1. Branching Processes of High-Level Petri Nets Victor Khomenko and Maciej Koutny University of Newcastle upon Tyne

2. 2 Talk Outline Motivation Unfoldings of coloured PNs Relationship between HL and LL unfoldings Extensions Future work

3. 3 Petri net unfoldings Partial-order semantics of PNs Alleviate the state space explosion problem Efficient model checking algorithms  Low-level PNs are not convenient for modelling

4. 4 Motivation Low-level PNs: Can be efficiently verified  Not convenient for modelling High-level descriptions: Convenient for modelling  Verification is hard Gap Coloured PNs: a good intermediate formalism

5. 5 Coloured PNs 1 2 w<u+v vu w {1,2} {1..4}

6. 6 Expansion 1 2 w<u+v v u w {1,2} {1..4} The expansion faithfully models the original net  Blow up in size

7. 7 Unfolding 1 2 w<u+v v u w {1,2} {1..4} 1 2 u=1 v=2 w=1 1 u=1 v=2 w=2 2

8. 8 Example: computing GCD 3 2 2 1 1 0 1 u=3, v=2 u=2, v=1 u=1 v0v0 m n v u%v u v 0 u u {0..100}

9. 9 Relationship diagram Coloured PNs unfolding Low-level prefix Coloured prefix unfolding Low-level PNs expansion ?

10. 10 ~ Relationship diagram Coloured PNs unfolding Low-level prefix Coloured prefix unfolding Low-level PNs expansion

11. 11 Relationship diagram 1 2 w<u+v v u w {1,2} {1..4} 1 2 u=1 v=2 w=1 12 u=1 v=2 w=2

12. 12 Relationship diagram Coloured PNs unfolding Prefix unfolding Low-level PNs expansion

13. 13 Benefits Avoiding an exponential blow up when building the expansion Definitions are similar to those for LL unfoldings, no new proofs All results and verification techniques for LL unfoldings are still applicable  Canonicity, completeness and finiteness results  Model checking algorithms

14. 14 Benefits Existing unfolding algorithms for LL PNs can easily be adapted  Usability of the total adequate order proposed in [ERV’96]  All the heuristics improving the efficiency can be employed (e.g. concurrency relation and preset trees)  Parallel unfolding algorithm [HKK’02]

15. 15 Extensions: infinite place types v0v0 m n v u%v u v 0 u u {0..100}

16. 16 Extensions: infinite place types v0v0 m n v u%v u v 0 u u N N N 3 2 2 1 1 0 1 u=3, v=2 u=2, v=1 u=1

17. 17 Extensions: infinite place types v0v0 m n v u%v u v 0 u u 3 2 2 1 1 0 1 u=3, v=2 u=2, v=1 u=1 {0..2} {1..3} {1}

18. 18 Refined expansion Coloured PNs unfolding Prefix unfolding Low-level PNs expansion

19. 19 Experimental results Tremendous improvements for colour- intensive PNs (e.g. GCD) Negligible slow-down (<0.5%) for control- intensive PNs (e.g. Lamport’s mutual exclusion algorithm)

20. 20 Future Work Partial-order verification for other PN classes (nets with read/inhibitor arcs, priorities etc.)