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Petri nets refresher

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. Elements. Rules. Connections are directed.

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Petri nets refresher

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  1. Petri netsrefresher 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. Elements

  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.

  4. Enabled • A transition is enabled if each of its input places contains at least one token. enabled Not enabled Not enabled

  5. 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

  6. Play “Token Game” • In the new state, make_picture is enabled. It will fire, etc.

  7. 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).

  8. Non-determinism Two transitions are enabled but only one can fire

  9. Example: Single traffic light

  10. Two traffic lights OR

  11. Problem

  12. Solution How to make them alternate?

  13. 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

  14. 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.

  15. 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.

  16. 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.)

  17. 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.

  18. Modeling problem

  19. Modeling problem (1): Zero testing • Transition t should fire if place p is empty. ? t p

  20. Solution • Only works if place is N-bounded t Initially there are N tokens N input and output arcs p’ p

  21. Modeling problem (2): Priority • Transition t1 has priority over t2 t1 ? t2 Hint: similar to Zero testing!

  22. 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.

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