1. Equivalence of open Petri nets Modeling and analysis with Petri net components. Marc Voorhoeve (AIS)

2. Open net definition Open net: 6-tuple satisfying Global set of external places A Addition and comparison of bags

3. Open net model External places Final marking indicators Internal places current int. marking Transitions weighted arcs t u v p q a bc r   2 2 2

4. Operational semantics Adding/removing external tokens Firing: External tokens cannot be “seen”; only added and removed! Internal tokens can be neither seen nor added or removed. Set of actions: external productions / consumptions / silent If Then Termination: 

5. Example

6. Properties Special case is OWF net (used by Reisig e.a. for SOA definition) - initial source and final sink place - path property WF nets: OWF nets without external places State oriented (like ExSpect and RoseRT) Abstraction of internal actions.

7. Interaction model (SOA) offer rdy claim handover

8. Temporal predicates offer rdy An active container can eventually become ready. offered but not readied yet #active: 012

9. Weak bisimilarity Relation between states, such that any sequence of additions and removals can be copied. Weakly bisimilar processes satisfy the same set of temporal predicates from a certain class. a b c a b c Nets not weakly bisimilar: b can be removed after addition of an a Weakly bisimilar, not branching bisimilar! Always brief appearance of b after consumption a and before production c a b c Production c after consumption a without b appearance is possible

10. Composition b c a Net: N Hide b within N: Rename c : d Hiding: Renaming:

11. Composition: merge + fusion b a b c Combine disjoint copies. Fuse places with same name. a c Deadlock possible No deadlock! d e

12. Composition semantics Token transfer is possible!

13. Compositional modeling Select open net subcomponents. Adapt interfaces by renaming and merge. Hide internal communication. b c a d e

14. Congruence Weak bisimilarity is congruence w.r.t. operators ab CC’ Component with subcomponent C Replace C with weakly bisimilar C’ Resulting components remain weakly bisimilar.

15. Caution with congruence Subnets at bottom are weakly bisimilar subcomponents. Hence the complete nets are weakly bisimilar too. External states differ! Left-hand net: b before c Weak bisimilarity is not a congruence for operators that directly access the state (inhibitor / reset arcs).

16. Research opportunities Verification of properties based on structure of process Example: active container predicate. a b e CD active objects in X : offered ( a ) and not yet claimed ( b ) = sum of active objects in C and D State space reduction!

17. The End