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Semantics of Petri Nets: A Comparison

Semantics of Petri Nets: A Comparison. Gabriel Juhás. WSC2007 , Washington, DC , 10th December 2007. Road plan: Present different semantics of Petri nets. Road plan: Present different semantics of Petri nets Associate each of them with a set of labelled partial order (LPOs).

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Semantics of Petri Nets: A Comparison

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  1. Semantics of Petri Nets: A Comparison Gabriel Juhás WSC2007, Washington, DC, 10th December 2007

  2. Road plan: Present different semantics of Petri nets

  3. Road plan: Present different semantics of Petri nets Associate each of them with a set of labelled partial order (LPOs)

  4. Road plan: Present different semantics of Petri nets Associate each of them with a set of labelled partial order (LPOs) Given two semantics associated with sets of LPOs A and B, investigate:

  5. Road plan: Present different semantics of Petri nets Associate each of them with a set of labelled partial order (LPOs) Given two semantics associated with sets of LPOs A and B, investigate: whether B  A

  6. Road plan: Present different semantics of Petri nets Associate each of them with a set of labelled partial order (LPOs) Given two semantics associated with sets of LPOs A and B, investigate: whether B  A whether each LPO from A is sequentialzation of an LPO from B (A B)

  7. Road plan: Sequential semantics: occurrence sequences

  8. Road plan: Sequential semantics: occurrence sequences Partial order based semantics: step semantics enabled labelled partial orders processes and runs

  9. Road plan: Sequential semantics: occurrence sequences Partial order based semantics: step semantics enabled labelled partial orders processes and runs Relationships

  10. Road plan: Sequential semantics: occurrence sequences Partial order based semantics: step semantics enabled labelled partial orders processes and runs Relationships Commutative Processes: Equivalence over sequences and processes

  11. Road plan: Sequential semantics: occurrence sequences Partial order based semantics: step semantics enabled labelled partial orders processes and runs Relationships Commutative Processes: Equivalence over sequences and processes Algebraic semantics: process terms and their equivalence classes Relationships

  12. Place/transition nets: p1 p2 a b 2 2 p3 2 c p4

  13. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a b 2 2 p3 2 c p4

  14. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a a a b 2 2 p3 2 c p4

  15. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a a a b 2 2 p3 2 c p4

  16. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a ab a b 2 2 b p3 2 c p4

  17. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a ab a b 2 2 b p3 2 c p4

  18. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a abc a b 2 2 b p3 2 c c p4

  19. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a abc a b 2 2 b p3 2 c c p4

  20. Place/transition nets: Sequential semantics: occurrence sequences p1 p2 a abcc a b 2 2 b p3 2 c c p4 c

  21. Place/transition nets: Step semantics: step sequences p1 p2 a b 2 2 p3 2 c p4

  22. Place/transition nets: Step semantics: step sequences p1 p2 a b 2 2 p3 2 c p4

  23. Place/transition nets: Step semantics: step sequences p1 p2 a b (a+b) a b 2 2 p3 2 c p4

  24. Place/transition nets: Step semantics: step sequences p1 p2 a b (a+b) a b 2 2 p3 2 c 2x p4

  25. Place/transition nets: Step semantics: step sequences p1 p2 a b (a+b) (2c) a b 2 2 p3 2 c c c p4

  26. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  27. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  28. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  29. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  30. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  31. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  32. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  33. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  34. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  35. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  36. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c p4

  37. An LPO is enabled iff for each cut of the LPO holds: When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled p1 p2 a b a b 2 2 p3 2 c c c 2x p4

  38. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 a b 2 2 p3 2 c p4

  39. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 a a b 2 2 a a p3 2 c p3 p3 p3 p3 p4

  40. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 a a b 2 2 a a p3 2 c p3 p3 p3 p3 p4

  41. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 ab a b 2 2 a a b b p3 2 c p3 p3 p3 p3 p3 p3 p3 p3 p4

  42. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 ab a b 2 2 a a b b p3 2 c p3 p3 p3 p3 p3 p3 p3 p3 p4

  43. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 abc a b 2 2 a a b b p3 2 c p3 p3 p3 p3 p3 p3 p3 p3 c c p4 p4 p4

  44. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 abc a b 2 2 a a b b p3 2 c p3 p3 p3 p3 p3 p3 p3 p3 c c p4 p4 p4

  45. Place/transition nets: Sequential executions: firing sequences Causal Structures: processes p1 p2 Process 1 Process 2 p1 p1 p2 p2 abcc a b 2 2 a a b b p3 2 c p3 p3 p3 p3 p3 p3 p3 p3 c c c c p4 p4 p4 p4 p4

  46. Process 1 Process 2 p1 p2 abcc p1 p1 p2 p2 a a b a b a b 2 2 p3 p3 p3 p3 p3 p3 p3 p3 p3 b 2 c c c c c c p4 p4 p4 p4 p4 Run 1 c Run 2 a b a b c c c c

  47. SEQ  STEPSEQ  ENABLED RUN  ENABLED SEQ  STEPSEQ  ENABLED  RUN Consequence MINENABLED = MINRUN

  48. Process 1 Process 2 p1 p1 p2 p2 p1 p2 a b a b a b p3 p3 p3 p3 p3 p3 p3 p3 2 2 c c p3 c c 2 c p4 p4 p4 p4 Run 1 p4 Run 2 a b a b c c c c abcc acbc

  49. Finest equivalence on sequences and processes relating them one-to-one: Exchange equivalence on sequences 0* Swapping equivalence on processes 1* p1 p2 a b 2 2 p3 2 Process 1 1 Process 2 p1 c p1 p2 p2 p4 a b a b abcc 0 bacc p3 p3 p3 p3 p3 p3 p3 p3 (a+b)cc c c c c p4 p4 p4 p4

  50. Algebraic semantics: process terms. p1 p2 pre(t) p1 p2 2p3 a b 2 2 t a b c p3 2 c p4 post(t) 2p3 2p3 p4

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