1. /faculteit technologie management PN-1 Petri nets refresher Prof.dr.ir. Wil van der Aalst Eindhoven University of Technology, Faculty of Technology Management, Department of Information and Technology, P.O.Box 513, NL-5600 MB, Eindhoven, The Netherlands.

2. /faculteit technologie management PN-2 Elements

3. /faculteit technologie management PN-3 Rules Connections are directed. No connections between two places or two transitions. Places may hold zero or more tokens. First, we consider the case of at most one arc between two nodes.

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5. /faculteit technologie management PN-5 Enabled A transition is enabled if each of its input places contains at least one token. enabled Not enabled

6. /faculteit technologie management PN-6 Firing An enabled transition can fire (i.e., it occurs). When it fires it consumes a token from each input place and produces a token for each output place. fired

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8. /faculteit technologie management PN-8 Play “Token Game” In the new state, make_picture is enabled. It will fire, etc.

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13. /faculteit technologie management PN-13 Remarks Firing is atomic. Multiple transitions may be enabled, but only one fires at a time, i.e., we assume interleaving semantics (cf. diamond rule). The number of tokens may vary if there are transitions for which the number of input places is not equal to the number of output places. The network is static. The state is represented by the distribution of tokens over places (also referred to as marking).

14. /faculteit technologie management PN-14 Non-determinism Two transitions are enabled but only one can fire

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16. /faculteit technologie management PN-16 Example: Single traffic light

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18. /faculteit technologie management PN-18 Two traffic lights OR

19. /faculteit technologie management PN-19 Problem

20. /faculteit technologie management PN-20 Solution How to make them alternate?

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22. /faculteit technologie management PN-22 Playing the “Token Game” on the Internet Applet to build your own Petri nets and execute them: http://is.tm.tue.nl/staff/wvdaalst/workflowcourse/ pn_applet/pn_applet.htm FLASH animations: www.workflowcourse.com

23. /faculteit technologie management PN-23 Exercise: Train system (1) Consider a circular railroad system with 4 (one- way) tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we do not care about the identities of the two trains.

24. /faculteit technologie management PN-24 Exercise: Train system (2) Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we want to distinguish the two trains.

25. /faculteit technologie management PN-25 Exercise: Train system (3) Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time. Moreover the next track should also be free to allow for a safe distance. (We do not care about train identities.)

26. /faculteit technologie management PN-26 Exercise: Train system (4) Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains. Tracks are free, busy or claimed. Trains need to claim the next track before entering.

27. /faculteit technologie management PN-27 Modeling problem

28. /faculteit technologie management PN-28 Modeling problem (1): Zero testing Transition t should fire if place p is empty. t ? p

29. /faculteit technologie management PN-29 Solution Only works if place is N-bounded t N input and output arcs Initially there are N tokens p p’

30. /faculteit technologie management PN-30 Modeling problem (2): Priority Transition t1 has priority over t2 t1 t2 ? Hint: similar to Zero testing!

31. /faculteit technologie management PN-31 A bit of theory Extensions have been proposed to tackle these problems, e.g., inhibitor arcs. These extensions extend the modeling power (Turing completeness*). Without such an extension not Turing complete. Still certain questions are difficult/expensive to answer or even undecidable (e.g., equivalence of two nets). * Turing completeness corresponds to the ability to execute any computation.