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Introduction to Petri Nets

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

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Introduction to Petri Nets

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

  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. 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. 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. Producer - Consumers PNET Static Structure

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

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

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

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

  10. Structure Dynamics Formal Model for PNets

  11. Occurrence Rule

  12. Occurrences and Reachability

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

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

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

  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/ • C. Ling. The Petri Net Method. http://www.utdallas.edu/~gupta/courses/semath/petri.ppt

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