Petri net - Knowledge

3387: 4826:, 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 4699:) 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. 3466: 2970: 89: 4844: 381: 373: 3382:{\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}}} 691:, 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. 4822:(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 4767:

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

3625: 4830:. 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. 3585:

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).

2955: 591:. 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 5410: 1933: 5178: 4799:

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

160:, 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. 5447:-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 3593:-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 5911:. 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. 3452: 1648: 2102: 5058: 4957: 2107: 1680: 866: 4623: 1598: 1106: 4397: 1186: 2329: 322: 5495:

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.).

3779: 1009: 4242: 4077: 3999: 3925: 85:. Unlike these standards, Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis. 4768:

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.

4278: 3682: 1334: 763: 3613: 1462: 1423: 133:. 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 4379: 4305: 4200: 4169: 4136: 4105: 4027: 3953: 3876: 3845: 3735: 3708: 3653: 2647: 2541: 2041: 1798: 1363: 5073: 3516:

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

2826: 2253: 1999: 2508: 2478: 2437: 1807: 6836:. Formal Methods for Industrial Critical Systems - 15th International Workshop, FMICS 2010. Lecture Notes in Computer Science. Vol. 6371. pp. 215–230. 4679:

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

7109:. Advanced Information Systems Engineering - 25th International Conference, CAiSE 2013. Lecture Notes in Computer Science. Vol. 7908. pp. 400–416. 3393: 6566: 4796: 701:

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

6056: 6916:"PetriNet Editor + PetriNet Engine: New Software Tool For Modelling and Control of Discrete Event Systems Using Petri Nets and Code Generation" 5420: 3601:, independently by Jerome Leroux and by Wojciech Czerwiński and Łukasz Orlikowski. These results thus close the long-standing complexity gap. 5066:

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

4107:-live if it can fire infinitely often, i.e. if there is some fixed (necessarily infinite) firing sequence in which for every positive integer 5428:

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

6595: 2219:{\displaystyle M_{0}{\underset {G,t_{1}}{\longrightarrow }}M_{1}\wedge \cdots \wedge M_{n-1}{\underset {G,t_{n}}{\longrightarrow }}M_{n}} 4860:(SM), every transition has one incoming arc, and one outgoing arc, and all markings have exactly one token. As a consequence, there can 2046: 1610: 6523:. Proc of the 3rd Workshop on Modelling of Objects, Components, and Agents. Vol. 571. Aarhus, Denmark: DAIMI PB. pp. 21–36. 4980: 4879: 1653: 7636:

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".

6411: 5906: 4710:, and all other extensions sketched in this section. The term is also used specifically for the type of coloured nets supported by 807: 7961: 7951: 4873: 4775:

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

4667:-bounded by adding a "counter-place" with flow opposite to that of the place, and adding tokens to make the total in both places 4588: 6289: 5761: 1037: 2795:

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

7976: 5577: 5508: 2332: 66: 2950:{\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)\}} 1948: 1689: 6678: 6500: 6389: 6356: 6238: 5924: 5809: 5743: 7851: 6376:. 2001 IEEE Pacific Rim Conference on Communications, Computers and signal Processing. Vol. 1. pp. 323–6. 6090:"New Software Tool for Modeling and Control of Discrete-Event and Hybrid Systems Using Timed Interpreted Petri Nets" 6151: 5993:

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).

5547: 3500: 43: 7917:"Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition in Considering Conflicting Opinions" 6600:

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".

4310: 1683: 167:(e.g. a strict ordering of transitions, describing precedence) is defined, the execution of Petri nets is 7981: 7068:"High-level Petri nets for the process description and control in service-oriented manufacturing systems" 5827: 3609: 1545:

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

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

168: 70: 6064: 6343:. 2001 IEEE International Conference on Systems, Man, and Cybernetics. Vol. 3. pp. 1554–8. 5726:

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.).

6431: 5592: 5532: 4809: 4537: 3524: 2357: 1247: 959:. In many textbooks, arcs can only have multiplicity 1. These texts often define Petri nets using 7650: 6612: 6598:[The Boolean differential calculus – A method for analysis and synthesis of Petri nets]. 6023: 5664: 5607: 5504: 5488: 4663:

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

3744: 976: 187: 148:. Any distribution of tokens over the places will represent a configuration of the net called a 105:, after whom such structures are named, analyzed Petri nets extensively in his 1962 dissertation 4695:

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.

5173:{\displaystyle \forall s\in S:(|s^{\bullet }|\leq 1)\vee ({}^{\bullet }(s^{\bullet })=\{s\})} 4805: 4783: 4761: 4250: 3658: 3605: 1301: 730: 1928:{\displaystyle R(N)\ {\stackrel {D}{=}}\ \left\{M'{\Bigg |}M_{0}{\xrightarrow{*}}M'\right\}} 1432: 1393: 662:. Once and only once a transition is enabled will the transition fire. In this example, the 7898:

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

7771: 6878: 6744: 6596:"Der Boolesche Differentialkalkül – eine Methode zur Analyse und Synthese von Petri-Netzen" 5698: 5654: 5634: 4827: 4819: 4801: 4357: 4283: 4178: 4147: 4114: 4083: 4005: 3931: 3854: 3823: 3713: 3686: 3631: 3536: 2625: 2519: 2510: 2406: 2019: 1776: 1773:, we are interested in the firings that can be performed starting with the initial marking 1341: 579:

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).

4764:, where the arc and guard expressions are restricted to make it easier to analyse the net. 2802: 2229: 1975: 8: 7971: 6452: 5567: 5460: 4965:(MG), every place has one incoming arc, and one outgoing arc. This means, that there can 4696: 3598: 2483: 2453: 2412: 152:. In an abstract sense relating to a Petri net diagram, a transition of a Petri net may 39: 7775: 6882: 6830:"Automatic Structure-Based Code Generation from Coloured Petri Nets: A Proof of Concept" 6748: 5930: 5702: 4644:, both places are assigned capacity 2, we obtain a Petri net with place capacities, say 3527:

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,

171:: when multiple transitions are enabled at the same time, they will fire in any order. 5887: 7966: 7901: 7879: 7857: 7853:

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

7611:"A new paradigm for uncertain knowledge representation by 'Plausible Petri nets'" 7448: 7115: 7083: 7066:

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

6841: 6757: 6695: 6622: 6487:. Lecture Notes in Computer Science. Vol. 2678. Springer. pp. 337–356. 6478:"Soundness and separability of workflow nets in the stepwise refinement approach" 6125: 5617: 5516: 4736: 4684: 968: 144:

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

102: 52: 47: 7408: 7360: 6891: 6867:"A Petri Net Neural Network Robust Control for New Paste Backfill Process Model" 6866: 5730:. Lecture Notes in Computer Science. Vol. 1491. Springer. pp. 12–121. 7067: 7042: 6829: 6794: 6381: 6292:. Department of Computer Science, University of Aarhus, Denmark. Archived from 5883: 4680: 3848: 971: 7784: 7759: 7713: 7627: 7610: 7563: 7519:"Modélisation des pannes d'une antenne active et modifications d'architecture" 7517:

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

5491:(DSM) can model process relations, and be utilized for process planning. The 4857: 4748:, every token has a value. In popular tools for coloured Petri nets such as 3496: 873: 799: 517: 88: 7553: 7210: 7172: 6991: 6948: 5801: 3612:

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).

6776: 5689: 5524: 4962: 4813: 4464: 4029:-live if it can fire arbitrarily often, i.e. if for every positive integer 129: 32: 7704: 7518: 7266: 5855: 2828:

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.

3465: 880:(or weight); note that no arc may connect two places or two transitions. 583:

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

7325: 7142:"Modeling shortest path games with Petri nets: a Lyapunov based theory" 6225:. Lecture Notes in Computer Science. Vol. 1443. pp. 103–115. 6170: 6165:. Lecture Notes in Computer Science. Vol. 1217. pp. 203–208. 5612: 3594: 3447:{\displaystyle M_{0}={\begin{bmatrix}1&0&2&1\end{bmatrix}}} 773: 7205: 2338:

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

7551: 7307: 7209:; Kishinevsky, M.; Kondratyev, A.; Lavagno, L.; Yakovlev, A. (2002). 4749: 4711: 4651: 4628: 3959:), if and only if it may fire, i.e. it is in some firing sequence in 2442: 2263:

A common variation is to disallow arc multiplicities and replace the

1880: 552:, so that the count (or weight) for each arc is a measure of the arc 78: 7672:

Formális módszerek az informatikában (Formal methods in informatics)

7349:"Strategies for Modeling Complex Processes Using Colored Petri Nets" 6036: 5759:

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

55:—at the age of 13—for the purpose of describing chemical processes. 6790: 5999: 5978: 5957: 5424: 3590: 2264: 869: 573: 549: 509: 7032: 4843: 3624: 2097:{\displaystyle {\vec {\sigma }}=\langle t_{1}\cdots t_{n}\rangle } 1643:{\displaystyle M{\overset {*}{\underset {G}{\longrightarrow }}}M'} 7355:. Lecture Notes in Computer Science. Vol. 7. pp. 6–55. 7146:

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".

3885:, if it can never fire, i.e. it is not in any firing sequence in 74: 7608: 5053:{\displaystyle \forall s\in S:|s^{\bullet }|=|{}^{\bullet }s|=1} 4952:{\displaystyle \forall t\in T:|t^{\bullet }|=|{}^{\bullet }t|=1} 7305: 6990:

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

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Transactions on Petri Nets and Other Models of Concurrency VII

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IEEE International Conference on Robotics and Automation, 2008

6989: 5423:(WF-nets) are a subclass of Petri nets intending to model the 4674: 380: 372: 7346: 7215:. Springer Series in Advanced Microelectronics. Vol. 8. 5861: 4744:

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

861:{\displaystyle W:(S\times T)\cup (T\times S)\to \mathbb {N} } 725: 201: 7171:

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

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property, indicating that a process with a start marking of

1188:. Definitions of pre- and postsets of places are analogous. 7251: 7212:

Logic Synthesis for Asynchronous Controllers and Interfaces

6670:

Modeling Business Processes - A Petri Net-Oriented Approach

6587: 6256:

IEEE Transactions on Systems, Man, and Cybernetics: Systems

5950: 4752:, the values of tokens are typed, and can be tested (using 4618:{\displaystyle C:P\rightarrow \!\!\!\shortmid \mathbb {N} } 7065: 6087: 4449:

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

181: 7516: 6574:

Handbook of Logic and the Foundations of Computer Science

6519:

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

5527:. Different models provide tradeoffs of concepts such as 4247:

Note that these are increasingly stringent requirements:

1101:{\displaystyle {}^{\bullet }t=\{s\in S\mid W(s,t)>0\}} 637:(bottom figure) be a Petri net with a marking configured 7306:

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

6475: 6220: 3510: 1229:. We say the marking assigns to each place a number of 560:

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

6950:

Process Mining - Data Science in Action, Second Edition

5068:

both concurrency and conflict, but not at the same time

1181:{\displaystyle t^{\bullet }=\{s\in S\mid W(t,s)>0\}} 7442: 7440: 6412:"The application of Petri nets to workflow management" 4808:

through adjustable randomness of the transitions. The

4314: 4313: 3415: 3260: 3126: 2992: 2450:

Its transition relation can be described as a pair of

2324:{\displaystyle F\subseteq (S\times T)\cup (T\times S)} 967:. When using this convention, a Petri net graph is a 623:(top figure) be a Petri net with a marking configured 317:{\displaystyle F\subseteq (P\times T)\cup (T\times P)} 7402: 7400: 7261:. Lecture Notes in Computer Science. Vol. 2549. 6827: 5888:"The Reachability Problem Requires Exponential Space" 5207: 5182:

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

5076: 4983: 4882: 4835:

simple net type possible for a given modelling task.

4591: 4540: 4483: 4453:

if it is bounded for every possible initial marking.

4360: 4286: 4253: 4208: 4181: 4150: 4117: 4086: 4043: 4008: 3965: 3934: 3891: 3857: 3826: 3787: 3747: 3716: 3689: 3661: 3634: 3553: 3396: 2973: 2841: 2805: 2742: 2655: 2628: 2549: 2522: 2486: 2456: 2415: 2360: 2281: 2232: 2110: 2049: 2022: 1978: 1951: 1810: 1779: 1722: 1692: 1656: 1613: 1568: 1488: 1435: 1396: 1344: 1304: 1250: 1201: 1122: 1040: 979: 894: 876:, i.e. it assigns to each arc a non-negative integer 810: 802:, i.e. no object can be both a place and a transition 733: 704:

The following formal definition is loosely based on (

564:

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

274: 209: 7255:; Yakovlev, Alex; Rozenberg, Grzegorz, eds. (2002). 7170: 6993:

Conformance Checking - Relating Processes and Models

6796:

Modeling in Systems Biology - The Petri Net Approach

6514: 6512: 5992: 4648:; its reachability graph is displayed on the right. 1593:{\displaystyle M{\underset {G}{\longrightarrow }}M'} 190:

that extend a class of nets called elementary nets.

7437: 6789: 6016:"Petri Nets: Properties, Analysis and Applications" 5725: 7397: 6666: 6576:. Vol. 4. OUP. pp. 1–148. Archived from 5404: 5172: 5052: 4951: 4617: 4577: 4526: 4373: 4343: 4299: 4272: 4236: 4194: 4163: 4130: 4099: 4071: 4021: 3993: 3947: 3919: 3870: 3839: 3812: 3773: 3729: 3702: 3676: 3647: 3574: 3446: 3381: 2949: 2820: 2784: 2722: 2641: 2613: 2535: 2502: 2472: 2431: 2397: 2323: 2247: 2218: 2096: 2035: 1993: 1964: 1927: 1792: 1765: 1713:; that is, if it is reachable in 0 or more steps. 1705: 1674: 1642: 1592: 1533: 1456: 1417: 1357: 1328: 1287: 1221: 1180: 1100: 1003: 951: 860: 757: 316: 239: 6799:. Computational Biology. Vol. 16. Springer. 6662: 6660: 6509: 6476:van Hee, K.; Sidorova, N.; Voorhoeve, M. (2003). 6339:Dawis, E. P.; Dawis, J. F.; Koo, Wei-Pin (2001). 5824: 5796:. LNCS. Vol. 2128. Springer. pp. 1–25. 4683:, and related to discrete, continuous and hybrid 4606: 4605: 4604: 3456: 2342:to be defined on places. This is discussed under 1965:{\displaystyle {\underset {G}{\longrightarrow }}} 1860: 1706:{\displaystyle {\underset {G}{\longrightarrow }}} 7943: 6733:"esyN: Network Building, Sharing and Publishing" 6469: 6119: 6117: 670:generates a map that has the marking configured 7308:"A Petri nets semantics for data flow networks" 6667:van der Aalst, W.M.P.; Stahl, C. (2011-05-27). 6593: 6483:. In van der Aalst, W. M. P.; Best, E. (eds.). 5856:"Decidability issues for Petri nets – a survey" 5468:includes every node in the sequence only once. 4721:Additional types of arcs; two common types are 4655:A two-bounded Petri net, obtained by extending 3539:for Petri nets is to decide, given a Petri net 2258: 616:is an output place to the same transition. Let 7589: 6834:Formal Methods for Industrial Critical Systems 6657: 5853: 5498: 4760:. A subsidiary of coloured Petri nets are the 7892: 7446: 7406: 7104: 6946: 6693: 6521:On 1-soundness and soundness of workflow nets 6409: 6405: 6403: 6401: 6338: 6156:"A brief introduction to coloured Petri Nets" 6114: 4717:A short list of possible extensions follows: 2226:. The set of firing sequences is denoted as 7870: 7732:Petri Net Theory and the Modeling of Systems 7483: 7072:International Journal of Production Research 6793:; Reisig, Wolfgang; Schreiber, Falk (2011). 6687: 6549: 5193:net (AC), concurrency and conflict (in sum, 5164: 5158: 3597:. In 2021, this problem was shown to be non- 2944: 2857: 2350:Formulation in terms of vectors and matrices 2091: 2065: 1175: 1136: 1095: 1056: 946: 901: 473:), which extends the elementary net so that 106: 77:for stepwise processes that include choice, 7758:Petri, Carl Adam; Reisig, Wolfgang (2008). 7757: 7689:Peterson, James Lyle (1977). "Petri Nets". 7590:Cardoso, Janette; Camargo, Heloisa (1999). 6828:Kristensen, L. M.; Westergaard, M. (2010). 6518: 6193: 6127:Discrete, continuous, and hybrid Petri Nets 5683:Petri, Carl Adam; Reisig, Wolfgang (2008). 5682: 4675:Discrete, continuous, and hybrid Petri nets 952:{\displaystyle F=\{(x,y)\mid W(x,y)>0\}} 6564: 6419:Journal of Circuits, Systems and Computers 6398: 6163:A brief introduction to colored Petri nets 5854:Esparza, Javier; Nielsen, Mogens (1995) . 5787: 4852:Petri nets are commonly used and studied: 7932: 7783: 7712: 7669: 7626: 7114: 6931: 6890: 6766: 6756: 6724: 6485:Application and Theory of Petri Nets 2003 6430: 6123: 6105: 5998: 5977: 5956: 5710: 4773:vector addition system with states (VASS) 4739:and implies existence of a universal net. 4611: 2723:{\displaystyle \forall s,t:W^{+}=W(t,s).} 1534:{\displaystyle \forall s:M(s)\geq W(s,t)} 1215: 854: 598:In the top figure (see right), the place 7729: 7688: 7635: 7139: 4842: 4650: 4627: 4395: 3623: 3608:is usually used in conjunction with the 2614:{\displaystyle \forall s,t:W^{-}=W(s,t)} 2441: 708:). Many alternative definitions exist. 705: 379: 371: 96: 87: 7914: 7849: 7830: 7811: 7488:. Andre Kleyner (5th ed.). Wiley. 7107:Active Flow and Combustion Control 2018 6253: 4459:Boundedness is decidable by looking at 376:A Petri net with an enabled transition. 182:Formal definition and basic terminology 7944: 7792: 7648: 7078:(6). Taylor & Francis: 1650–1665. 7037:. Pasadena, CA, USA. pp. 1372–7. 6730: 6150: 6013: 5971: 5904: 5882: 5758: 4344:{\textstyle \textstyle {j\in {1,2,3}}} 3589:In fact, this problem was shown to be 1373: 7748: 7525:. Sécurité des Systèmes Industriels. 6864: 6371: 6083: 6081: 5443:tokens in its sink place (defined as 3511:Mathematical properties of Petri nets 1972:restricted to its reachable markings 1425:tokens from each of its input places 16:Model to describe distributed systems 7753:(Ph. D. thesis). University of Bonn. 6290:"Very Brief Introduction to CP-nets" 5541: 4756:expressions) and manipulated with a 4202:-live in every reachable marking in 4175:) if it may always fire, i.e. it is 3460: 1464:tokens in each of its output places 7856:. LAP LAMBERT Academic Publishing. 6124:David, René; Alla, Hassane (2005). 5578:Diagnosis (artificial intelligence) 5509:communicating finite-state machines 3499:and Montanari considered a kind of 2883: is a firing sequence of 2409:of non-negative integers of length 1369:, a marking of the Petri net graph. 1222:{\displaystyle M:S\to \mathbb {N} } 114: 67:Business Process Model and Notation 13: 7812:Riemann, Robert-Christoph (1999). 7583: 6554:. World Scientific. pp. 3–67. 6078: 5290: 5208: 5077: 4984: 4883: 2866: 2785:{\displaystyle W^{T}=-W^{-}+W^{+}} 2656: 2550: 2335:as both can represent each other. 1489: 14: 7993: 7486:Practical reliability engineering 4412:if it does not contain more than 4408:A place in a Petri net is called 4037:times in some firing sequence in 1244:by some, see above) is a 4-tuple 178:behavior of distributed systems. 7484:O'Connor, Patrick D. T. (2012). 7455:. Springer. pp. 4716–4717. 7453:Encyclopedia of Database Systems 7415:. Springer. pp. 4717–4718. 7413:Encyclopedia of Database Systems 6700:Encyclopedia of Database Systems 6410:van der Aalst, W. M. P. (1998). 6315:"LLPN - Linear Logic Petri Nets" 5415: 4475:Petri nets with place capacities 3628:A Petri net in which transition 3464: 605:is an input place of transition 92:(a) Petri net trajectory example 58:Like industry standards such as 7900:. World Scientific Publishing. 7545: 7510: 7477: 7340: 7299: 7258:Concurrency and Hardware Design 7245: 7199: 7164: 7133: 7098: 7059: 7026: 6983: 6940: 6907: 6858: 6821: 6783: 6645:from the original on 2017-10-16 6558: 6543: 6365: 6332: 6307: 6282: 6247: 6214: 6187: 6144: 6049: 6007: 5986: 5965: 5905:Küngas, P. (July 26–29, 2005). 4838: 4810:exponential random distribution 4758:functional programming language 4527:{\displaystyle (S,T,W,C,M_{0})} 4437:A (marked) Petri net is called 3530: 1766:{\displaystyle N=(S,T,W,M_{0})} 724:by some, but see below) is a 3- 7962:Concurrency (computer science) 7952:Formal specification languages 7896:; Venkatesh, Kurapati (1998). 7878:. Kluwer Academic Publishers. 7461:10.1007/978-1-4614-8265-9_1476 7447:van der Aalst, W.M.P. (2018). 7407:van der Aalst, W.M.P. (2018). 7174:Hardware Design and Petri Nets 6947:van der Aalst, W.M.P. (2016). 6708:10.1007/978-1-4614-8265-9_1179 6702:. Springer. pp. 370–374. 6694:van der Aalst, W.M.P. (2018). 5944: 5898: 5876: 5847: 5818: 5781: 5752: 5719: 5676: 5399: 5396: 5352: 5346: 5302: 5299: 5296: 5293: 5243: 5167: 5152: 5139: 5127: 5121: 5111: 5096: 5092: 5040: 5023: 5015: 5000: 4939: 4922: 4914: 4899: 4601: 4572: 4541: 4521: 4484: 4391: 4231: 4212: 4066: 4047: 3988: 3969: 3914: 3895: 3807: 3788: 3569: 3563: 3457:Category-theoretic formulation 2941: 2935: 2851: 2845: 2815: 2809: 2714: 2702: 2693: 2681: 2608: 2596: 2587: 2575: 2496: 2488: 2466: 2458: 2425: 2417: 2392: 2361: 2318: 2306: 2300: 2288: 2271:with a simple set, called the 2242: 2236: 2183: 2123: 2056: 1988: 1982: 1954: 1900: 1882: 1820: 1814: 1760: 1729: 1695: 1660: 1620: 1574: 1528: 1516: 1507: 1501: 1451: 1439: 1412: 1400: 1323: 1305: 1282: 1251: 1211: 1166: 1154: 1086: 1074: 998: 980: 937: 925: 916: 904: 850: 847: 835: 829: 817: 752: 734: 311: 299: 293: 281: 234: 216: 101:The German computer scientist 1: 7730:Peterson, James Lyle (1981). 7421:10.1007/978-1-4614-8265-9_826 6696:"Business Process Management" 6673:. MIT Press. pp. 1–400. 5670: 5548:Boolean differential calculus 4690: 4578:{\displaystyle (S,T,W,M_{0})} 3878:-live, where a transition is 3501:symmetric monoidal categories 2398:{\displaystyle (S,T,W,M_{0})} 2043:is a sequence of transitions 1288:{\displaystyle (S,T,W,M_{0})} 44:discrete event dynamic system 7795:A Primer in Petri Net Design 7116:10.1007/978-3-642-38709-8_26 7084:10.1080/00207543.2011.575892 6865:Gao, X.; Hu, Xinyan (2020). 6842:10.1007/978-3-642-15898-8_14 6758:10.1371/journal.pone.0106035 6623:10.1524/auto.1991.39.112.226 6208:10.1016/0304-3975(76)90067-0 6196:Theoretical Computer Science 6061:www.techfak.uni-bielefeld.de 6014:Murata, Tadao (April 1989). 5841:10.1016/0890-5401(90)90013-8 5775:10.1016/0304-3975(91)90203-e 5762:Theoretical Computer Science 2354:The markings of a Petri net 2259:Variations on the definition 1684:reflexive transitive closure 258:are disjoint finite sets of 46:. A Petri net is a directed 7: 7751:Kommunikation mit Automaten 7361:10.1007/978-3-642-38143-0_2 6892:10.1109/ACCESS.2020.2968510 6731:Favrin, Bean (2014-09-02). 5894:. Yale University: 305–329. 5828:Information and Computation 5628: 5623:Workflow management systems 5499:Other models of concurrency 4847:Petri net types graphically 4640:For example, if in the net 3851:all of its transitions are 3774:{\displaystyle L_{1}-L_{4}} 3619: 2012:for a Petri net with graph 1945:is the transition relation 1004:{\displaystyle (S\cup T,F)} 108:Kommunikation mit Automaten 71:event-driven process chains 10: 7998: 7977:Software modeling language 7874:; Dicesare, Frank (1993). 7670:Pataricza, András (2004). 7638:Przegląd Elektrotechniczny 7043:10.1109/ROBOT.2008.4543394 6382:10.1109/PACRIM.2001.953588 4420:if it is 1-bounded; it is 4400:The reachability graph of 4237:{\displaystyle R(N,M_{0})} 4072:{\displaystyle L(N,M_{0})} 3994:{\displaystyle L(N,M_{0})} 3920:{\displaystyle L(N,M_{0})} 7793:Reisig, Wolfgang (1992). 7785:10.4249/scholarpedia.6477 7749:Petri, Carl Adam (1962). 7628:10.1016/j.ins.2018.04.029 7564:10.1007/978-3-642-03121-2 7523:Techniques de l'Ingénieur 7449:"Workflow Model Analysis" 7221:10.1007/978-3-642-55989-1 7183:10.1007/978-1-4757-3143-9 7002:10.1007/978-3-319-99414-7 6959:10.1007/978-3-662-49851-4 6805:10.1007/978-1-84996-474-6 6606:(7). Stuttgart, Germany: 6565:Winskel, G.; Nielsen, M. 6441:10.1142/s0218126698000043 6349:10.1109/ICSMC.2001.973505 6268:10.1109/TSMC.2012.2237549 5712:10.4249/scholarpedia.6477 5645:Petri Net Markup Language 5553:Business process modeling 4477:can be defined as tuples 3813:{\displaystyle (N,M_{0})} 3575:{\displaystyle M\in R(N)} 2961:It must be required that 1716:For a (marked) Petri net 711: 684:and results in Petri net 240:{\displaystyle N=(P,T,F)} 7850:Zaitsev, Dmitry (2013). 7831:Störrle, Harald (2000). 7140:Clempner, Julio (2006). 6567:"Models for Concurrency" 6493:10.1007/3-540-44919-1_22 5736:10.1007/3-540-65306-6_14 5583:Discrete process control 4632:An unbounded Petri net, 3523:, with decidability and 188:state-transition systems 119:A Petri net consists of 7592:Fuzziness in Petri Nets 6024:Proceedings of the IEEE 5802:10.1007/3-540-45541-8_1 5665:Vector addition systems 5608:Reliability engineering 5505:vector addition systems 5489:design structure matrix 4273:{\displaystyle L_{j+1}} 3677:{\displaystyle j>0,} 3655:is dead, while for all 3614:semi-decision technique 1329:{\displaystyle (S,T,W)} 758:{\displaystyle (S,T,W)} 644:. The configuration of 524:extends the concept of 324:is a set of (directed) 38:for the description of 7816:. Herbert Utz Verlag. 7621:(July 2018): 323–345. 5792:; et al. (eds.). 5563:Concurrent programming 5406: 5184:transformed into an FC 5174: 5054: 4953: 4848: 4824:transformation marking 4784:Prioritised Petri nets 4777:finite state automaton 4762:well-formed Petri nets 4660: 4659:with "counter-places". 4637: 4619: 4579: 4528: 4463:, by constructing the 4451:(structurally) bounded 4405: 4375: 4345: 4301: 4274: 4238: 4196: 4165: 4132: 4101: 4073: 4023: 3995: 3949: 3921: 3872: 3841: 3814: 3775: 3738: 3731: 3704: 3678: 3649: 3576: 3448: 3383: 2951: 2822: 2786: 2733:Then their difference 2724: 2643: 2615: 2537: 2504: 2474: 2447: 2433: 2399: 2331:. This does not limit 2325: 2249: 2220: 2098: 2037: 1995: 1966: 1929: 1910: 1794: 1767: 1707: 1676: 1644: 1594: 1548:We say that a marking 1535: 1458: 1457:{\displaystyle W(t,s)} 1419: 1418:{\displaystyle W(s,t)} 1359: 1330: 1289: 1223: 1182: 1102: 1005: 953: 862: 759: 385: 377: 318: 241: 107: 93: 7957:Models of computation 7705:10.1145/356698.356702 7692:ACM Computing Surveys 7649:Jensen, Kurt (1997). 7267:10.1007/3-540-36190-1 6372:Dawis, E. P. (2001). 5598:Kahn process networks 5558:Computational biology 5513:Kahn process networks 5407: 5175: 5055: 4954: 4846: 4806:nondeterministic time 4802:stochastic Petri nets 4654: 4631: 4620: 4580: 4529: 4399: 4376: 4374:{\displaystyle L_{0}} 4346: 4302: 4300:{\displaystyle L_{j}} 4275: 4239: 4197: 4195:{\displaystyle L_{1}} 4166: 4164:{\displaystyle L_{4}} 4133: 4131:{\displaystyle L_{3}} 4102: 4100:{\displaystyle L_{3}} 4074: 4033:, it occurs at least 4024: 4022:{\displaystyle L_{2}} 3996: 3950: 3948:{\displaystyle L_{1}} 3922: 3873: 3871:{\displaystyle L_{k}} 3842: 3840:{\displaystyle L_{k}} 3815: 3776: 3732: 3730:{\displaystyle L_{j}} 3705: 3703:{\displaystyle t_{j}} 3679: 3650: 3648:{\displaystyle t_{0}} 3627: 3606:linear temporal logic 3577: 3449: 3384: 2952: 2823: 2787: 2725: 2644: 2642:{\displaystyle W^{+}} 2616: 2538: 2536:{\displaystyle W^{-}} 2505: 2475: 2446:(b) Petri net example 2445: 2434: 2400: 2326: 2250: 2221: 2099: 2038: 2036:{\displaystyle M_{0}} 1996: 1967: 1930: 1876: 1795: 1793:{\displaystyle M_{0}} 1768: 1708: 1677: 1645: 1595: 1536: 1459: 1420: 1360: 1358:{\displaystyle M_{0}} 1336:is a Petri net graph; 1331: 1290: 1224: 1183: 1103: 1011:with node partitions 1006: 954: 863: 760: 609:; whereas, the place 457:is a net of the form 395:is a net of the form 383: 375: 319: 242: 97:Historical background 91: 73:, Petri nets offer a 31:), is one of several 7915:Xue-Guo, Xu (2019). 7615:Information Sciences 7531:10.51257/a-v1-se1221 6608:R. Oldenbourg Verlag 5655:Process architecture 5635:Finite-state machine 5205: 5199:not symmetrically: m 5074: 4981: 4880: 4828:process architecture 4820:Dualistic Petri Nets 4704:high-level Petri net 4589: 4538: 4481: 4358: 4311: 4284: 4251: 4206: 4179: 4148: 4115: 4084: 4041: 4006: 3963: 3957:potentially fireable 3932: 3889: 3855: 3824: 3785: 3745: 3714: 3687: 3659: 3632: 3551: 3537:reachability problem 3519:An overview of such 3394: 2971: 2839: 2821:{\displaystyle o(w)} 2803: 2740: 2653: 2626: 2547: 2520: 2484: 2454: 2413: 2358: 2279: 2248:{\displaystyle L(N)} 2230: 2108: 2047: 2020: 2016:and initial marking 1994:{\displaystyle R(N)} 1976: 1949: 1808: 1777: 1720: 1690: 1654: 1611: 1566: 1486: 1433: 1394: 1382:firing a transition 1342: 1302: 1248: 1199: 1120: 1038: 977: 892: 888:is the set of arcs: 808: 731: 568:that map to 1 under 328:(or flow relations). 272: 207: 83:concurrent execution 42:. It is a class of 25:place/transition net 7835:. Books on Demand. 7797:. Springer-Verlag. 7776:2008SchpJ...3.6477P 7655:. Springer Verlag. 7652:Coloured Petri Nets 7409:"Workflow Patterns" 6933:10.3390/app10217662 6883:2020IEEEA...818420G 6749:2014PLoSO...9j6035B 6231:10.1007/11527862_11 6107:10.3390/app10155027 5917:10.1007/11527862_11 5892:Technical Report 62 5794:Unifying Petri Nets 5703:2008SchpJ...3.6477P 5568:Control engineering 5480:is not a WF-net). 4973:, but there can be 4876:): mathematically, 4868:, but there can be 4697:coloured Petri nets 3599:primitive recursive 2503:{\displaystyle |T|} 2473:{\displaystyle |S|} 2432:{\displaystyle |S|} 2405:can be regarded as 1909: 1904: 1374:Execution semantics 785:is a finite set of 141:of the transition. 40:distributed systems 7982:Modeling languages 7934:10.3390/app9050983 7714:10338.dmlcz/135597 7594:. Physica-Verlag. 7326:10.1007/BF01178383 6552:The Book of Traces 6171:10.1007/BFb0035389 5865:(Revised ed.) 5660:Slicing Petri nets 5402: 5170: 5070:: mathematically, 5050: 4949: 4849: 4791:non-deterministic. 4746:coloured Petri net 4661: 4638: 4615: 4575: 4524: 4406: 4371: 4341: 4340: 4297: 4280:-liveness implies 4270: 4234: 4192: 4161: 4128: 4097: 4069: 4019: 3991: 3945: 3917: 3868: 3837: 3810: 3771: 3739: 3727: 3700: 3674: 3645: 3572: 3476:. You can help by 3444: 3438: 3379: 3373: 3230: 3096: 2947: 2818: 2782: 2720: 2639: 2611: 2533: 2500: 2470: 2448: 2429: 2395: 2321: 2245: 2216: 2204: 2144: 2094: 2033: 1991: 1962: 1960: 1939:reachability graph 1925: 1802:reachable markings 1790: 1763: 1703: 1701: 1672: 1666: 1640: 1626: 1602:is reachable from 1590: 1580: 1531: 1454: 1415: 1355: 1326: 1285: 1219: 1178: 1112:is the set of its 1098: 1030:is the set of its 1001: 949: 858: 755: 386: 378: 314: 237: 94: 75:graphical notation 36:modeling languages 23:, also known as a 7907:978-981-02-3029-6 7885:978-0-7923-9289-7 7863:978-3-659-42228-7 7842:978-3-8311-1330-9 7823:978-3-89675-629-9 7804:978-3-540-52044-3 7741:978-0-13-661983-3 7734:. Prentice Hall. 7681:978-963-9548-08-4 7674:. TYPOTEX Kiadó. 7662:978-3-540-62867-5 7601:978-3-7908-1158-2 7573:978-3-642-03122-9 7495:978-1-119-96126-0 7470:978-1-4614-8266-6 7430:978-1-4614-8266-6 7370:978-3-642-38142-3 7276:978-3-540-00199-7 7253:Cortadella, Jordi 7230:978-3-642-62776-7 7192:978-1-4419-4969-1 7126:978-3-319-98176-5 7052:978-1-4244-1646-2 7011:978-3-319-99413-0 6968:978-3-662-49850-7 6851:978-3-642-15897-1 6814:978-1-84996-473-9 6717:978-1-4614-8266-6 6180:978-3-540-62790-6 6137:978-3-540-22480-8 5542:Application areas 5451:-sound for every 5431:WF-nets have the 5197:) may occur, but 5191:asymmetric choice 4111:, the transition 3521:decision problems 3494: 3493: 3241: 3107: 2899: 2895: 2891: 2884: 2877: 2832:. Then, we have 2182: 2122: 2059: 1953: 1844: 1839: 1825: 1694: 1670: 1659: 1630: 1619: 1600:; we say that it 1573: 1553:is reachable from 63:activity diagrams 7989: 7938: 7936: 7921:Applied Sciences 7911: 7889: 7867: 7846: 7827: 7808: 7789: 7787: 7754: 7745: 7726: 7716: 7685: 7666: 7645: 7632: 7630: 7605: 7578: 7577: 7549: 7543: 7542: 7514: 7508: 7507: 7481: 7475: 7474: 7444: 7435: 7434: 7404: 7395: 7394: 7388: 7384: 7382: 7374: 7344: 7338: 7337: 7313:Acta Informatica 7303: 7297: 7296: 7249: 7243: 7242: 7203: 7197: 7196: 7168: 7162: 7161: 7137: 7131: 7130: 7118: 7102: 7096: 7095: 7063: 7057: 7056: 7030: 7024: 7023: 6987: 6981: 6980: 6944: 6938: 6937: 6935: 6920:Applied Sciences 6911: 6905: 6904: 6894: 6862: 6856: 6855: 6825: 6819: 6818: 6787: 6781: 6780: 6770: 6760: 6728: 6722: 6721: 6691: 6685: 6684: 6664: 6655: 6653: 6651: 6650: 6616: 6591: 6585: 6584: 6582: 6571: 6562: 6556: 6555: 6547: 6541: 6540: 6516: 6507: 6506: 6482: 6473: 6467: 6466: 6464: 6463: 6457: 6451:. Archived from 6434: 6416: 6407: 6396: 6395: 6369: 6363: 6362: 6336: 6330: 6329: 6327: 6326: 6317:. Archived from 6311: 6305: 6304: 6302: 6301: 6286: 6280: 6279: 6251: 6245: 6244: 6218: 6212: 6211: 6191: 6185: 6184: 6160: 6148: 6142: 6141: 6121: 6112: 6111: 6109: 6094:Applied Sciences 6085: 6076: 6075: 6073: 6072: 6063:. Archived from 6053: 6047: 6046: 6044: 6043: 6020: 6011: 6005: 6004: 6002: 5990: 5984: 5983: 5981: 5969: 5963: 5962: 5960: 5948: 5942: 5941: 5939: 5938: 5929:. Archived from 5902: 5896: 5895: 5880: 5874: 5873: 5871: 5870: 5860:Bulletin of the 5851: 5845: 5844: 5822: 5816: 5815: 5785: 5779: 5778: 5756: 5750: 5749: 5723: 5717: 5716: 5714: 5680: 5640:Machine learning 5603:Process modeling 5535:, and locality. 5529:compositionality 5411: 5409: 5408: 5403: 5395: 5394: 5389: 5386: 5385: 5373: 5372: 5367: 5364: 5363: 5345: 5344: 5339: 5336: 5335: 5323: 5322: 5317: 5314: 5313: 5286: 5285: 5280: 5277: 5276: 5264: 5263: 5258: 5255: 5254: 5233: 5232: 5220: 5219: 5179: 5177: 5176: 5171: 5151: 5150: 5138: 5137: 5132: 5114: 5109: 5108: 5099: 5059: 5057: 5056: 5051: 5043: 5035: 5034: 5029: 5026: 5018: 5013: 5012: 5003: 4977:mathematically, 4958: 4956: 4955: 4950: 4942: 4934: 4933: 4928: 4925: 4917: 4912: 4911: 4902: 4797:timed Petri nets 4708:Nets within Nets 4624: 4622: 4621: 4616: 4614: 4585:is a Petri net, 4584: 4582: 4581: 4576: 4571: 4570: 4533: 4531: 4530: 4525: 4520: 4519: 4380: 4378: 4377: 4372: 4370: 4369: 4350: 4348: 4347: 4342: 4339: 4338: 4306: 4304: 4303: 4298: 4296: 4295: 4279: 4277: 4276: 4271: 4269: 4268: 4243: 4241: 4240: 4235: 4230: 4229: 4201: 4199: 4198: 4193: 4191: 4190: 4170: 4168: 4167: 4162: 4160: 4159: 4141: 4138:occurs at least 4137: 4135: 4134: 4129: 4127: 4126: 4110: 4106: 4104: 4103: 4098: 4096: 4095: 4078: 4076: 4075: 4070: 4065: 4064: 4036: 4032: 4028: 4026: 4025: 4020: 4018: 4017: 4000: 3998: 3997: 3992: 3987: 3986: 3954: 3952: 3951: 3946: 3944: 3943: 3926: 3924: 3923: 3918: 3913: 3912: 3877: 3875: 3874: 3869: 3867: 3866: 3846: 3844: 3843: 3838: 3836: 3835: 3819: 3817: 3816: 3811: 3806: 3805: 3780: 3778: 3777: 3772: 3770: 3769: 3757: 3756: 3736: 3734: 3733: 3728: 3726: 3725: 3709: 3707: 3706: 3701: 3699: 3698: 3683: 3681: 3680: 3675: 3654: 3652: 3651: 3646: 3644: 3643: 3581: 3579: 3578: 3573: 3505:Petri categories 3489: 3486: 3468: 3461: 3453: 3451: 3450: 3445: 3443: 3442: 3406: 3405: 3388: 3386: 3385: 3380: 3378: 3377: 3251: 3250: 3239: 3235: 3234: 3117: 3116: 3105: 3101: 3100: 2983: 2982: 2964: 2956: 2954: 2953: 2948: 2928: 2927: 2915: 2914: 2897: 2896: 2893: 2889: 2885: 2882: 2875: 2831: 2827: 2825: 2824: 2819: 2798: 2791: 2789: 2788: 2783: 2781: 2780: 2768: 2767: 2752: 2751: 2729: 2727: 2726: 2721: 2680: 2679: 2648: 2646: 2645: 2640: 2638: 2637: 2620: 2618: 2617: 2612: 2574: 2573: 2542: 2540: 2539: 2534: 2532: 2531: 2509: 2507: 2506: 2501: 2499: 2491: 2479: 2477: 2476: 2471: 2469: 2461: 2438: 2436: 2435: 2430: 2428: 2420: 2404: 2402: 2401: 2396: 2391: 2390: 2333:expressive power 2330: 2328: 2327: 2322: 2254: 2252: 2251: 2246: 2225: 2223: 2222: 2217: 2215: 2214: 2205: 2203: 2202: 2201: 2180: 2179: 2155: 2154: 2145: 2143: 2142: 2141: 2120: 2119: 2103: 2101: 2100: 2095: 2090: 2089: 2077: 2076: 2061: 2060: 2052: 2042: 2040: 2039: 2034: 2032: 2031: 2015: 2000: 1998: 1997: 1992: 1971: 1969: 1968: 1963: 1961: 1944: 1934: 1932: 1931: 1926: 1924: 1920: 1919: 1911: 1903: 1874: 1873: 1864: 1863: 1857: 1842: 1841: 1840: 1838: 1833: 1828: 1823: 1799: 1797: 1796: 1791: 1789: 1788: 1772: 1770: 1769: 1764: 1759: 1758: 1712: 1710: 1709: 1704: 1702: 1681: 1679: 1678: 1673: 1671: 1658: 1649: 1647: 1646: 1641: 1639: 1631: 1618: 1605: 1599: 1597: 1596: 1591: 1589: 1581: 1558: 1551: 1540: 1538: 1537: 1532: 1481: 1470:a transition is 1467: 1463: 1461: 1460: 1455: 1428: 1424: 1422: 1421: 1416: 1389: 1385: 1364: 1362: 1361: 1356: 1354: 1353: 1335: 1333: 1332: 1327: 1294: 1292: 1291: 1286: 1281: 1280: 1242:marked Petri net 1228: 1226: 1225: 1220: 1218: 1187: 1185: 1184: 1179: 1132: 1131: 1107: 1105: 1104: 1099: 1049: 1048: 1043: 1026:of a transition 1010: 1008: 1007: 1002: 958: 956: 955: 950: 878:arc multiplicity 867: 865: 864: 859: 857: 764: 762: 761: 756: 677:in the image of 547: 507: 441: 368: 323: 321: 320: 315: 257: 253: 246: 244: 243: 238: 169:nondeterministic 165:execution policy 115:Petri net basics 110: 7997: 7996: 7992: 7991: 7990: 7988: 7987: 7986: 7942: 7941: 7908: 7886: 7864: 7843: 7824: 7805: 7742: 7682: 7663: 7602: 7586: 7584:Further reading 7581: 7574: 7550: 7546: 7515: 7511: 7496: 7482: 7478: 7471: 7445: 7438: 7431: 7405: 7398: 7386: 7385: 7376: 7375: 7371: 7345: 7341: 7304: 7300: 7277: 7250: 7246: 7231: 7204: 7200: 7193: 7169: 7165: 7138: 7134: 7127: 7103: 7099: 7064: 7060: 7053: 7031: 7027: 7012: 6988: 6984: 6969: 6945: 6941: 6912: 6908: 6877:: 18420–18425. 6863: 6859: 6852: 6826: 6822: 6815: 6788: 6784: 6729: 6725: 6718: 6692: 6688: 6681: 6665: 6658: 6648: 6646: 6610: 6592: 6588: 6580: 6569: 6563: 6559: 6548: 6544: 6517: 6510: 6503: 6480: 6474: 6470: 6461: 6459: 6455: 6414: 6408: 6399: 6392: 6370: 6366: 6359: 6337: 6333: 6324: 6322: 6313: 6312: 6308: 6299: 6297: 6288: 6287: 6283: 6252: 6248: 6241: 6219: 6215: 6192: 6188: 6181: 6158: 6149: 6145: 6138: 6122: 6115: 6086: 6079: 6070: 6068: 6055: 6054: 6050: 6041: 6039: 6037:10.1109/5.24143 6018: 6012: 6008: 5991: 5987: 5970: 5966: 5949: 5945: 5936: 5934: 5927: 5903: 5899: 5881: 5877: 5868: 5866: 5852: 5848: 5823: 5819: 5812: 5786: 5782: 5757: 5753: 5746: 5724: 5720: 5681: 5677: 5673: 5631: 5618:Software design 5593:Hardware design 5544: 5517:process algebra 5501: 5466:elementary path 5418: 5390: 5388: 5387: 5381: 5377: 5368: 5366: 5365: 5359: 5355: 5340: 5338: 5337: 5331: 5327: 5318: 5316: 5315: 5309: 5305: 5281: 5279: 5278: 5272: 5268: 5259: 5257: 5256: 5250: 5246: 5228: 5224: 5215: 5211: 5206: 5203: 5202: 5201:athematically, 5146: 5142: 5133: 5131: 5130: 5110: 5104: 5100: 5095: 5075: 5072: 5071: 5039: 5030: 5028: 5027: 5022: 5014: 5008: 5004: 4999: 4982: 4979: 4978: 4938: 4929: 4927: 4926: 4921: 4913: 4907: 4903: 4898: 4881: 4878: 4877: 4841: 4737:Turing complete 4693: 4677: 4610: 4590: 4587: 4586: 4566: 4562: 4539: 4536: 4535: 4515: 4511: 4482: 4479: 4478: 4394: 4365: 4361: 4359: 4356: 4355: 4322: 4315: 4312: 4309: 4308: 4307:-liveness, for 4291: 4287: 4285: 4282: 4281: 4258: 4254: 4252: 4249: 4248: 4225: 4221: 4207: 4204: 4203: 4186: 4182: 4180: 4177: 4176: 4155: 4151: 4149: 4146: 4145: 4139: 4122: 4118: 4116: 4113: 4112: 4108: 4091: 4087: 4085: 4082: 4081: 4060: 4056: 4042: 4039: 4038: 4034: 4030: 4013: 4009: 4007: 4004: 4003: 3982: 3978: 3964: 3961: 3960: 3939: 3935: 3933: 3930: 3929: 3908: 3904: 3890: 3887: 3886: 3862: 3858: 3856: 3853: 3852: 3831: 3827: 3825: 3822: 3821: 3801: 3797: 3786: 3783: 3782: 3765: 3761: 3752: 3748: 3746: 3743: 3742: 3721: 3717: 3715: 3712: 3711: 3694: 3690: 3688: 3685: 3684: 3660: 3657: 3656: 3639: 3635: 3633: 3630: 3629: 3622: 3552: 3549: 3548: 3533: 3513: 3490: 3484: 3481: 3474:needs expansion 3459: 3437: 3436: 3431: 3426: 3421: 3411: 3410: 3401: 3397: 3395: 3392: 3391: 3372: 3371: 3366: 3361: 3352: 3351: 3343: 3338: 3329: 3328: 3320: 3315: 3306: 3305: 3300: 3292: 3283: 3282: 3274: 3266: 3256: 3255: 3246: 3242: 3229: 3228: 3223: 3218: 3209: 3208: 3203: 3198: 3189: 3188: 3183: 3178: 3169: 3168: 3163: 3158: 3149: 3148: 3140: 3132: 3122: 3121: 3112: 3108: 3095: 3094: 3089: 3084: 3075: 3074: 3069: 3064: 3055: 3054: 3049: 3044: 3035: 3034: 3029: 3024: 3015: 3014: 3006: 2998: 2988: 2987: 2978: 2974: 2972: 2969: 2968: 2962: 2923: 2919: 2910: 2906: 2894: and 2892: 2881: 2840: 2837: 2836: 2829: 2804: 2801: 2800: 2796: 2776: 2772: 2763: 2759: 2747: 2743: 2741: 2738: 2737: 2675: 2671: 2654: 2651: 2650: 2633: 2629: 2627: 2624: 2623: 2569: 2565: 2548: 2545: 2544: 2527: 2523: 2521: 2518: 2517: 2495: 2487: 2485: 2482: 2481: 2465: 2457: 2455: 2452: 2451: 2424: 2416: 2414: 2411: 2410: 2386: 2382: 2359: 2356: 2355: 2352: 2280: 2277: 2276: 2261: 2231: 2228: 2227: 2210: 2206: 2197: 2193: 2186: 2181: 2169: 2165: 2150: 2146: 2137: 2133: 2126: 2121: 2115: 2111: 2109: 2106: 2105: 2085: 2081: 2072: 2068: 2051: 2050: 2048: 2045: 2044: 2027: 2023: 2021: 2018: 2017: 2013: 2010:firing sequence 1977: 1974: 1973: 1952: 1950: 1947: 1946: 1942: 1912: 1881: 1875: 1869: 1865: 1859: 1858: 1850: 1849: 1845: 1834: 1829: 1827: 1826: 1809: 1806: 1805: 1784: 1780: 1778: 1775: 1774: 1754: 1750: 1721: 1718: 1717: 1693: 1691: 1688: 1687: 1657: 1655: 1652: 1651: 1632: 1617: 1612: 1609: 1608: 1603: 1582: 1572: 1567: 1564: 1563: 1556: 1549: 1487: 1484: 1483: 1479: 1465: 1434: 1431: 1430: 1429:, and produces 1426: 1395: 1392: 1391: 1387: 1383: 1376: 1367:initial marking 1349: 1345: 1343: 1340: 1339: 1303: 1300: 1299: 1276: 1272: 1249: 1246: 1245: 1214: 1200: 1197: 1196: 1127: 1123: 1121: 1118: 1117: 1044: 1042: 1041: 1039: 1036: 1035: 978: 975: 974: 893: 890: 889: 853: 809: 806: 805: 732: 729: 728: 718:Petri net graph 714: 690: 683: 676: 650: 643: 636: 629: 622: 615: 604: 535: 495: 433: 360: 273: 270: 269: 266:, respectively. 255: 251: 208: 205: 204: 186:Petri nets are 184: 117: 103:Carl Adam Petri 99: 53:Carl Adam Petri 48:bipartite graph 17: 12: 11: 5: 7995: 7985: 7984: 7979: 7974: 7969: 7964: 7959: 7954: 7940: 7939: 7912: 7906: 7890: 7884: 7868: 7862: 7847: 7841: 7828: 7822: 7809: 7803: 7790: 7755: 7746: 7740: 7727: 7699:(3): 223–252. 7686: 7680: 7667: 7661: 7646: 7633: 7606: 7600: 7585: 7582: 7580: 7579: 7572: 7544: 7509: 7494: 7476: 7469: 7436: 7429: 7396: 7387:|journal= 7369: 7339: 7320:(4): 347–374. 7298: 7275: 7244: 7229: 7207:Cortadella, J. 7198: 7191: 7163: 7152:(3): 387–397. 7132: 7125: 7097: 7058: 7051: 7025: 7010: 6982: 6967: 6939: 6906: 6857: 6850: 6820: 6813: 6782: 6743:(9): e106035. 6723: 6716: 6686: 6679: 6656: 6586: 6583:on 2020-05-04. 6557: 6542: 6508: 6501: 6468: 6432:10.1.1.30.3125 6397: 6390: 6364: 6357: 6331: 6306: 6281: 6246: 6239: 6213: 6186: 6179: 6143: 6136: 6113: 6077: 6048: 6031:(4): 541–558. 6006: 5985: 5964: 5943: 5925: 5897: 5875: 5846: 5835:(2): 105–155. 5817: 5810: 5790:Ehrig, Hartmut 5780: 5751: 5744: 5718: 5674: 5672: 5669: 5668: 5667: 5662: 5657: 5652: 5647: 5642: 5637: 5630: 5627: 5626: 5625: 5620: 5615: 5610: 5605: 5600: 5595: 5590: 5585: 5580: 5575: 5570: 5565: 5560: 5555: 5550: 5543: 5540: 5500: 5497: 5417: 5414: 5413: 5412: 5401: 5398: 5393: 5384: 5380: 5376: 5371: 5362: 5358: 5354: 5351: 5348: 5343: 5334: 5330: 5326: 5321: 5312: 5308: 5304: 5301: 5298: 5295: 5292: 5289: 5284: 5275: 5271: 5267: 5262: 5253: 5249: 5245: 5242: 5239: 5236: 5231: 5227: 5223: 5218: 5214: 5210: 5187: 5180: 5169: 5166: 5163: 5160: 5157: 5154: 5149: 5145: 5141: 5136: 5129: 5126: 5123: 5120: 5117: 5113: 5107: 5103: 5098: 5094: 5091: 5088: 5085: 5082: 5079: 5060: 5049: 5046: 5042: 5038: 5033: 5025: 5021: 5017: 5011: 5007: 5002: 4998: 4995: 4992: 4989: 4986: 4959: 4948: 4945: 4941: 4937: 4932: 4924: 4920: 4916: 4910: 4906: 4901: 4897: 4894: 4891: 4888: 4885: 4874:nondeterminism 4840: 4837: 4832: 4831: 4817: 4792: 4781: 4769: 4765: 4742: 4741: 4740: 4729: 4692: 4689: 4681:control theory 4676: 4673: 4613: 4609: 4603: 4600: 4597: 4594: 4574: 4569: 4565: 4561: 4558: 4555: 4552: 4549: 4546: 4543: 4523: 4518: 4514: 4510: 4507: 4504: 4501: 4498: 4495: 4492: 4489: 4486: 4467:–Miller Tree. 4393: 4390: 4384:as a term for 4368: 4364: 4337: 4334: 4331: 4328: 4325: 4321: 4318: 4294: 4290: 4267: 4264: 4261: 4257: 4245: 4244: 4233: 4228: 4224: 4220: 4217: 4214: 4211: 4189: 4185: 4158: 4154: 4143: 4125: 4121: 4094: 4090: 4079: 4068: 4063: 4059: 4055: 4052: 4049: 4046: 4016: 4012: 4001: 3990: 3985: 3981: 3977: 3974: 3971: 3968: 3942: 3938: 3927: 3916: 3911: 3907: 3903: 3900: 3897: 3894: 3865: 3861: 3849:if and only if 3834: 3830: 3809: 3804: 3800: 3796: 3793: 3790: 3781:. A Petri net 3768: 3764: 3760: 3755: 3751: 3724: 3720: 3697: 3693: 3673: 3670: 3667: 3664: 3642: 3638: 3621: 3618: 3610:tableau method 3571: 3568: 3565: 3562: 3559: 3556: 3543:and a marking 3532: 3529: 3512: 3509: 3492: 3491: 3485:September 2022 3471: 3469: 3458: 3455: 3441: 3435: 3432: 3430: 3427: 3425: 3422: 3420: 3417: 3416: 3414: 3409: 3404: 3400: 3376: 3370: 3367: 3365: 3362: 3360: 3357: 3354: 3353: 3350: 3347: 3344: 3342: 3339: 3337: 3334: 3331: 3330: 3327: 3324: 3321: 3319: 3316: 3314: 3311: 3308: 3307: 3304: 3301: 3299: 3296: 3293: 3291: 3288: 3285: 3284: 3281: 3278: 3275: 3273: 3270: 3267: 3265: 3262: 3261: 3259: 3254: 3249: 3245: 3238: 3233: 3227: 3224: 3222: 3219: 3217: 3214: 3211: 3210: 3207: 3204: 3202: 3199: 3197: 3194: 3191: 3190: 3187: 3184: 3182: 3179: 3177: 3174: 3171: 3170: 3167: 3164: 3162: 3159: 3157: 3154: 3151: 3150: 3147: 3144: 3141: 3139: 3136: 3133: 3131: 3128: 3127: 3125: 3120: 3115: 3111: 3104: 3099: 3093: 3090: 3088: 3085: 3083: 3080: 3077: 3076: 3073: 3070: 3068: 3065: 3063: 3060: 3057: 3056: 3053: 3050: 3048: 3045: 3043: 3040: 3037: 3036: 3033: 3030: 3028: 3025: 3023: 3020: 3017: 3016: 3013: 3010: 3007: 3005: 3002: 2999: 2997: 2994: 2993: 2991: 2986: 2981: 2977: 2959: 2958: 2946: 2943: 2940: 2937: 2934: 2931: 2926: 2922: 2918: 2913: 2909: 2905: 2902: 2888: 2880: 2874: 2871: 2868: 2865: 2862: 2859: 2856: 2853: 2850: 2847: 2844: 2817: 2814: 2811: 2808: 2793: 2792: 2779: 2775: 2771: 2766: 2762: 2758: 2755: 2750: 2746: 2731: 2730: 2719: 2716: 2713: 2710: 2707: 2704: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2678: 2674: 2670: 2667: 2664: 2661: 2658: 2636: 2632: 2621: 2610: 2607: 2604: 2601: 2598: 2595: 2592: 2589: 2586: 2583: 2580: 2577: 2572: 2568: 2564: 2561: 2558: 2555: 2552: 2530: 2526: 2498: 2494: 2490: 2468: 2464: 2460: 2427: 2423: 2419: 2394: 2389: 2385: 2381: 2378: 2375: 2372: 2369: 2366: 2363: 2351: 2348: 2320: 2317: 2314: 2311: 2308: 2305: 2302: 2299: 2296: 2293: 2290: 2287: 2284: 2260: 2257: 2244: 2241: 2238: 2235: 2213: 2209: 2200: 2196: 2192: 2189: 2185: 2178: 2175: 2172: 2168: 2164: 2161: 2158: 2153: 2149: 2140: 2136: 2132: 2129: 2125: 2118: 2114: 2093: 2088: 2084: 2080: 2075: 2071: 2067: 2064: 2058: 2055: 2030: 2026: 1990: 1987: 1984: 1981: 1959: 1956: 1923: 1918: 1915: 1908: 1902: 1899: 1896: 1893: 1890: 1887: 1884: 1879: 1872: 1868: 1862: 1856: 1853: 1848: 1837: 1832: 1822: 1819: 1816: 1813: 1800:. Its set of 1787: 1783: 1762: 1757: 1753: 1749: 1746: 1743: 1740: 1737: 1734: 1731: 1728: 1725: 1700: 1697: 1669: 1665: 1662: 1638: 1635: 1629: 1625: 1622: 1616: 1588: 1585: 1579: 1576: 1571: 1543: 1542: 1530: 1527: 1524: 1521: 1518: 1515: 1512: 1509: 1506: 1503: 1500: 1497: 1494: 1491: 1468: 1453: 1450: 1447: 1444: 1441: 1438: 1414: 1411: 1408: 1405: 1402: 1399: 1375: 1372: 1371: 1370: 1352: 1348: 1337: 1325: 1322: 1319: 1316: 1313: 1310: 1307: 1284: 1279: 1275: 1271: 1268: 1265: 1262: 1259: 1256: 1253: 1217: 1213: 1210: 1207: 1204: 1177: 1174: 1171: 1168: 1165: 1162: 1159: 1156: 1153: 1150: 1147: 1144: 1141: 1138: 1135: 1130: 1126: 1097: 1094: 1091: 1088: 1085: 1082: 1079: 1076: 1073: 1070: 1067: 1064: 1061: 1058: 1055: 1052: 1047: 1000: 997: 994: 991: 988: 985: 982: 972:directed graph 948: 945: 942: 939: 936: 933: 930: 927: 924: 921: 918: 915: 912: 909: 906: 903: 900: 897: 882: 881: 856: 852: 849: 846: 843: 840: 837: 834: 831: 828: 825: 822: 819: 816: 813: 803: 789: 780: 754: 751: 748: 745: 742: 739: 736: 713: 710: 688: 681: 674: 666:of transition 648: 641: 634: 627: 620: 613: 602: 558: 557: 533: 493: 448: 447: 427: 393:elementary net 330: 329: 313: 310: 307: 304: 301: 298: 295: 292: 289: 286: 283: 280: 277: 267: 236: 233: 230: 227: 224: 221: 218: 215: 212: 183: 180: 116: 113: 98: 95: 15: 9: 6: 4: 3: 2: 7994: 7983: 7980: 7978: 7975: 7973: 7970: 7968: 7965: 7963: 7960: 7958: 7955: 7953: 7950: 7949: 7947: 7935: 7930: 7926: 7922: 7918: 7913: 7909: 7903: 7899: 7895: 7894:Zhou, Mengchu 7891: 7887: 7881: 7877: 7873: 7872:Zhou, Mengchu 7869: 7865: 7859: 7855: 7854: 7848: 7844: 7838: 7834: 7829: 7825: 7819: 7815: 7810: 7806: 7800: 7796: 7791: 7786: 7781: 7777: 7773: 7769: 7765: 7761: 7756: 7752: 7747: 7743: 7737: 7733: 7728: 7724: 7720: 7715: 7710: 7706: 7702: 7698: 7694: 7693: 7687: 7683: 7677: 7673: 7668: 7664: 7658: 7654: 7653: 7647: 7644:(12a): 47–50. 7643: 7639: 7634: 7629: 7624: 7620: 7616: 7612: 7607: 7603: 7597: 7593: 7588: 7587: 7575: 7569: 7565: 7561: 7557: 7556: 7548: 7540: 7536: 7532: 7528: 7524: 7520: 7513: 7505: 7501: 7497: 7491: 7487: 7480: 7472: 7466: 7462: 7458: 7454: 7450: 7443: 7441: 7432: 7426: 7422: 7418: 7414: 7410: 7403: 7401: 7392: 7380: 7372: 7366: 7362: 7358: 7354: 7350: 7343: 7335: 7331: 7327: 7323: 7319: 7315: 7314: 7309: 7302: 7294: 7290: 7286: 7282: 7278: 7272: 7268: 7264: 7260: 7259: 7254: 7248: 7240: 7236: 7232: 7226: 7222: 7218: 7214: 7213: 7208: 7202: 7194: 7188: 7184: 7180: 7176: 7175: 7167: 7159: 7155: 7151: 7147: 7143: 7136: 7128: 7122: 7117: 7112: 7108: 7101: 7093: 7089: 7085: 7081: 7077: 7073: 7069: 7062: 7054: 7048: 7044: 7040: 7036: 7029: 7021: 7017: 7013: 7007: 7003: 6999: 6995: 6994: 6986: 6978: 6974: 6970: 6964: 6960: 6956: 6952: 6951: 6943: 6934: 6929: 6925: 6921: 6917: 6910: 6902: 6898: 6893: 6888: 6884: 6880: 6876: 6872: 6868: 6861: 6853: 6847: 6843: 6839: 6835: 6831: 6824: 6816: 6810: 6806: 6802: 6798: 6797: 6792: 6786: 6778: 6774: 6769: 6764: 6759: 6754: 6750: 6746: 6742: 6738: 6734: 6727: 6719: 6713: 6709: 6705: 6701: 6697: 6690: 6682: 6680:9780262015387 6676: 6672: 6671: 6663: 6661: 6644: 6640: 6636: 6632: 6628: 6624: 6620: 6614: 6609: 6605: 6602:(in German). 6601: 6597: 6590: 6579: 6575: 6568: 6561: 6553: 6546: 6538: 6534: 6530: 6526: 6522: 6515: 6513: 6504: 6502:3-540-44919-1 6498: 6494: 6490: 6486: 6479: 6472: 6458:on 2016-11-19 6454: 6450: 6446: 6442: 6438: 6433: 6428: 6424: 6420: 6413: 6406: 6404: 6402: 6393: 6391:0-7803-7080-5 6387: 6383: 6379: 6375: 6368: 6360: 6358:0-7803-7087-2 6354: 6350: 6346: 6342: 6335: 6321:on 2005-11-03 6320: 6316: 6310: 6296:on 2010-10-28 6295: 6291: 6285: 6277: 6273: 6269: 6265: 6261: 6257: 6250: 6242: 6240:3-540-68681-9 6236: 6232: 6228: 6224: 6217: 6209: 6205: 6202:(1): 85–104. 6201: 6197: 6190: 6182: 6176: 6172: 6168: 6164: 6157: 6153: 6147: 6139: 6133: 6129: 6128: 6120: 6118: 6108: 6103: 6099: 6095: 6091: 6084: 6082: 6067:on 2011-09-27 6066: 6062: 6058: 6052: 6038: 6034: 6030: 6026: 6025: 6017: 6010: 6001: 5996: 5989: 5980: 5975: 5968: 5959: 5954: 5947: 5933:on 2012-02-09 5932: 5928: 5926:3-540-31882-8 5922: 5918: 5914: 5910: 5909: 5901: 5893: 5889: 5885: 5879: 5864: 5863: 5857: 5850: 5842: 5838: 5834: 5830: 5829: 5821: 5813: 5811:9783540430674 5807: 5803: 5799: 5795: 5791: 5784: 5776: 5772: 5768: 5764: 5763: 5755: 5747: 5745:3-540-65306-6 5741: 5737: 5733: 5729: 5722: 5713: 5708: 5704: 5700: 5696: 5692: 5691: 5686: 5679: 5675: 5666: 5663: 5661: 5658: 5656: 5653: 5651: 5648: 5646: 5643: 5641: 5638: 5636: 5633: 5632: 5624: 5621: 5619: 5616: 5614: 5611: 5609: 5606: 5604: 5601: 5599: 5596: 5594: 5591: 5589: 5586: 5584: 5581: 5579: 5576: 5574: 5573:Data analysis 5571: 5569: 5566: 5564: 5561: 5559: 5556: 5554: 5551: 5549: 5546: 5545: 5539: 5536: 5534: 5530: 5526: 5522: 5518: 5514: 5510: 5506: 5496: 5494: 5490: 5485: 5481: 5477: 5474: 5469: 5467: 5463: 5462: 5456: 5454: 5450: 5446: 5442: 5438: 5434: 5429: 5426: 5422: 5421:Workflow nets 5416:Workflow nets 5391: 5382: 5378: 5374: 5369: 5360: 5356: 5349: 5341: 5332: 5328: 5324: 5319: 5310: 5306: 5287: 5282: 5273: 5269: 5265: 5260: 5251: 5247: 5240: 5237: 5234: 5229: 5225: 5221: 5216: 5212: 5200: 5196: 5192: 5188: 5185: 5181: 5161: 5155: 5147: 5143: 5134: 5124: 5118: 5115: 5105: 5101: 5089: 5086: 5083: 5080: 5069: 5065: 5061: 5047: 5044: 5036: 5031: 5019: 5009: 5005: 4996: 4993: 4990: 4987: 4976: 4972: 4968: 4964: 4960: 4946: 4943: 4935: 4930: 4918: 4908: 4904: 4895: 4892: 4889: 4886: 4875: 4871: 4867: 4863: 4859: 4858:state machine 4855: 4854: 4853: 4845: 4836: 4829: 4825: 4821: 4818: 4815: 4811: 4807: 4803: 4798: 4793: 4790: 4785: 4782: 4778: 4774: 4770: 4766: 4763: 4759: 4755: 4751: 4747: 4743: 4738: 4734: 4733:inhibitor arc 4730: 4727: 4723: 4722: 4720: 4719: 4718: 4715: 4713: 4709: 4705: 4700: 4698: 4688: 4686: 4682: 4672: 4670: 4666: 4658: 4653: 4649: 4647: 4643: 4635: 4630: 4626: 4607: 4598: 4595: 4592: 4567: 4563: 4559: 4556: 4553: 4550: 4547: 4544: 4516: 4512: 4508: 4505: 4502: 4499: 4496: 4493: 4490: 4487: 4476: 4471: 4468: 4466: 4462: 4457: 4454: 4452: 4448: 4444: 4440: 4435: 4433: 4429: 4425: 4424: 4419: 4415: 4411: 4403: 4398: 4389: 4387: 4383: 4366: 4362: 4352: 4335: 4332: 4329: 4326: 4323: 4319: 4316: 4292: 4288: 4265: 4262: 4259: 4255: 4226: 4222: 4218: 4215: 4209: 4187: 4183: 4174: 4156: 4152: 4144: 4123: 4119: 4092: 4088: 4080: 4061: 4057: 4053: 4050: 4044: 4014: 4010: 4002: 3983: 3979: 3975: 3972: 3966: 3958: 3940: 3936: 3928: 3909: 3905: 3901: 3898: 3892: 3884: 3881: 3880: 3879: 3863: 3859: 3850: 3832: 3828: 3802: 3798: 3794: 3791: 3766: 3762: 3758: 3753: 3749: 3722: 3718: 3695: 3691: 3671: 3668: 3665: 3662: 3640: 3636: 3626: 3617: 3615: 3611: 3607: 3602: 3600: 3596: 3592: 3587: 3583: 3566: 3560: 3557: 3554: 3546: 3542: 3538: 3528: 3526: 3522: 3517: 3508: 3506: 3502: 3498: 3488: 3479: 3475: 3472:This section 3470: 3467: 3463: 3462: 3454: 3439: 3433: 3428: 3423: 3418: 3412: 3407: 3402: 3398: 3389: 3374: 3368: 3363: 3358: 3355: 3348: 3345: 3340: 3335: 3332: 3325: 3322: 3317: 3312: 3309: 3302: 3297: 3294: 3289: 3286: 3279: 3276: 3271: 3268: 3263: 3257: 3252: 3247: 3243: 3236: 3231: 3225: 3220: 3215: 3212: 3205: 3200: 3195: 3192: 3185: 3180: 3175: 3172: 3165: 3160: 3155: 3152: 3145: 3142: 3137: 3134: 3129: 3123: 3118: 3113: 3109: 3102: 3097: 3091: 3086: 3081: 3078: 3071: 3066: 3061: 3058: 3051: 3046: 3041: 3038: 3031: 3026: 3021: 3018: 3011: 3008: 3003: 3000: 2995: 2989: 2984: 2979: 2975: 2966: 2938: 2932: 2929: 2924: 2920: 2916: 2911: 2907: 2903: 2900: 2886: 2878: 2872: 2869: 2863: 2860: 2854: 2848: 2842: 2835: 2834: 2833: 2812: 2806: 2777: 2773: 2769: 2764: 2760: 2756: 2753: 2748: 2744: 2736: 2735: 2734: 2717: 2711: 2708: 2705: 2699: 2696: 2690: 2687: 2684: 2676: 2672: 2668: 2665: 2662: 2659: 2649:, defined by 2634: 2630: 2622: 2605: 2602: 2599: 2593: 2590: 2584: 2581: 2578: 2570: 2566: 2562: 2559: 2556: 2553: 2543:, defined by 2528: 2524: 2516: 2515: 2514: 2512: 2492: 2462: 2444: 2440: 2421: 2408: 2387: 2383: 2379: 2376: 2373: 2370: 2367: 2364: 2347: 2345: 2341: 2336: 2334: 2315: 2312: 2309: 2303: 2297: 2294: 2291: 2285: 2282: 2274: 2273:flow relation 2270: 2266: 2256: 2239: 2233: 2211: 2207: 2198: 2194: 2190: 2187: 2176: 2173: 2170: 2166: 2162: 2159: 2156: 2151: 2147: 2138: 2134: 2130: 2127: 2116: 2112: 2086: 2082: 2078: 2073: 2069: 2062: 2053: 2028: 2024: 2011: 2006: 2004: 2001:. It is the 1985: 1979: 1957: 1940: 1935: 1921: 1916: 1913: 1906: 1897: 1894: 1891: 1888: 1885: 1877: 1870: 1866: 1854: 1851: 1846: 1835: 1830: 1817: 1811: 1803: 1785: 1781: 1755: 1751: 1747: 1744: 1741: 1738: 1735: 1732: 1726: 1723: 1714: 1698: 1685: 1667: 1663: 1636: 1633: 1627: 1623: 1614: 1606: 1586: 1583: 1577: 1569: 1561: 1554: 1546: 1525: 1522: 1519: 1513: 1510: 1504: 1498: 1495: 1492: 1477: 1473: 1469: 1448: 1445: 1442: 1436: 1409: 1406: 1403: 1397: 1386:in a marking 1381: 1380: 1379: 1368: 1350: 1346: 1338: 1320: 1317: 1314: 1311: 1308: 1298: 1297: 1296: 1277: 1273: 1269: 1266: 1263: 1260: 1257: 1254: 1243: 1239: 1234: 1232: 1208: 1205: 1202: 1194: 1189: 1172: 1169: 1163: 1160: 1157: 1151: 1148: 1145: 1142: 1139: 1133: 1128: 1124: 1115: 1114:output places 1111: 1092: 1089: 1083: 1080: 1077: 1071: 1068: 1065: 1062: 1059: 1053: 1050: 1045: 1033: 1029: 1025: 1020: 1018: 1014: 995: 992: 989: 986: 983: 973: 970: 966: 962: 943: 940: 934: 931: 928: 922: 919: 913: 910: 907: 898: 895: 887: 886:flow relation 879: 875: 871: 844: 841: 838: 832: 826: 823: 820: 814: 811: 804: 801: 797: 793: 790: 788: 784: 781: 779: 775: 771: 768: 767: 766: 749: 746: 743: 740: 737: 727: 723: 719: 709: 707: 706:Peterson 1981 702: 700: 696: 692: 687: 680: 673: 669: 665: 661: 657: 653: 647: 640: 633: 626: 619: 612: 608: 601: 596: 594: 590: 584: 582: 578: 575: 571: 567: 563: 555: 551: 546: 542: 538: 534: 531: 527: 526:configuration 523: 519: 518:countable set 515: 511: 506: 502: 498: 494: 491: 487: 483: 479: 476: 475: 474: 472: 468: 464: 460: 456: 452: 451:Definition 4. 445: 444:configuration 440: 436: 432:is such that 431: 428: 425: 421: 417: 413: 410: 409: 408: 406: 402: 398: 394: 390: 389:Definition 3. 382: 374: 370: 367: 363: 358: 354: 353:configuration 350: 346: 342: 338: 334: 333:Definition 2. 327: 308: 305: 302: 296: 290: 287: 284: 278: 275: 268: 265: 261: 250: 249: 248: 231: 228: 225: 222: 219: 213: 210: 203: 199: 195: 194:Definition 1. 191: 189: 179: 177: 172: 170: 166: 161: 159: 155: 151: 147: 142: 140: 139:output places 136: 132: 131: 126: 122: 112: 109: 104: 90: 86: 84: 80: 76: 72: 68: 64: 61: 56: 54: 49: 45: 41: 37: 34: 30: 26: 22: 7924: 7920: 7897: 7875: 7852: 7832: 7813: 7794: 7767: 7764:Scholarpedia 7763: 7750: 7731: 7696: 7690: 7671: 7651: 7641: 7637: 7618: 7614: 7591: 7554: 7547: 7522: 7512: 7485: 7479: 7452: 7412: 7352: 7342: 7317: 7311: 7301: 7257: 7247: 7211: 7201: 7173: 7166: 7149: 7145: 7135: 7106: 7100: 7075: 7071: 7061: 7034: 7028: 6996:. Springer. 6992: 6985: 6953:. Springer. 6949: 6942: 6926:(21): 7662. 6923: 6919: 6909: 6874: 6870: 6860: 6833: 6823: 6795: 6785: 6740: 6736: 6726: 6699: 6689: 6669: 6647:. Retrieved 6603: 6599: 6589: 6578:the original 6573: 6560: 6551: 6545: 6520: 6484: 6471: 6460:. Retrieved 6453:the original 6425:(1): 21–66. 6422: 6418: 6373: 6367: 6340: 6334: 6323:. Retrieved 6319:the original 6309: 6298:. Retrieved 6294:the original 6284: 6259: 6255: 6249: 6222: 6216: 6199: 6195: 6189: 6162: 6152:Jensen, Kurt 6146: 6130:. Springer. 6126: 6100:(15): 5027. 6097: 6093: 6069:. Retrieved 6065:the original 6060: 6057:"Petri Nets" 6051: 6040:. Retrieved 6028: 6022: 6009: 5988: 5967: 5946: 5935:. Retrieved 5931:the original 5907: 5900: 5891: 5878: 5867:. Retrieved 5859: 5849: 5832: 5826: 5820: 5793: 5783: 5766: 5760: 5754: 5727: 5721: 5694: 5690:Scholarpedia 5688: 5678: 5537: 5525:trace theory 5502: 5492: 5486: 5482: 5478: 5473:well-handled 5472: 5470: 5465: 5459: 5457: 5452: 5448: 5444: 5440: 5436: 5432: 5430: 5419: 5198: 5194: 5190: 5183: 5067: 5063: 4975:concurrency: 4974: 4970: 4966: 4963:marked graph 4869: 4865: 4861: 4850: 4839:Restrictions 4833: 4823: 4814:Markov chain 4788: 4753: 4732: 4725: 4716: 4701: 4694: 4678: 4668: 4664: 4662: 4656: 4645: 4641: 4639: 4633: 4474: 4472: 4469: 4458: 4455: 4450: 4446: 4442: 4438: 4436: 4431: 4427: 4421: 4417: 4413: 4409: 4407: 4401: 4385: 4381: 4353: 4246: 4172: 3956: 3882: 3740: 3603: 3588: 3584: 3544: 3540: 3534: 3531:Reachability 3518: 3514: 3504: 3495: 3482: 3478:adding to it 3473: 3390: 2967: 2960: 2794: 2732: 2449: 2353: 2343: 2339: 2337: 2272: 2268: 2262: 2009: 2007: 2005:of the net. 1938: 1936: 1801: 1715: 1601: 1559: 1552: 1547: 1544: 1475: 1471: 1377: 1366: 1241: 1237: 1235: 1230: 1192: 1190: 1113: 1109: 1032:input places 1031: 1027: 1023: 1021: 1016: 1012: 964: 960: 885: 883: 877: 795: 791: 786: 782: 777: 769: 721: 717: 715: 703: 698: 694: 693: 685: 678: 671: 667: 663: 659: 655: 651: 645: 638: 631: 624: 617: 610: 606: 599: 597: 592: 588: 585: 580: 576: 569: 565: 561: 559: 554:multiplicity 553: 544: 540: 536: 529: 525: 521: 513: 504: 500: 496: 489: 485: 481: 477: 470: 466: 462: 458: 454: 450: 449: 443: 438: 434: 429: 423: 419: 415: 411: 404: 400: 396: 392: 388: 387: 365: 361: 356: 352: 348: 344: 340: 336: 335:Given a net 332: 331: 325: 263: 259: 197: 193: 192: 185: 173: 164: 162: 157: 153: 149: 145: 143: 138: 135:input places 134: 128: 124: 120: 118: 100: 57: 33:mathematical 28: 24: 20: 18: 7770:(4): 6477. 7760:"Petri net" 6871:IEEE Access 6617:: 226–233. 6611: [ 5769:(1): 1–34. 5697:(4): 6477. 5685:"Petri net" 5650:Petriscript 5588:Game theory 5521:actor model 5458:A directed 5064:free choice 4866:concurrency 4392:Boundedness 2003:state space 1804:is the set 1560:in one step 963:instead of 787:transitions 654:transition 508:is a place 492:) is a net. 426:) is a net. 264:transitions 125:transitions 7972:Petri nets 7946:Categories 7927:(5): 983. 6649:2017-10-16 6462:2015-04-02 6325:2006-01-06 6300:2007-08-22 6071:2011-04-13 6042:2024-05-26 6000:2104.13866 5979:2104.12695 5958:1809.07115 5937:2008-07-10 5884:Lipton, R. 5869:2014-05-14 5671:References 5613:Simulation 5533:modularity 4702:The term 4691:Extensions 4441:-bounded, 3820:is called 3595:ELEMENTARY 3547:, whether 3525:complexity 2344:extensions 2340:capacities 2104:such that 1555:a marking 774:finite set 548:is an arc 176:concurrent 163:Unless an 7539:245057775 7504:862121371 7389:ignored ( 7379:cite book 7285:0302-9743 7239:1437-0387 7158:1641-876X 6901:210994447 6791:Koch, Ina 6654:(8 pages) 6631:0178-2312 6537:872760679 6529:0105-8517 6449:248401501 6427:CiteSeerX 6262:: 47–58. 5433:soundness 5392:∙ 5375:⊆ 5370:∙ 5350:∨ 5342:∙ 5325:⊆ 5320:∙ 5297:→ 5291:∅ 5288:≠ 5283:∙ 5266:∩ 5261:∙ 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