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Petri Nets: Properties and Analysis

Mikael Harkjær Mø[email protected]

Department of Computer ScienceAalborg University

23. september 2009

Based on: Tadao Murata. ”Petri Nets: Properties, Analysis andApplications.“ Proc. of the IEEE, 77(4), 1989.

IntroductionDefinition and Example

Behavioral PropertiesAnalysis

Outline

1 Introduction

2 Definition and Example

3 Behavioral Properties

4 Analysis

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IntroductionExample

Introduction to Petri nets

First presented by Carl Adam Petri in his dissertation (1962)

Petri nets are both a graphical and mathematical modelingtool.

Can model many different kinds of systems.

Capable of modeling, concurrency, nondeterminism andasynchronous system.

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IntroductionExample

Example of simple Petri net

Directed graph and a marking.p1

p2

p3t1

Input places Transistions Output places

Precondition Event PostconditionInput data Computation Output data

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IntroductionExample

Example of simple Petri net

Directed graph and a marking.p1

p2

p3t1

Input places Transistions Output places

Precondition Event PostconditionInput data Computation Output data

22

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Petri NetTransition (Firing) ruleExamples

Definition: Petri Net

A Petri net is a 5-tuple, PN = (P,T ,F ,w ,M0) where,

P = {p1, p2, . . . , pm}, is a finite set of places.

T = {t1, t2, . . . , tn}, is a finite set of transitions.

F ⊆ (PxT ) ∪ (TxP), is a set of arcs (Flow relation).

w : F → N, is the weight function.

M0 : P → N0 is the initial marking.

P ∩ T = ∅ and P ∪ T 6= ∅.

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Firing rule

A transition t is enabled if each input place p has at leastw(p, t) tokens.

A transition may fire or not, there is no urgency in Petri nets.

Firing an enabled transition t, removes w(p, t) tokens fromeach input place p and adds w(t, p′) tokens at each outputplace p′.

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Example 1

Simple example showing the chemical reaction 2H2 + O2 → 2H2O.

H2

O2

H2Ot22

Transitions t is enabled.

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Example 1

Simple example showing the chemical reaction 2H2 + O2 → 2H2O.

H2

O2

H2Ot22

Transitions t has been fired.

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Example 2

Example of a Petri net representing a finite-state machine of acoffee machine.

0 kr.

1 kr.

2 kr.

Dep. 1kr

Dep. 2krDep. 1kr

Get Tea

Get Coffee

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Example 2

Example of a Petri net representing a finite-state machine of acoffee machine.

Decision

0 kr.

1 kr.

2 kr.

Dep. 1kr

Dep. 2krDep. 1kr

Get Tea

Get Coffee

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Example 3

Example of a Petrinet expressing concurrency.

p1

p2

p3

p4

p5

t1

t2

t3

t4

Par Begin Par End

Note: this net is a marked graph.

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Example 3


Concurrent

p1

p2

p3

p4

p5

t1

t2

t3

t4

Par Begin Par End

Note: this net is a marked graph.

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Example 3


Concurrent

p1

p2

p3

p4

p5

t1

t2

t3

t4

Par Begin Par End

Note: this net is a marked graph. 9 / 18



ReachabilityBoundedness and CoverabilityLiveness

Behavioral Properties

Reachability

Boundedness and Coverability

Liveness

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Reachability

A marking M is said to be reachable from M0 if there exists asequence of firings of transitions σ that transforms M0 to M.

This is denoted M0[σ〉M, where σ = t1t2 · · · tm.

The set of all reachable markings from M0 is written R(N,M0) orsimply R(M0).

The reachability problem is shown to be decidable, but inEXPTIME.

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Boundedness and Coverability

Boundedness

A Petri net (N,M0) is said to be k-bounded,if

M(p) ≤ k for all p ∈ P and M ∈ R(M0).

If a net is 1-bounded is said to be safe.

Coverability

A marking M is said to be coverable from M0 if there exists amarking M ′ ∈ R(M0) such that M(p) ≤ M ′(p) for all places p inthe net.

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Liveness

A Petri net is said to be live if it is deadlock-free. We definedifferent levels of liveness. A transition t is said to be:

Dead (L0-live) if t can never be fired.

L1-live if t can be fired at least once in some firing sequence.

L2-live if given any k ∈ N, then t can be fired at least k-timesin some firing sequence.

L3-live if t appears infinitely often in some firing sequence.

L4-live(live) if t is L1-live for every marking M ∈ R(M0).

A Petri net is said to be Lk-live if every t ∈ T is Lk-live, wherek ∈ {0, 1, 2, 3, 4}.

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Coverability tree (infinite)Coverability tree (finite)ExampleProperties of Coverability tree

Coverability tree (infinite)

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1}

t1

{1, 1, 0}

{0, 1, 1}

{0, 0, 1}

t2

t1

{1, 2, 0}

{0, 2, 1}

{0, 1, 1}

.

.

.

t2

t2

t1

.

.

.

t3

t3

t3

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Coverability tree (finite)

To keep the tree finite, we introduce the symbol ω. It can bethought of as ”infinity“, and has the properties for each integer n

ω > n, ω ± n = ω and ω ≤ ω.

Main idea

Whenever you reach a marking M, if there exists a marking M ′ onthe path from M0 to M such that M(p) ≥ M ′(p) for every placeand M 6= M ′, i.e. M’ is coverable, then replace M(p) with ω foreach p where M(p) > M ′(p).

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}new

t1 t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} new

t1 t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} new

t1

{1, ω, 0} new

t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} dead-end

t1

{1, ω, 0} new

t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} dead-end

t1

{1, ω, 0}

t1 t3

t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} dead-end

t1

{1, ω, 0}

{0, ω, 1} new

t1

{1, ω, 0} new

t3

t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} dead-end

t1

{1, ω, 0}

{0, ω, 1}

{0, ω, 1} new

t2

t1

{1, ω, 0} new

t3

t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} dead-end

t1

{1, ω, 0}

{0, ω, 1}

{0, ω, 1} old

t2

t1

{1, ω, 0} new

t3

t3

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Example

p1

p2

p3

t0t1

t3

t2

{1, 0, 0}

{0, 0, 1} dead-end

t1

{1, ω, 0}

{0, ω, 1}

{0, ω, 1} old

t2

t1

{1, ω, 0} old

t3

t3

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Properties of Coverability tree

Let T be a coverability tree for a Petri net (N,M0) then

The net is bounded and thus R(M0) is finite iff ω does notappear in node labels of T .

The net is safe iff only 1’s and 0’s appear in node labels of T .

A transition in N is dead iff it does not appear as an edgelabel in T .

M is reachable if there exists a node labeled M ′ such thatM ≤ M ′.

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Thank you for your attention

Questions?

These slides are available athttp://www.cs.aau.dk/~mikaelhm/slides/spfs1.pdf

http://www.cs.aau.dk/~mikaelhm/slides/spfs1.pdf

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