1. HELSINKI UNIVERSITY OF TECHNOLOGY *Laboratory for Theoretical Computer Science Helsinki University of Technology **Department of Computing Science University of Newcastle upon Tyne Parallelization of the Petri Net Unfolding Algorithm K.Heljanko*, V.Khomenko**, and M.Koutny**

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

3. 3 Unf  places from M 0 pe  transitions enabled by M 0 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) end while add cut-off events and their postsets to Unf The ERV unfolding algorithm

4. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8

5. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 P7P7 P8P8 P9P9 T6T6

6. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P7P7 P8P8 P9P9 T6T6

7. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8

8. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8

9. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8

10. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 12

11. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9

12. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P6P6 T5T5 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9

13. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P6P6 T5T5 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9 P 14 T 10

14. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P6P6 T5T5 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9 P 14 T 10

15. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P6P6 T5T5 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9 P 14 T 10

16. T1T1 P3P3 T3T3 P5P5 P2P2 T2T2 P1P1 T5T5 P6P6 T4T4 P4P4 P7P7 P8P8 P9P9 P 11 P 10 P 13 P 14 P 12 T9T9 T7T7 T 10 T6T6 T8T8 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P6P6 T5T5 P1P1 P7P7 P8P8 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9 P 14 T 10 P9P9 P7P7 P8P8

17. 17 while pe   extract appropriate non-empty Sl  pe for all e  Sl in any order refining do 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) end for end while Step 1: Unfolding algorithm with slices

18. 18 Problem 1 The order in which the events are processed may be inconsistent with ! Can be fixed by imposing the constraint: for every e  Sl and every f e:  f  pe\Sl and  pe does not contain causal predecessors of f

19. 19 Theorem: Let Pref' and Pref'' be the prefixes of the unfolding of a bounded net system, produced by arbitrary runs of the basic and slicing algorithms respectively. Then Pref' and Pref'' are isomorphic. Correctness

20. 20 Problem 2 How to choose slices to satisfy the imposed condition? For orders refining the McMillan’s adequate order C 1 C 2  |C 1 | < |C 2 | a good choice is to take Sl = { e | [e] = k }, where k = min { |[e]| | e  pe }.

21. 21 T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P6P6 T5T5 P1P1 P7P7 P8P8 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9 P 14 T 10 P9P9 P7P7 P8P8 Example

22. 22 Step 2: Parallelisation The events in a slice can be inserted into the prefix all together, and their possible extensions can be computed in parallel!

23. 23 Problem 3 The same possible extensions can be computed for several times! T1T1 P1P1 T2T2 T3T3 P2P2 P3P3 P4P4 P5P5 T4T4 P7P7 P8P8 P9P9 T6T6 T7T7 P 10 P 11 T8T8 P 13 P 12 T9T9 T9T9 T4T4

24. 24 Sl Restricting the scope

25. 25 Restricting the scope

26. 26 Restricting the scope

27. 27 Restricting the scope

28. 28 Problem 4 How to get rid of the ordering in the for all loop? for all e  Sl in any order refining do if e is a cut-off event then cut-off  cut-off  {e} else UpdatePotExt(pe, Unf, e) end for If there are no cut-offs in the slice Sl then the order in which the events are processed is irrelevant.

29. 29 Cut-offs “in advance” One can check the cut-off criterion as soon as a new possible extension is computed Advantages:  No cut-offs in a slice (fixes Problem 4)  The cut-off criterion is checked in UpdatePotExt(pe, Unf, e) – the part of the algorithm which is computed in parallel

30. 30 The queue of possible extensions  Can be represented as a sequence Sl 1,Sl 2,Sl 3,… where Sl i contains events whose local configurations have the size i  Insertion an event e into the queue is reduced to adding it to the set Sl |[e]|  Choosing a slice is reduced to detaching the first non-empty set Sl i from the queue No comparisons of configurations are involved!

31. 31 The total number of comparisons of configurations performed by the parallel algorithm is equal to |E cut |, i.e. there are no redundant comparisons! In contrast, the ERV unfolding algorithm performs O(|E|log|E|) comparisons. Comparisons of configurations

32. 32 Experimental results Processors:234 Speedup:1.682.142.27 The speedup is real, but not linear due to limited memory bandwidth (“bus contention”) 4  Pentium TM III 500MHz 512K cache processors, 512M 133MHz RAM

33. 33 Conclusions The algorithm is faster even on a uniprocessor The size of slices is usually large, which allows for good parallelization More than 95% of time is spent in the parallel sections of the algorithm Can be efficiently implemented even on distributed memory architectures Linear speedup for most of the examples (in theory)  Limited memory bandwidth (“bus contention”)