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Towards a n Efficient Algorithm for Unfolding Petri Nets

This paper presents an efficient algorithm for unfolding Petri nets, addressing the partial order semantics and mitigating the state space explosion problem in model checking. The ERV unfolding algorithm enhances performance by employing cut-off events and effectively managing transitions. Experimental results demonstrate significant improvements in processing small to large presets, showcasing the algorithm's capability in handling complex Petri net structures. This work contributes to the advancement of model checking methodologies essential for verifying system properties.

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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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