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Towards a n Efficient Algorithm for Unfolding Petri Nets

This research focuses on developing an efficient algorithm for unfolding Petri nets to address the state space explosion problem, emphasizing model checking. The ERV unfolding algorithm is detailed and its approach illustrated with examples. Results show improvements for small, medium, and large presets.

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Towards a n Efficient Algorithm for Unfolding Petri Nets

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  1. Towards an Efficient Algorithm for Unfolding Petri Nets Victor Khomenko andMaciejKoutny Department of Computing Science University of Newcastleupon Tyne

  2. Motivation • Partial order semantics of Petri nets • Alleviate the state space explosion problem • Efficient model checking algorithms

  3. The ERV unfolding algorithm Unfplaces from M0 pe transitions enabled by M0 cut-off  while pe extract eminpe if e is a cut-off event then cut-off  cut-off  {e} else add e and its postset into Unf UpdatePotExt(pe,Unf,e) add cut-off events and their postsets into Unf

  4. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6

  5. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 P5 P3 P1

  6. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 P6 P5 T3 P3 P1

  7. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 T4 P6 P5 P5 T3 P4 P3 T2 P1

  8. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 T4 P6 P5 P5 T3 P4 P3 P3 T2 T1 P1 P2

  9. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 T4 P6 P6 P5 P5 T3 T3 P4 P3 P3 T2 T1 P1 P2

  10. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 T4 P6 P6 P5 P5 T3 T3 P4 P3 P3 T2 T1 P1 P2 P1

  11. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 T4 T4 P6 P6 P5 P5 P5 T3 T3 P4 P3 P3 P4 T2 T1 P1 P2 P1

  12. P1 P3 P5 · · · T1 T2 T3 T4 P2 P4 P6 T4 T4 T4 P6 P6 P5 P5 P5 P6 T3 T3 P4 P3 P3 P4 T2 T1 P1 P2 P1

  13. The ERV unfolding algorithm Unfplaces from M0 pe transitions enabled by M0 cut-off  while pe extract eminpe if e is a cut-off event then cut-off  cut-off  {e} else add e and its postset into Unf UpdatePotExt(pe,Unf,e) add cut-off events and their postsets into Unf

  14. P7 T6 P6 T5 P5 T4 P4 T3 P3 T2 P2 T1 P1

  15. P7 T6 P6 T5 P5 T4 P4 T3 P3 T2 P2 {P1} T1 P1

  16. P7 T6 P6 T5 P5 T4 P4 T3 P3 T2 P2 {P1} T1 P1

  17. P7 T6 P6 T5 P5 T4 P4  T3 P3 T2 P2 {P1} T1 P1

  18. P7 T6 P6 T5 P5 {P5} T4 P4  T3 P3 T2 P2 {P1} T1 P1

  19. P7 T6 P6 T5 {P5,P6} P5 {P5} T4 P4  T3 P3 T2 P2 {P1} T1 P1

  20. P7 T6 {P5,P7} P6 T5 {P5,P6} P5 {P5} T4 P4  T3 P3 T2 P2 {P1} T1 P1

  21. T1:{P1} T3: T4:{P5} T5:{P5,P6} T6:{P5,P7} Preset Trees T5:{P6} T6:{P7} T1:{P1} T4:{P5} T3: Weight = || + |{P1}| + |{P5}| + |{P6}| + |{P7}| = 4

  22. T1:{P1} T3: T4:{P5} T5:{P5,P6} T6:{P5,P7} Preset Trees T1:{P1} T4:{P5} T5:{P5,P6} T6:{P5,P7} T3: Weight = || + |{P1}| + |{P5}| + |{P5,P6}| + |{P5,P7}| = 6

  23. Proposition (P.Rossmanith).Building a minimal-weightpreset tree is an NP-complete problem in the sizeof a Petri net, even if all transition presets have the size 3.

  24. Proposition (P.Rossmanith).Building a minimal-weightpreset tree is an NP-complete problem in the sizeof a Petri net, even if all transition presets have the size 3.

  25. Proposition (P.Rossmanith).Building a minimal-weightpreset tree is an NP-complete problem in the sizeof a Petri net, even if all transition presets have the size 3. P2 P3 P1 P4

  26. Proposition (P.Rossmanith).Building a minimal-weightpreset tree is an NP-complete problem in the sizeof a Petri net, even if all transition presets have the size 3. T2 P2 P3 T5 T1 T3 P1 P4 T4

  27. Proposition (P.Rossmanith).Building a minimal-weightpreset tree is an NP-complete problem in the sizeof a Petri net, even if all transition presets have the size 3. T2 P2 P3 T5 T1 T3 P1 P4 T4

  28. T1:{P1,P2} T2:{P2,P3} T3:{P3,P4} T4:{P1,P4} T5:{P2,P4} T1:{P1} T2:{P3} T5:{P4} T4:{P1} T3:{P3} {P2} {P4}

  29. {P1,P2,P3} {P1,P2,P4} {P1,P2,P5} {P1,P3,P4} {P1,P4} 

  30. {P1,P2,P3} {P1,P2,P4} {P1,P2,P5} {P1,P3,P4} {P1,P4}  {P1,P2}

  31. {P3} {P4} {P5} {P1,P3,P4} {P1,P4}  {P1,P2}

  32. {P3} {P4} {P5} {P1,P3,P4} {P1,P4}  {P1,P2} {P1,P4}

  33. {P3} {P4} {P5} {P3}  {P1,P2} {P1,P4}

  34. {P3} {P4} {P5} {P3}  {P1,P2} {P1,P4} {P1}

  35. {P3} {P4} {P5} {P3}  {P2} {P4} {P1}

  36. {P3} {P4} {P5} {P3}  {P2} {P4} {P1} 

  37. {P3} {P4} {P5} {P3} {P2} {P4} {P1} 

  38. Building Preset Trees function BuildTree({A1,...,Ak}) TS {Tree(A1,),…,Tree(Ak,)} while|TS|>1 choose Tree(A',·)TSand Tree(A'',·)TS such thatA‘A'' and |A'A''| is maximal I A'A'‘ T {Tree(B\I,ts) | Tree(B,ts) TS and IB} T= {ts | Tree(I,ts) TS and ts } TS  TS\{Tree(B, ·) TS | IB} TS  TS {Tree(I, T T=)} /* |TS|=1 */ return the remaining tree Tr  TS

  39. Experimental Results • Small presets (2-3 places): • improvements for some examples • Medium-size presets (5-6 places): • improvements for all examples • Large presets (7 and more places): • significant improvements for all examples

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