyle L_{4}} 4144: 4142:{\displaystyle L_{3}} 4113: 4111:{\displaystyle L_{3}} 4085: 4044:, it occurs at least 4035: 4033:{\displaystyle L_{2}} 4007: 3961: 3959:{\displaystyle L_{1}} 3933: 3884: 3882:{\displaystyle L_{k}} 3853: 3851:{\displaystyle L_{k}} 3826: 3787: 3743: 3741:{\displaystyle L_{j}} 3716: 3714:{\displaystyle t_{j}} 3690: 3661: 3659:{\displaystyle t_{0}} 3638: 3617:linear temporal logic 3588: 3460: 3395: 2963: 2834: 2798: 2736: 2655: 2653:{\displaystyle W^{+}} 2627: 2549: 2547:{\displaystyle W^{-}} 2516: 2486: 2457:(b) Petri net example 2456: 2445: 2411: 2337: 2261: 2232: 2110: 2049: 2047:{\displaystyle M_{0}} 2007: 1978: 1941: 1887: 1806: 1804:{\displaystyle M_{0}} 1779: 1719: 1688: 1656: 1606: 1547: 1470: 1431: 1371: 1369:{\displaystyle M_{0}} 1347:is a Petri net graph; 1342: 1301: 1235: 1194: 1114: 1022:with node partitions 1017: 965: 874: 771: 620:; whereas, the place 468:is a net of the form 406:is a net of the form 394: 386: 330: 253: 108:Historical background 102: 84:, Petri nets offer a 42:), is one of several 7926:Xue-Guo, Xu (2019). 7626:Information Sciences 7542:10.51257/a-v1-se1221 6619:R. Oldenbourg Verlag 5666:Process architecture 5646:Finite-state machine 5216: 5210:not symmetrically: m 5085: 4992: 4891: 4839:process architecture 4831:Dualistic Petri Nets 4715:high-level Petri net 4600: 4549: 4492: 4369: 4322: 4295: 4262: 4217: 4190: 4159: 4126: 4095: 4052: 4017: 3974: 3968:potentially fireable 3943: 3900: 3866: 3835: 3796: 3756: 3725: 3698: 3670: 3643: 3562: 3548:reachability problem 3530:An overview of such 3405: 2982: 2850: 2832:{\displaystyle o(w)} 2814: 2751: 2664: 2637: 2558: 2531: 2495: 2465: 2424: 2369: 2290: 2259:{\displaystyle L(N)} 2241: 2119: 2058: 2031: 2027:and initial marking 2005:{\displaystyle R(N)} 1987: 1960: 1819: 1788: 1731: 1701: 1665: 1622: 1577: 1497: 1444: 1405: 1393:firing a transition 1353: 1313: 1259: 1210: 1131: 1049: 988: 903: 899:is the set of arcs: 819: 742: 579:that map to 1 under 339:(or flow relations). 283: 218: 94:concurrent execution 53:. It is a class of 36:place/transition net 7846:. Books on Demand. 7808:. Springer-Verlag. 7787:2008SchpJ...3.6477P 7666:. Springer Verlag. 7663:Coloured Petri Nets 7420:"Workflow Patterns" 6944:10.3390/app10217662 6894:2020IEEEA...818420G 6760:2014PLoSO...9j6035B 6242:10.1007/11527862_11 6118:10.3390/app10155027 5928:10.1007/11527862_11 5903:Technical Report 62 5805:Unifying Petri Nets 5714:2008SchpJ...3.6477P 5579:Control engineering 5491:is not a WF-net). 4984:, but there can be 4887:): mathematically, 4879:, but there can be 4708:coloured Petri nets 3610:primitive recursive 2514:{\displaystyle |T|} 2484:{\displaystyle |S|} 2443:{\displaystyle |S|} 2416:can be regarded as 1920: 1915: 1385:Execution semantics 796:is a finite set of 152:of the transition. 51:distributed systems 7993:Modeling languages 7945:10.3390/app9050983 7725:10338.dmlcz/135597 7605:. Physica-Verlag. 7337:10.1007/BF01178383 6563:The Book of Traces 6182:10.1007/BFb0035389 5876:(Revised ed.) 5671:Slicing Petri nets 5413: 5181: 5081:: mathematically, 5061: 4960: 4860: 4802:non-deterministic. 4757:coloured Petri net 4672: 4649: 4626: 4586: 4535: 4417: 4382: 4352: 4351: 4308: 4291:-liveness implies 4281: 4245: 4203: 4172: 4139: 4108: 4080: 4030: 4002: 3956: 3928: 3879: 3848: 3821: 3782: 3750: 3738: 3711: 3685: 3656: 3583: 3487:. You can help by 3455: 3449: 3390: 3384: 3241: 3107: 2958: 2829: 2793: 2731: 2650: 2622: 2544: 2511: 2481: 2459: 2440: 2406: 2332: 2256: 2227: 2215: 2155: 2105: 2044: 2002: 1973: 1971: 1950:reachability graph 1936: 1813:reachable markings 1801: 1774: 1714: 1712: 1683: 1677: 1651: 1637: 1613:is reachable from 1601: 1591: 1542: 1465: 1426: 1366: 1337: 1296: 1230: 1189: 1123:is the set of its 1109: 1041:is the set of its 1012: 960: 869: 766: 397: 389: 325: 248: 105: 86:graphical notation 47:modeling languages 34:, also known as a 7918:978-981-02-3029-6 7896:978-0-7923-9289-7 7874:978-3-659-42228-7 7853:978-3-8311-1330-9 7834:978-3-89675-629-9 7815:978-3-540-52044-3 7752:978-0-13-661983-3 7745:. Prentice Hall. 7692:978-963-9548-08-4 7685:. TYPOTEX Kiadó. 7673:978-3-540-62867-5 7612:978-3-7908-1158-2 7584:978-3-642-03122-9 7506:978-1-119-96126-0 7481:978-1-4614-8266-6 7441:978-1-4614-8266-6 7381:978-3-642-38142-3 7287:978-3-540-00199-7 7264:Cortadella, Jordi 7241:978-3-642-62776-7 7203:978-1-4419-4969-1 7137:978-3-319-98176-5 7063:978-1-4244-1646-2 7022:978-3-319-99413-0 6979:978-3-662-49850-7 6862:978-3-642-15897-1 6825:978-1-84996-473-9 6728:978-1-4614-8266-6 6191:978-3-540-62790-6 6148:978-3-540-22480-8 5553:Application areas 5462:-sound for every 5442:WF-nets have the 5208:) may occur, but 5202:asymmetric choice 4122:, the transition 3532:decision problems 3505: 3504: 3252: 3118: 2910: 2906: 2902: 2895: 2888: 2843:. Then, we have 2193: 2133: 2070: 1964: 1855: 1850: 1836: 1705: 1681: 1670: 1641: 1630: 1611:; we say that it 1584: 1564:is reachable from 74:activity diagrams 16:(Redirected from 8000: 7949: 7947: 7932:Applied Sciences 7922: 7900: 7878: 7857: 7838: 7819: 7800: 7798: 7765: 7756: 7737: 7727: 7696: 7677: 7656: 7643: 7641: 7616: 7589: 7588: 7560: 7554: 7553: 7525: 7519: 7518: 7492: 7486: 7485: 7455: 7446: 7445: 7415: 7406: 7405: 7399: 7395: 7393: 7385: 7355: 7349: 7348: 7324:Acta Informatica 7314: 7308: 7307: 7260: 7254: 7253: 7214: 7208: 7207: 7179: 7173: 7172: 7148: 7142: 7141: 7129: 7113: 7107: 7106: 7074: 7068: 7067: 7041: 7035: 7034: 6998: 6992: 6991: 6955: 6949: 6948: 6946: 6931:Applied Sciences 6922: 6916: 6915: 6905: 6873: 6867: 6866: 6836: 6830: 6829: 6798: 6792: 6791: 6781: 6771: 6739: 6733: 6732: 6702: 6696: 6695: 6675: 6666: 6664: 6662: 6661: 6627: 6602: 6596: 6595: 6593: 6582: 6573: 6567: 6566: 6558: 6552: 6551: 6527: 6518: 6517: 6493: 6484: 6478: 6477: 6475: 6474: 6468: 6462:. Archived from 6445: 6427: 6418: 6407: 6406: 6380: 6374: 6373: 6347: 6341: 6340: 6338: 6337: 6328:. Archived from 6322: 6316: 6315: 6313: 6312: 6297: 6291: 6290: 6262: 6256: 6255: 6229: 6223: 6222: 6202: 6196: 6195: 6171: 6159: 6153: 6152: 6132: 6123: 6122: 6120: 6105:Applied Sciences 6096: 6087: 6086: 6084: 6083: 6074:. Archived from 6064: 6058: 6057: 6055: 6054: 6031: 6022: 6016: 6015: 6013: 6001: 5995: 5994: 5992: 5980: 5974: 5973: 5971: 5959: 5953: 5952: 5950: 5949: 5940:. Archived from 5913: 5907: 5906: 5891: 5885: 5884: 5882: 5881: 5871:Bulletin of the 5862: 5856: 5855: 5833: 5827: 5826: 5796: 5790: 5789: 5767: 5761: 5760: 5734: 5728: 5727: 5725: 5691: 5651:Machine learning 5614:Process modeling 5546:, and locality. 5540:compositionality 5422: 5420: 5419: 5414: 5406: 5405: 5400: 5397: 5396: 5384: 5383: 5378: 5375: 5374: 5356: 5355: 5350: 5347: 5346: 5334: 5333: 5328: 5325: 5324: 5297: 5296: 5291: 5288: 5287: 5275: 5274: 5269: 5266: 5265: 5244: 5243: 5231: 5230: 5190: 5188: 5187: 5182: 5162: 5161: 5149: 5148: 5143: 5125: 5120: 5119: 5110: 5070: 5068: 5067: 5062: 5054: 5046: 5045: 5040: 5037: 5029: 5024: 5023: 5014: 4988:mathematically, 4969: 4967: 4966: 4961: 4953: 4945: 4944: 4939: 4936: 4928: 4923: 4922: 4913: 4808:timed Petri nets 4719:Nets within Nets 4635: 4633: 4632: 4627: 4625: 4596:is a Petri net, 4595: 4593: 4592: 4587: 4582: 4581: 4544: 4542: 4541: 4536: 4531: 4530: 4391: 4389: 4388: 4383: 4381: 4380: 4361: 4359: 4358: 4353: 4350: 4349: 4317: 4315: 4314: 4309: 4307: 4306: 4290: 4288: 4287: 4282: 4280: 4279: 4254: 4252: 4251: 4246: 4241: 4240: 4212: 4210: 4209: 4204: 4202: 4201: 4181: 4179: 4178: 4173: 4171: 4170: 4152: 4149:occurs at least 4148: 4146: 4145: 4140: 4138: 4137: 4121: 4117: 4115: 4114: 4109: 4107: 4106: 4089: 4087: 4086: 4081: 4076: 4075: 4047: 4043: 4039: 4037: 4036: 4031: 4029: 4028: 4011: 4009: 4008: 4003: 3998: 3997: 3965: 3963: 3962: 3957: 3955: 3954: 3937: 3935: 3934: 3929: 3924: 3923: 3888: 3886: 3885: 3880: 3878: 3877: 3857: 3855: 3854: 3849: 3847: 3846: 3830: 3828: 3827: 3822: 3817: 3816: 3791: 3789: 3788: 3783: 3781: 3780: 3768: 3767: 3747: 3745: 3744: 3739: 3737: 3736: 3720: 3718: 3717: 3712: 3710: 3709: 3694: 3692: 3691: 3686: 3665: 3663: 3662: 3657: 3655: 3654: 3592: 3590: 3589: 3584: 3516:Petri categories 3500: 3497: 3479: 3472: 3464: 3462: 3461: 3456: 3454: 3453: 3417: 3416: 3399: 3397: 3396: 3391: 3389: 3388: 3262: 3261: 3250: 3246: 3245: 3128: 3127: 3116: 3112: 3111: 2994: 2993: 2975: 2967: 2965: 2964: 2959: 2939: 2938: 2926: 2925: 2908: 2907: 2904: 2900: 2896: 2893: 2886: 2842: 2838: 2836: 2835: 2830: 2809: 2802: 2800: 2799: 2794: 2792: 2791: 2779: 2778: 2763: 2762: 2740: 2738: 2737: 2732: 2691: 2690: 2659: 2657: 2656: 2651: 2649: 2648: 2631: 2629: 2628: 2623: 2585: 2584: 2553: 2551: 2550: 2545: 2543: 2542: 2520: 2518: 2517: 2512: 2510: 2502: 2490: 2488: 2487: 2482: 2480: 2472: 2449: 2447: 2446: 2441: 2439: 2431: 2415: 2413: 2412: 2407: 2402: 2401: 2344:expressive power 2341: 2339: 2338: 2333: 2265: 2263: 2262: 2257: 2236: 2234: 2233: 2228: 2226: 2225: 2216: 2214: 2213: 2212: 2191: 2190: 2166: 2165: 2156: 2154: 2153: 2152: 2131: 2130: 2114: 2112: 2111: 2106: 2101: 2100: 2088: 2087: 2072: 2071: 2063: 2053: 2051: 2050: 2045: 2043: 2042: 2026: 2011: 2009: 2008: 2003: 1982: 1980: 1979: 1974: 1972: 1955: 1945: 1943: 1942: 1937: 1935: 1931: 1930: 1922: 1914: 1885: 1884: 1875: 1874: 1868: 1853: 1852: 1851: 1849: 1844: 1839: 1834: 1810: 1808: 1807: 1802: 1800: 1799: 1783: 1781: 1780: 1775: 1770: 1769: 1723: 1721: 1720: 1715: 1713: 1692: 1690: 1689: 1684: 1682: 1669: 1660: 1658: 1657: 1652: 1650: 1642: 1629: 1616: 1610: 1608: 1607: 1602: 1600: 1592: 1569: 1562: 1551: 1549: 1548: 1543: 1492: 1481:a transition is 1478: 1474: 1472: 1471: 1466: 1439: 1435: 1433: 1432: 1427: 1400: 1396: 1375: 1373: 1372: 1367: 1365: 1364: 1346: 1344: 1343: 1338: 1305: 1303: 1302: 1297: 1292: 1291: 1253:marked Petri net 1239: 1237: 1236: 1231: 1229: 1198: 1196: 1195: 1190: 1143: 1142: 1118: 1116: 1115: 1110: 1060: 1059: 1054: 1037:of a transition 1021: 1019: 1018: 1013: 969: 967: 966: 961: 889:arc multiplicity 878: 876: 875: 870: 868: 775: 773: 772: 767: 688:in the image of 558: 518: 452: 379: 334: 332: 331: 326: 268: 264: 257: 255: 254: 249: 180:nondeterministic 176:execution policy 126:Petri net basics 121: 21: 8008: 8007: 8003: 8002: 8001: 7999: 7998: 7997: 7953: 7952: 7925: 7919: 7903: 7897: 7881: 7875: 7860: 7854: 7841: 7835: 7822: 7816: 7803: 7768: 7759: 7753: 7740: 7699: 7693: 7680: 7674: 7659: 7646: 7619: 7613: 7600: 7597: 7595:Further reading 7592: 7585: 7562: 7561: 7557: 7527: 7526: 7522: 7507: 7494: 7493: 7489: 7482: 7457: 7456: 7449: 7442: 7417: 7416: 7409: 7396: 7386: 7382: 7357: 7356: 7352: 7316: 7315: 7311: 7288: 7262: 7261: 7257: 7242: 7216: 7215: 7211: 7204: 7181: 7180: 7176: 7150: 7149: 7145: 7138: 7115: 7114: 7110: 7076: 7075: 7071: 7064: 7043: 7042: 7038: 7023: 7000: 6999: 6995: 6980: 6957: 6956: 6952: 6924: 6923: 6919: 6888:: 18420–18425. 6875: 6874: 6870: 6863: 6838: 6837: 6833: 6826: 6800: 6799: 6795: 6741: 6740: 6736: 6729: 6704: 6703: 6699: 6692: 6677: 6676: 6669: 6659: 6657: 6621: 6604: 6603: 6599: 6591: 6580: 6575: 6574: 6570: 6560: 6559: 6555: 6529: 6528: 6521: 6514: 6491: 6486: 6485: 6481: 6472: 6470: 6466: 6425: 6420: 6419: 6410: 6403: 6382: 6381: 6377: 6370: 6349: 6348: 6344: 6335: 6333: 6324: 6323: 6319: 6310: 6308: 6299: 6298: 6294: 6264: 6263: 6259: 6252: 6231: 6230: 6226: 6204: 6203: 6199: 6192: 6169: 6161: 6160: 6156: 6149: 6134: 6133: 6126: 6098: 6097: 6090: 6081: 6079: 6066: 6065: 6061: 6052: 6050: 6048:10.1109/5.24143 6029: 6024: 6023: 6019: 6003: 6002: 5998: 5982: 5981: 5977: 5961: 5960: 5956: 5947: 5945: 5938: 5915: 5914: 5910: 5893: 5892: 5888: 5879: 5877: 5864: 5863: 5859: 5835: 5834: 5830: 5823: 5798: 5797: 5793: 5769: 5768: 5764: 5757: 5736: 5735: 5731: 5693: 5692: 5688: 5684: 5642: 5629:Software design 5604:Hardware design 5555: 5528:process algebra 5512: 5477:elementary path 5429: 5398: 5388: 5376: 5366: 5348: 5338: 5326: 5316: 5289: 5279: 5267: 5257: 5235: 5222: 5214: 5213: 5212:athematically, 5153: 5141: 5111: 5083: 5082: 5038: 5015: 4990: 4989: 4937: 4914: 4889: 4888: 4852: 4748:Turing complete 4704: 4688: 4598: 4597: 4573: 4547: 4546: 4522: 4490: 4489: 4405: 4372: 4367: 4366: 4320: 4319: 4318:-liveness, for 4298: 4293: 4292: 4265: 4260: 4259: 4232: 4215: 4214: 4193: 4188: 4187: 4162: 4157: 4156: 4150: 4129: 4124: 4123: 4119: 4098: 4093: 4092: 4067: 4050: 4049: 4045: 4041: 4020: 4015: 4014: 3989: 3972: 3971: 3946: 3941: 3940: 3915: 3898: 3897: 3869: 3864: 3863: 3838: 3833: 3832: 3808: 3794: 3793: 3772: 3759: 3754: 3753: 3728: 3723: 3722: 3701: 3696: 3695: 3668: 3667: 3646: 3641: 3640: 3633: 3560: 3559: 3544: 3524: 3501: 3495: 3492: 3485:needs expansion 3470: 3448: 3447: 3442: 3437: 3432: 3422: 3408: 3403: 3402: 3383: 3382: 3377: 3372: 3363: 3362: 3354: 3349: 3340: 3339: 3331: 3326: 3317: 3316: 3311: 3303: 3294: 3293: 3285: 3277: 3267: 3253: 3240: 3239: 3234: 3229: 3220: 3219: 3214: 3209: 3200: 3199: 3194: 3189: 3180: 3179: 3174: 3169: 3160: 3159: 3151: 3143: 3133: 3119: 3106: 3105: 3100: 3095: 3086: 3085: 3080: 3075: 3066: 3065: 3060: 3055: 3046: 3045: 3040: 3035: 3026: 3025: 3017: 3009: 2999: 2985: 2980: 2979: 2973: 2930: 2917: 2905: and 2848: 2847: 2840: 2812: 2811: 2807: 2783: 2770: 2754: 2749: 2748: 2682: 2662: 2661: 2640: 2635: 2634: 2576: 2556: 2555: 2534: 2529: 2528: 2493: 2492: 2463: 2462: 2422: 2421: 2393: 2367: 2366: 2363: 2288: 2287: 2272: 2239: 2238: 2217: 2204: 2197: 2176: 2157: 2144: 2137: 2122: 2117: 2116: 2092: 2079: 2056: 2055: 2034: 2029: 2028: 2024: 2021:firing sequence 1985: 1984: 1958: 1957: 1953: 1923: 1876: 1861: 1860: 1856: 1817: 1816: 1791: 1786: 1785: 1761: 1729: 1728: 1699: 1698: 1663: 1662: 1643: 1620: 1619: 1614: 1593: 1575: 1574: 1567: 1560: 1495: 1494: 1490: 1476: 1442: 1441: 1440:, and produces 1437: 1403: 1402: 1398: 1394: 1387: 1378:initial marking 1356: 1351: 1350: 1311: 1310: 1283: 1257: 1256: 1208: 1207: 1134: 1129: 1128: 1052: 1047: 1046: 986: 985: 901: 900: 817: 816: 740: 739: 729:Petri net graph 725: 701: 694: 687: 661: 654: 647: 640: 633: 626: 615: 546: 506: 444: 371: 281: 280: 277:, respectively. 266: 262: 216: 215: 197:Petri nets are 195: 128: 114:Carl Adam Petri 110: 64:Carl Adam Petri 59:bipartite graph 28: 23: 22: 15: 12: 11: 5: 8006: 8004: 7996: 7995: 7990: 7985: 7980: 7975: 7970: 7965: 7955: 7954: 7951: 7950: 7923: 7917: 7901: 7895: 7879: 7873: 7858: 7852: 7839: 7833: 7820: 7814: 7801: 7766: 7757: 7751: 7738: 7710:(3): 223–252. 7697: 7691: 7678: 7672: 7657: 7644: 7617: 7611: 7596: 7593: 7591: 7590: 7583: 7555: 7520: 7505: 7487: 7480: 7447: 7440: 7407: 7398:|journal= 7380: 7350: 7331:(4): 347–374. 7309: 7286: 7255: 7240: 7218:Cortadella, J. 7209: 7202: 7174: 7163:(3): 387–397. 7143: 7136: 7108: 7069: 7062: 7036: 7021: 6993: 6978: 6950: 6917: 6868: 6861: 6831: 6824: 6793: 6754:(9): e106035. 6734: 6727: 6697: 6690: 6667: 6597: 6594:on 2020-05-04. 6568: 6553: 6519: 6512: 6479: 6443:10.1.1.30.3125 6408: 6401: 6375: 6368: 6342: 6317: 6292: 6257: 6250: 6224: 6197: 6190: 6154: 6147: 6124: 6088: 6059: 6042:(4): 541–558. 6017: 5996: 5975: 5954: 5936: 5908: 5886: 5857: 5846:(2): 105–155. 5828: 5821: 5801:Ehrig, Hartmut 5791: 5762: 5755: 5729: 5685: 5683: 5680: 5679: 5678: 5673: 5668: 5663: 5658: 5653: 5648: 5641: 5638: 5637: 5636: 5631: 5626: 5621: 5616: 5611: 5606: 5601: 5596: 5591: 5586: 5581: 5576: 5571: 5566: 5561: 5554: 5551: 5511: 5508: 5428: 5425: 5424: 5423: 5412: 5409: 5404: 5395: 5391: 5387: 5382: 5373: 5369: 5365: 5362: 5359: 5354: 5345: 5341: 5337: 5332: 5323: 5319: 5315: 5312: 5309: 5306: 5303: 5300: 5295: 5286: 5282: 5278: 5273: 5264: 5260: 5256: 5253: 5250: 5247: 5242: 5238: 5234: 5229: 5225: 5221: 5198: 5191: 5180: 5177: 5174: 5171: 5168: 5165: 5160: 5156: 5152: 5147: 5140: 5137: 5134: 5131: 5128: 5124: 5118: 5114: 5109: 5105: 5102: 5099: 5096: 5093: 5090: 5071: 5060: 5057: 5053: 5049: 5044: 5036: 5032: 5028: 5022: 5018: 5013: 5009: 5006: 5003: 5000: 4997: 4970: 4959: 4956: 4952: 4948: 4943: 4935: 4931: 4927: 4921: 4917: 4912: 4908: 4905: 4902: 4899: 4896: 4885:nondeterminism 4851: 4848: 4843: 4842: 4828: 4803: 4792: 4780: 4776: 4753: 4752: 4751: 4740: 4703: 4700: 4692:control theory 4687: 4684: 4624: 4620: 4614: 4611: 4608: 4605: 4585: 4580: 4576: 4572: 4569: 4566: 4563: 4560: 4557: 4554: 4534: 4529: 4525: 4521: 4518: 4515: 4512: 4509: 4506: 4503: 4500: 4497: 4478:–Miller Tree. 4404: 4401: 4395:as a term for 4379: 4375: 4348: 4345: 4342: 4339: 4336: 4332: 4329: 4305: 4301: 4278: 4275: 4272: 4268: 4256: 4255: 4244: 4239: 4235: 4231: 4228: 4225: 4222: 4200: 4196: 4169: 4165: 4154: 4136: 4132: 4105: 4101: 4090: 4079: 4074: 4070: 4066: 4063: 4060: 4057: 4027: 4023: 4012: 4001: 3996: 3992: 3988: 3985: 3982: 3979: 3953: 3949: 3938: 3927: 3922: 3918: 3914: 3911: 3908: 3905: 3876: 3872: 3860:if and only if 3845: 3841: 3820: 3815: 3811: 3807: 3804: 3801: 3792:. A Petri net 3779: 3775: 3771: 3766: 3762: 3735: 3731: 3708: 3704: 3684: 3681: 3678: 3675: 3653: 3649: 3632: 3629: 3621:tableau method 3582: 3579: 3576: 3573: 3570: 3567: 3554:and a marking 3543: 3540: 3523: 3520: 3503: 3502: 3496:September 2022 3482: 3480: 3469: 3466: 3452: 3446: 3443: 3441: 3438: 3436: 3433: 3431: 3428: 3427: 3425: 3420: 3415: 3411: 3387: 3381: 3378: 3376: 3373: 3371: 3368: 3365: 3364: 3361: 3358: 3355: 3353: 3350: 3348: 3345: 3342: 3341: 3338: 3335: 3332: 3330: 3327: 3325: 3322: 3319: 3318: 3315: 3312: 3310: 3307: 3304: 3302: 3299: 3296: 3295: 3292: 3289: 3286: 3284: 3281: 3278: 3276: 3273: 3272: 3270: 3265: 3260: 3256: 3249: 3244: 3238: 3235: 3233: 3230: 3228: 3225: 3222: 3221: 3218: 3215: 3213: 3210: 3208: 3205: 3202: 3201: 3198: 3195: 3193: 3190: 3188: 3185: 3182: 3181: 3178: 3175: 3173: 3170: 3168: 3165: 3162: 3161: 3158: 3155: 3152: 3150: 3147: 3144: 3142: 3139: 3138: 3136: 3131: 3126: 3122: 3115: 3110: 3104: 3101: 3099: 3096: 3094: 3091: 3088: 3087: 3084: 3081: 3079: 3076: 3074: 3071: 3068: 3067: 3064: 3061: 3059: 3056: 3054: 3051: 3048: 3047: 3044: 3041: 3039: 3036: 3034: 3031: 3028: 3027: 3024: 3021: 3018: 3016: 3013: 3010: 3008: 3005: 3004: 3002: 2997: 2992: 2988: 2970: 2969: 2957: 2954: 2951: 2948: 2945: 2942: 2937: 2933: 2929: 2924: 2920: 2916: 2913: 2899: 2891: 2885: 2882: 2879: 2876: 2873: 2870: 2867: 2864: 2861: 2858: 2855: 2828: 2825: 2822: 2819: 2804: 2803: 2790: 2786: 2782: 2777: 2773: 2769: 2766: 2761: 2757: 2742: 2741: 2730: 2727: 2724: 2721: 2718: 2715: 2712: 2709: 2706: 2703: 2700: 2697: 2694: 2689: 2685: 2681: 2678: 2675: 2672: 2669: 2647: 2643: 2632: 2621: 2618: 2615: 2612: 2609: 2606: 2603: 2600: 2597: 2594: 2591: 2588: 2583: 2579: 2575: 2572: 2569: 2566: 2563: 2541: 2537: 2509: 2505: 2501: 2479: 2475: 2471: 2438: 2434: 2430: 2405: 2400: 2396: 2392: 2389: 2386: 2383: 2380: 2377: 2374: 2362: 2359: 2331: 2328: 2325: 2322: 2319: 2316: 2313: 2310: 2307: 2304: 2301: 2298: 2295: 2271: 2268: 2255: 2252: 2249: 2246: 2224: 2220: 2211: 2207: 2203: 2200: 2196: 2189: 2186: 2183: 2179: 2175: 2172: 2169: 2164: 2160: 2151: 2147: 2143: 2140: 2136: 2129: 2125: 2104: 2099: 2095: 2091: 2086: 2082: 2078: 2075: 2069: 2066: 2041: 2037: 2001: 1998: 1995: 1992: 1970: 1967: 1934: 1929: 1926: 1919: 1913: 1910: 1907: 1904: 1901: 1898: 1895: 1890: 1883: 1879: 1873: 1867: 1864: 1859: 1848: 1843: 1833: 1830: 1827: 1824: 1811:. Its set of 1798: 1794: 1773: 1768: 1764: 1760: 1757: 1754: 1751: 1748: 1745: 1742: 1739: 1736: 1711: 1708: 1680: 1676: 1673: 1649: 1646: 1640: 1636: 1633: 1627: 1599: 1596: 1590: 1587: 1582: 1554: 1553: 1541: 1538: 1535: 1532: 1529: 1526: 1523: 1520: 1517: 1514: 1511: 1508: 1505: 1502: 1479: 1464: 1461: 1458: 1455: 1452: 1449: 1425: 1422: 1419: 1416: 1413: 1410: 1386: 1383: 1382: 1381: 1363: 1359: 1348: 1336: 1333: 1330: 1327: 1324: 1321: 1318: 1295: 1290: 1286: 1282: 1279: 1276: 1273: 1270: 1267: 1264: 1228: 1224: 1221: 1218: 1215: 1188: 1185: 1182: 1179: 1176: 1173: 1170: 1167: 1164: 1161: 1158: 1155: 1152: 1149: 1146: 1141: 1137: 1108: 1105: 1102: 1099: 1096: 1093: 1090: 1087: 1084: 1081: 1078: 1075: 1072: 1069: 1066: 1063: 1058: 1011: 1008: 1005: 1002: 999: 996: 993: 983:directed graph 959: 956: 953: 950: 947: 944: 941: 938: 935: 932: 929: 926: 923: 920: 917: 914: 911: 908: 893: 892: 867: 863: 860: 857: 854: 851: 848: 845: 842: 839: 836: 833: 830: 827: 824: 814: 800: 791: 765: 762: 759: 756: 753: 750: 747: 724: 721: 699: 692: 685: 677:of transition 659: 652: 645: 638: 631: 624: 613: 569: 568: 544: 504: 459: 458: 438: 404:elementary net 341: 340: 324: 321: 318: 315: 312: 309: 306: 303: 300: 297: 294: 291: 288: 278: 247: 244: 241: 238: 235: 232: 229: 226: 223: 194: 191: 127: 124: 109: 106: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8005: 7994: 7991: 7989: 7986: 7984: 7981: 7979: 7976: 7974: 7971: 7969: 7966: 7964: 7961: 7960: 7958: 7946: 7941: 7937: 7933: 7929: 7924: 7920: 7914: 7910: 7906: 7905:Zhou, Mengchu 7902: 7898: 7892: 7888: 7884: 7883:Zhou, Mengchu 7880: 7876: 7870: 7866: 7865: 7859: 7855: 7849: 7845: 7840: 7836: 7830: 7826: 7821: 7817: 7811: 7807: 7802: 7797: 7792: 7788: 7784: 7780: 7776: 7772: 7767: 7763: 7758: 7754: 7748: 7744: 7739: 7735: 7731: 7726: 7721: 7717: 7713: 7709: 7705: 7704: 7698: 7694: 7688: 7684: 7679: 7675: 7669: 7665: 7664: 7658: 7655:(12a): 47–50. 7654: 7650: 7645: 7640: 7635: 7631: 7627: 7623: 7618: 7614: 7608: 7604: 7599: 7598: 7594: 7586: 7580: 7576: 7572: 7568: 7567: 7559: 7556: 7551: 7547: 7543: 7539: 7535: 7531: 7524: 7521: 7516: 7512: 7508: 7502: 7498: 7491: 7488: 7483: 7477: 7473: 7469: 7465: 7461: 7454: 7452: 7448: 7443: 7437: 7433: 7429: 7425: 7421: 7414: 7412: 7408: 7403: 7391: 7383: 7377: 7373: 7369: 7365: 7361: 7354: 7351: 7346: 7342: 7338: 7334: 7330: 7326: 7325: 7320: 7313: 7310: 7305: 7301: 7297: 7293: 7289: 7283: 7279: 7275: 7271: 7270: 7265: 7259: 7256: 7251: 7247: 7243: 7237: 7233: 7229: 7225: 7224: 7219: 7213: 7210: 7205: 7199: 7195: 7191: 7187: 7186: 7178: 7175: 7170: 7166: 7162: 7158: 7154: 7147: 7144: 7139: 7133: 7128: 7123: 7119: 7112: 7109: 7104: 7100: 7096: 7092: 7088: 7084: 7080: 7073: 7070: 7065: 7059: 7055: 7051: 7047: 7040: 7037: 7032: 7028: 7024: 7018: 7014: 7010: 7006: 7005: 6997: 6994: 6989: 6985: 6981: 6975: 6971: 6967: 6963: 6962: 6954: 6951: 6945: 6940: 6936: 6932: 6928: 6921: 6918: 6913: 6909: 6904: 6899: 6895: 6891: 6887: 6883: 6879: 6872: 6869: 6864: 6858: 6854: 6850: 6846: 6842: 6835: 6832: 6827: 6821: 6817: 6813: 6809: 6808: 6803: 6797: 6794: 6789: 6785: 6780: 6775: 6770: 6765: 6761: 6757: 6753: 6749: 6745: 6738: 6735: 6730: 6724: 6720: 6716: 6712: 6708: 6701: 6698: 6693: 6691:9780262015387 6687: 6683: 6682: 6674: 6672: 6668: 6655: 6651: 6647: 6643: 6639: 6635: 6631: 6625: 6620: 6616: 6613:(in German). 6612: 6608: 6601: 6598: 6590: 6586: 6579: 6572: 6569: 6564: 6557: 6554: 6549: 6545: 6541: 6537: 6533: 6526: 6524: 6520: 6515: 6513:3-540-44919-1 6509: 6505: 6501: 6497: 6490: 6483: 6480: 6469:on 2016-11-19 6465: 6461: 6457: 6453: 6449: 6444: 6439: 6435: 6431: 6424: 6417: 6415: 6413: 6409: 6404: 6402:0-7803-7080-5 6398: 6394: 6390: 6386: 6379: 6376: 6371: 6369:0-7803-7087-2 6365: 6361: 6357: 6353: 6346: 6343: 6332:on 2005-11-03 6331: 6327: 6321: 6318: 6307:on 2010-10-28 6306: 6302: 6296: 6293: 6288: 6284: 6280: 6276: 6272: 6268: 6261: 6258: 6253: 6251:3-540-68681-9 6247: 6243: 6239: 6235: 6228: 6225: 6220: 6216: 6213:(1): 85–104. 6212: 6208: 6201: 6198: 6193: 6187: 6183: 6179: 6175: 6168: 6164: 6158: 6155: 6150: 6144: 6140: 6139: 6131: 6129: 6125: 6119: 6114: 6110: 6106: 6102: 6095: 6093: 6089: 6078:on 2011-09-27 6077: 6073: 6069: 6063: 6060: 6049: 6045: 6041: 6037: 6036: 6028: 6021: 6018: 6012: 6007: 6000: 5997: 5991: 5986: 5979: 5976: 5970: 5965: 5958: 5955: 5944:on 2012-02-09 5943: 5939: 5937:3-540-31882-8 5933: 5929: 5925: 5921: 5920: 5912: 5909: 5904: 5900: 5896: 5890: 5887: 5875: 5874: 5868: 5861: 5858: 5853: 5849: 5845: 5841: 5840: 5832: 5829: 5824: 5822:9783540430674 5818: 5814: 5810: 5806: 5802: 5795: 5792: 5787: 5783: 5779: 5775: 5774: 5766: 5763: 5758: 5756:3-540-65306-6 5752: 5748: 5744: 5740: 5733: 5730: 5724: 5719: 5715: 5711: 5707: 5703: 5702: 5697: 5690: 5687: 5681: 5677: 5674: 5672: 5669: 5667: 5664: 5662: 5659: 5657: 5654: 5652: 5649: 5647: 5644: 5643: 5639: 5635: 5632: 5630: 5627: 5625: 5622: 5620: 5617: 5615: 5612: 5610: 5607: 5605: 5602: 5600: 5597: 5595: 5592: 5590: 5587: 5585: 5584:Data analysis 5582: 5580: 5577: 5575: 5572: 5570: 5567: 5565: 5562: 5560: 5557: 5556: 5552: 5550: 5547: 5545: 5541: 5537: 5533: 5529: 5525: 5521: 5517: 5509: 5507: 5505: 5501: 5496: 5492: 5488: 5485: 5480: 5478: 5474: 5473: 5467: 5465: 5461: 5457: 5453: 5449: 5445: 5440: 5437: 5433: 5432:Workflow nets 5427:Workflow nets 5426: 5402: 5393: 5389: 5385: 5380: 5371: 5367: 5360: 5352: 5343: 5339: 5335: 5330: 5321: 5317: 5298: 5293: 5284: 5280: 5276: 5271: 5262: 5258: 5251: 5248: 5245: 5240: 5236: 5232: 5227: 5223: 5211: 5207: 5203: 5199: 5196: 5192: 5172: 5166: 5158: 5154: 5145: 5135: 5129: 5126: 5116: 5112: 5100: 5097: 5094: 5091: 5080: 5076: 5072: 5058: 5055: 5047: 5042: 5030: 5020: 5016: 5007: 5004: 5001: 4998: 4987: 4983: 4979: 4975: 4971: 4957: 4954: 4946: 4941: 4929: 4919: 4915: 4906: 4903: 4900: 4897: 4886: 4882: 4878: 4874: 4870: 4869:state machine 4866: 4865: 4864: 4856: 4849: 4847: 4840: 4836: 4832: 4829: 4826: 4822: 4818: 4814: 4809: 4804: 4801: 4796: 4793: 4789: 4785: 4781: 4777: 4774: 4770: 4766: 4762: 4758: 4754: 4749: 4745: 4744:inhibitor arc 4741: 4738: 4734: 4733: 4731: 4730: 4729: 4726: 4724: 4720: 4716: 4711: 4709: 4701: 4699: 4697: 4693: 4685: 4683: 4681: 4677: 4669: 4664: 4660: 4658: 4654: 4646: 4641: 4637: 4609: 4606: 4603: 4578: 4574: 4570: 4567: 4564: 4561: 4558: 4555: 4527: 4523: 4519: 4516: 4513: 4510: 4507: 4504: 4501: 4498: 4487: 4482: 4479: 4477: 4473: 4468: 4465: 4463: 4459: 4455: 4451: 4446: 4444: 4440: 4436: 4435: 4430: 4426: 4422: 4414: 4409: 4402: 4400: 4398: 4394: 4377: 4373: 4363: 4346: 4343: 4340: 4337: 4334: 4330: 4327: 4303: 4299: 4276: 4273: 4270: 4266: 4237: 4233: 4229: 4226: 4220: 4198: 4194: 4185: 4167: 4163: 4155: 4134: 4130: 4103: 4099: 4091: 4072: 4068: 4064: 4061: 4055: 4025: 4021: 4013: 3994: 3990: 3986: 3983: 3977: 3969: 3951: 3947: 3939: 3920: 3916: 3912: 3909: 3903: 3895: 3892: 3891: 3890: 3874: 3870: 3861: 3843: 3839: 3813: 3809: 3805: 3802: 3777: 3773: 3769: 3764: 3760: 3733: 3729: 3706: 3702: 3682: 3679: 3676: 3673: 3651: 3647: 3637: 3630: 3628: 3626: 3622: 3618: 3613: 3611: 3607: 3603: 3598: 3594: 3577: 3571: 3568: 3565: 3557: 3553: 3549: 3541: 3539: 3537: 3533: 3528: 3521: 3519: 3517: 3513: 3509: 3499: 3490: 3486: 3483:This section 3481: 3478: 3474: 3473: 3467: 3465: 3450: 3444: 3439: 3434: 3429: 3423: 3418: 3413: 3409: 3400: 3385: 3379: 3374: 3369: 3366: 3359: 3356: 3351: 3346: 3343: 3336: 3333: 3328: 3323: 3320: 3313: 3308: 3305: 3300: 3297: 3290: 3287: 3282: 3279: 3274: 3268: 3263: 3258: 3254: 3247: 3242: 3236: 3231: 3226: 3223: 3216: 3211: 3206: 3203: 3196: 3191: 3186: 3183: 3176: 3171: 3166: 3163: 3156: 3153: 3148: 3145: 3140: 3134: 3129: 3124: 3120: 3113: 3108: 3102: 3097: 3092: 3089: 3082: 3077: 3072: 3069: 3062: 3057: 3052: 3049: 3042: 3037: 3032: 3029: 3022: 3019: 3014: 3011: 3006: 3000: 2995: 2990: 2986: 2977: 2949: 2943: 2940: 2935: 2931: 2927: 2922: 2918: 2914: 2911: 2897: 2889: 2883: 2880: 2874: 2871: 2865: 2859: 2853: 2846: 2845: 2844: 2823: 2817: 2788: 2784: 2780: 2775: 2771: 2767: 2764: 2759: 2755: 2747: 2746: 2745: 2728: 2722: 2719: 2716: 2710: 2707: 2701: 2698: 2695: 2687: 2683: 2679: 2676: 2673: 2670: 2660:, defined by 2645: 2641: 2633: 2616: 2613: 2610: 2604: 2601: 2595: 2592: 2589: 2581: 2577: 2573: 2570: 2567: 2564: 2554:, defined by 2539: 2535: 2527: 2526: 2525: 2523: 2503: 2473: 2455: 2451: 2432: 2419: 2398: 2394: 2390: 2387: 2384: 2381: 2378: 2375: 2360: 2358: 2356: 2352: 2347: 2345: 2326: 2323: 2320: 2314: 2308: 2305: 2302: 2296: 2293: 2285: 2284:flow relation 2281: 2277: 2269: 2267: 2250: 2244: 2222: 2218: 2209: 2205: 2201: 2198: 2187: 2184: 2181: 2177: 2173: 2170: 2167: 2162: 2158: 2149: 2145: 2141: 2138: 2127: 2123: 2097: 2093: 2089: 2084: 2080: 2073: 2064: 2039: 2035: 2022: 2017: 2015: 2012:. It is the 1996: 1990: 1968: 1951: 1946: 1932: 1927: 1924: 1917: 1908: 1905: 1902: 1899: 1896: 1888: 1881: 1877: 1865: 1862: 1857: 1846: 1841: 1828: 1822: 1814: 1796: 1792: 1766: 1762: 1758: 1755: 1752: 1749: 1746: 1743: 1737: 1734: 1725: 1709: 1696: 1678: 1674: 1647: 1644: 1638: 1634: 1625: 1617: 1597: 1594: 1588: 1580: 1572: 1565: 1557: 1536: 1533: 1530: 1524: 1521: 1515: 1509: 1506: 1503: 1488: 1484: 1480: 1459: 1456: 1453: 1447: 1420: 1417: 1414: 1408: 1397:in a marking 1392: 1391: 1390: 1384: 1379: 1361: 1357: 1349: 1331: 1328: 1325: 1322: 1319: 1309: 1308: 1307: 1288: 1284: 1280: 1277: 1274: 1271: 1268: 1265: 1254: 1250: 1245: 1243: 1219: 1216: 1213: 1205: 1200: 1183: 1180: 1174: 1171: 1168: 1162: 1159: 1156: 1153: 1150: 1144: 1139: 1135: 1126: 1125:output places 1122: 1103: 1100: 1094: 1091: 1088: 1082: 1079: 1076: 1073: 1070: 1064: 1061: 1056: 1044: 1040: 1036: 1031: 1029: 1025: 1006: 1003: 1000: 997: 994: 984: 981: 977: 973: 954: 951: 945: 942: 939: 933: 930: 924: 921: 918: 909: 906: 898: 897:flow relation 890: 886: 882: 855: 852: 849: 843: 837: 834: 831: 825: 822: 815: 812: 808: 804: 801: 799: 795: 792: 790: 786: 782: 779: 778: 777: 760: 757: 754: 751: 748: 738: 734: 730: 722: 720: 718: 717:Peterson 1981 713: 711: 707: 703: 698: 691: 684: 680: 676: 672: 668: 664: 658: 651: 644: 637: 630: 623: 619: 612: 607: 605: 601: 595: 593: 589: 586: 582: 578: 574: 566: 562: 557: 553: 549: 545: 542: 538: 537:configuration 534: 530: 529:countable set 526: 522: 517: 513: 509: 505: 502: 498: 494: 490: 487: 486: 485: 483: 479: 475: 471: 467: 463: 462:Definition 4. 456: 455:configuration 451: 447: 443:is such that 442: 439: 436: 432: 428: 424: 421: 420: 419: 417: 413: 409: 405: 401: 400:Definition 3. 393: 385: 381: 378: 374: 369: 365: 364:configuration 361: 357: 353: 349: 345: 344:Definition 2. 338: 319: 316: 313: 307: 301: 298: 295: 289: 286: 279: 276: 272: 261: 260: 259: 242: 239: 236: 233: 230: 224: 221: 214: 210: 206: 205:Definition 1. 202: 200: 192: 190: 188: 183: 181: 177: 172: 170: 166: 162: 158: 153: 151: 150:output places 147: 143: 142: 137: 133: 125: 123: 120: 115: 107: 101: 97: 95: 91: 87: 83: 79: 75: 72: 67: 65: 60: 56: 52: 48: 45: 41: 37: 33: 19: 7935: 7931: 7908: 7886: 7863: 7843: 7824: 7805: 7778: 7775:Scholarpedia 7774: 7761: 7742: 7707: 7701: 7682: 7662: 7652: 7648: 7629: 7625: 7602: 7565: 7558: 7533: 7523: 7496: 7490: 7463: 7423: 7363: 7353: 7328: 7322: 7312: 7268: 7258: 7222: 7212: 7184: 7177: 7160: 7156: 7146: 7117: 7111: 7086: 7082: 7072: 7045: 7039: 7007:. Springer. 7003: 6996: 6964:. Springer. 6960: 6953: 6937:(21): 7662. 6934: 6930: 6920: 6885: 6881: 6871: 6844: 6834: 6806: 6796: 6751: 6747: 6737: 6710: 6700: 6680: 6658:. Retrieved 6614: 6610: 6600: 6589:the original 6584: 6571: 6562: 6556: 6531: 6495: 6482: 6471:. Retrieved 6464:the original 6436:(1): 21–66. 6433: 6429: 6384: 6378: 6351: 6345: 6334:. Retrieved 6330:the original 6320: 6309:. Retrieved 6305:the original 6295: 6270: 6266: 6260: 6233: 6227: 6210: 6206: 6200: 6173: 6163:Jensen, Kurt 6157: 6141:. Springer. 6137: 6111:(15): 5027. 6108: 6104: 6080:. Retrieved 6076:the original 6071: 6068:"Petri Nets" 6062: 6051:. Retrieved 6039: 6033: 6020: 5999: 5978: 5957: 5946:. Retrieved 5942:the original 5918: 5911: 5902: 5889: 5878:. Retrieved 5870: 5860: 5843: 5837: 5831: 5804: 5794: 5777: 5771: 5765: 5738: 5732: 5705: 5701:Scholarpedia 5699: 5689: 5548: 5536:trace theory 5513: 5503: 5497: 5493: 5489: 5484:well-handled 5483: 5481: 5476: 5470: 5468: 5463: 5459: 5455: 5451: 5447: 5443: 5441: 5430: 5209: 5205: 5201: 5194: 5078: 5074: 4986:concurrency: 4985: 4981: 4977: 4974:marked graph 4880: 4876: 4872: 4861: 4850:Restrictions 4844: 4834: 4825:Markov chain 4799: 4764: 4743: 4736: 4727: 4712: 4705: 4689: 4679: 4675: 4673: 4667: 4656: 4652: 4650: 4644: 4485: 4483: 4480: 4469: 4466: 4461: 4457: 4453: 4449: 4447: 4442: 4438: 4432: 4428: 4424: 4420: 4418: 4412: 4396: 4392: 4364: 4257: 4183: 3967: 3893: 3751: 3614: 3599: 3595: 3555: 3551: 3545: 3542:Reachability 3529: 3525: 3515: 3506: 3493: 3489:adding to it 3484: 3401: 2978: 2971: 2805: 2743: 2460: 2364: 2354: 2350: 2348: 2283: 2279: 2273: 2020: 2018: 2016:of the net. 1949: 1947: 1812: 1726: 1612: 1570: 1563: 1558: 1555: 1486: 1482: 1388: 1377: 1252: 1248: 1246: 1241: 1203: 1201: 1124: 1120: 1043:input places 1042: 1038: 1034: 1032: 1027: 1023: 975: 971: 896: 894: 888: 806: 802: 797: 793: 788: 780: 732: 728: 726: 714: 709: 705: 704: 696: 689: 682: 678: 674: 670: 666: 662: 656: 649: 642: 635: 628: 621: 617: 610: 608: 603: 599: 596: 591: 587: 580: 576: 572: 570: 565:multiplicity 564: 555: 551: 547: 540: 536: 532: 524: 515: 511: 507: 500: 496: 492: 488: 481: 477: 473: 469: 465: 461: 460: 454: 449: 445: 440: 434: 430: 426: 422: 415: 411: 407: 403: 399: 398: 376: 372: 367: 363: 359: 355: 351: 347: 346:Given a net 343: 342: 336: 274: 270: 208: 204: 203: 196: 184: 175: 173: 168: 164: 160: 156: 154: 149: 146:input places 145: 139: 135: 131: 129: 111: 68: 44:mathematical 39: 35: 31: 29: 7781:(4): 6477. 7771:"Petri net" 6882:IEEE Access 6628:: 226–233. 6622: [ 5780:(1): 1–34. 5708:(4): 6477. 5696:"Petri net" 5661:Petriscript 5599:Game theory 5532:actor model 5469:A directed 5075:free choice 4877:concurrency 4403:Boundedness 2014:state space 1815:is the set 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