1. Introduction to Petri Nets Hugo Andrés López lopez@dit.unitn.it

2. 2 Plan for lectures 6th November ‘07 Informal Introduction, Intuitions. Formal definition Properties for PNets. 8th November ’07 Examples on specifications. Applications. Petri Nets Variants. Advantages, Limitations.

3. 3 A little of History C.A. Petri proposes a new model for information flow (early 60’s). Main ideas: Modelling Systems with asynchronous and Concurrent executions as graphs. Holt and Petri: “net theory” (mid 70’s) MIT and ADR: Research in Petri Net Properties and Relations with Automata Theory Nowadays: Event Structures, Bigraphs, (new) flowcharts, relations with Process algebra.

4. 4 Intuitions A Petri Net (PN) is a formalism for representing concurrent programs in terms of events and transitions. Defining Static Properties (Structural). Dynamic Properties (Behavioural).

5. 5 Producer - Consumers PNET Static Structure

6. 6 Producer - Consumers PNET Markings introduce the dynamics of the system Initial Marking

7. 7 Producer - Consumers PNET (Non-Determinism) Firing T1 again will lead to multiple production of tokens in P3

8. 8 Producer - Consumers PNET (Non-Determinism) Firing T6 disable T3Firing T3 disable T6

9. 9 PNet Evolutions Resembles a board game. A transition can be fired if their input events are marked. Possible Scenarios: Concurrent ExecutionConflicting Execution

10. 10 Formal Model for PNets St ru ct ur e DynamicsDynamics

11. 11 Occurrence Rule

12. 12 Occurrences and Reachability

13. 13 Example: An Scheduler Resources: A buffer of input processes with k=4. A dual-core processor. Buffer of Results with k=4. Processes are independent.

14. 14 Behavioural Properties of Marked Petri Nets A marked p/t-net is terminating terminating – if there is no infinite occurrence sequence deadlock-free deadlock-free – if each reachable marking enables a transition live live – if each reachable marking enables an occurrence sequence containing all transitions bounded bounded - if, for each place p, there is a bound b(p) s.t. m(p) <= b(p) for every reachable marking m 1-Safe 1-Safe - if b(s) = 1 is a bound for each place s Reversible Reversible – if m0 is reachable from each other reachable marking

15. 15 A vending machine -VM is Deadlock- free. Every marking generated from m 0 enables a transition -VM is Live. The occurrences generated by m 0 contains all the transitions -VM is bounded by 1 (1-safe) there are no induced tokens, the constraints used in m0 holds for the system. -VM is Reversible It is possible to go back to m 0 from every marking derived from m 0

16. 16 Exercise Show by inspection (or other methods) the properties that holds for the scheduler example.

17. 17 Bibliography J.L. Peterson. “Petri Nets”. Computing Surveys, Vol. 9 No. 3, 1977. A. Kondratyev et al. “The use of Petri nets for the design and verification of asynchronous circuits and systems”. Journal of Circuits Systems and Computers. 1998. Balbo et al. Lecture notes of the 21st. Int. Conference on Application and Theory of Petri Nets. 2000. The World of Petri nets: http://www.daimi.au.dk/PetriNets/ http://www.daimi.au.dk/PetriNets/ C. Ling. The Petri Net Method. http://www.utdallas.edu/~gupta/courses/semath/petri.pptwww.utdallas.edu/~gupta/courses/semath/petri.ppt