Introduction to Petri Nets

Hugo Andrés Lópezlopez@dit.unitn.it

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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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A little of HistoryC.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.

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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).

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Producer - Consumers PNET

Static Structure

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Producer - Consumers PNET

Markings introducethe dynamics of

the system

Initial Marking

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Producer - Consumers PNET (Non-Determinism)

Firing T1 again will leadto multiple production

of tokens in P3

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Producer - Consumers PNET (Non-Determinism)

Firing T6 disable T3Firing T3 disable T6

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PNet Evolutions

Resembles a board game.

A transition can be fired if their input events are marked.

Possible Scenarios:

Concurrent Execution Conflicting Execution

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Formal Model for PNets

Structure

Dynamics

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Occurrence Rule

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Occurrences and Reachability

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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.

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Behavioural Properties of Marked Petri Nets

•A marked p/t-net is

terminatingterminating – if there is no infinite occurrence sequence deadlock-freedeadlock-free – if each reachable marking enables a transition livelive – if each reachable marking enables an occurrence sequence containing all transitions boundedbounded - if, for each place p, there is a bound b(p) s.t. m(p) <= b(p) for every reachable marking m1-Safe1-Safe - if b(s) = 1 is a bound for each place sReversibleReversible – if m0 is reachable from each other reachable marking

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A vending machine

5c

Take 15c bar

Deposit 5c

0c

Deposit 10c

Deposit 5c

10c

Deposit 10c

Deposit5c

Deposit 10c20c

Deposit5c

15c

Take 20c bar

-VM is Deadlock-free.

t8

t1

p1

t2

p2

t3

p3

t4

t5

t6 p5

t7

p4

t9

Every marking generated from m0 enables a transition

-VM is Live.

The occurrences generated by m0 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 m0 from every

marking derived from

m0

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Exercise

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

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BibliographyJ.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