Tutorial Perti Nets Geeraerts

8/13/2019 Tutorial Perti Nets Geeraerts

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Introduction

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Introduction

Concurrency: propertyof a system in whichmany entities actat the same time and

interact. Often found in many application: Computer science (e.g.: parallel computing)

Workflow Manufacturing systems

....

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Introduction

Concurrenc

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Introduction

Concurrenc

Work in parallel4


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Introduction

Concurrenc

Work in parallel

Must wait for thetwo other machines

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Can write orread on the DB

Can write or

read on the DB

Introduction

Concurrenc

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Introduction

ConcurrencBoss

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Introduction

ConcurrencBoss

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Introduction

ConcurrencBoss

Employees: work in parallel6


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Introduction

ConcurrencBoss

Employees: work in parallel

gives work

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Introduction

ConcurrencBoss

Employees: work in parallel

gives workreceives creditfor the results

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Introduction

Petri netsare a tool to modelconcurrent systems andreasonabout them.

Inventedin 1962 by C.A.Petri.

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The aim of the talk

Introduceyou to Petri nets (and some oftheir extensions)

Explain several analysis methods for PN i.e., what can you ask about a PN ?

Give a rough idea of the researchin theverification groupat ULB...

... and foster new collaborations?

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How I use Petri nets

template T Max(T a,

T b)

{

return a < b ? b : a;

}

#include

int main() // fonction main

{

int i = Max(3, 5);

char c = Max('e', 'b');

std::string s = Max(std::string

("hello"), std::string("world"));

float f = Max(1, 2.2f);

p1

p2

p4

p5

p3

p6 p7

t1

t3

t4

t5 t6t2

t7 t8

2

Analysis methodof PN

abstraction

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Howyoumight use PN

p1

p2

p4

p5

p3

p6 p7

t1

t3

t4

t5 t6t2

t7 t8

2

Analysis methodof PN

abstraction

Your favorite

application

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Intuitions

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Ingredients

A Petri net is made up of...

Places

Transitions

Tokens

= some type of resource

consume and produceresources

= one unity of acertain resource

Tokens live in the places12


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Firing a transition

2

3

Transitions consumetokens from the inputplacesand produce tokens in the outputplaces

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Firing a transition

2

3

Transitions consumetokens from the inputplacesand produce tokens in the outputplaces

Now, the transition

cannot be fired anymore15


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Example 1

Can write or

read on the DB

Can write orread on the DB

The two machines cannot write at the

same time16


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Example1

read

write

idle

write

read

idle

The tokentells us the stateof the process17


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Example1

read

write

idle

write

read

idle



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Example1

read

write

idle

write

read

idle



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Example1

read

write

idle

write

read

idle



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Example1

read

write

idle

write

read

idle



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Example 1read

write

idle

write

read

idle

Add a lockto ensure mutual exclusion22


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Example 1read

write

idle

write

idle

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Example 2

p1

p2

p3

t1

t3 t2

mutexM ;

Process P {

repeat {

take M ;

critical;

release M ; }

}

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Example 2

p1

p2

p3

t1

t3 t2

mutexM ;

Process P {

repeat {

take M ;

critical;

release M ; }

}

Here, we have applied a counting abstraction24


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Plan of the talk

Preliminaries Toolsfor the analysis of PN

reachability tree and reachability graph

place invariants Karp & Miller and the coverability set

The coverability problem Moreon PN: extensions... Conclusion

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Plan of the talk

Preliminaries Toolsfor the analysis of PN

reachability tree and reachability graph

place invariants Karp & Miller and the coverability set

The coverability problem Moreon PN: extensions... Conclusion

Detailedcoverage

Survey

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Preliminaries

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Formal definition

A Petri netis a tuple P,Twhere:

Pis the (finite) set of places

Tis the (finite) set of transitions. Eachtransition tis a tuple I,Owhere:

I: is a function s.t. tconsumesI(p) tokensin each place p

Ois a function s.t. tproducesO(p)tokens in each place p

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Example

2

3

tp1

p2

p5

p4

p3

I(p1)=2 I(p2)=1 I(p3)=0 I(p4)=0 I(p5)=0O(p1)=0 O(p2)=0 O(p3)=1 O(p4)=3 O(p5)=1

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Markings

The distribution of the tokens in the placesis formalised by the notion of marking, which

can be seen:

either as a functionm, s.t. m(p) is thenumber of tokensin place p

or as a vectorm=m1, m2,... mnwheremi is the number of tokensin place pi

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Example

2

3

tp1

p2

p5

p4

p3

m =1,1,1,2,0m = p1, p2, p3, 2p4

m(p1)=1, m(p2)=1, m(p3)=1, m(p4)=2, m(p5)=0

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Firing a transition

A transition t = I,Ocan be firedfrom mifffor any place p:

m(p) !I(p)

The firing transformsthe marking minto amarking ms.t. for any place p:

m(p) = m(p) - I(p) + O(p)

Notation: m!m

Notation: Post(m) = {m| m!m}

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Example

p1

p2

p3

t1

t3 t2

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Example

p1

p2

p3

t1

t3 t2

Post(1, 1, 0)={2, 1, 0,0, 0, 1}

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Example

p1

p2

p3

t1

t3 t2

Post(1, 1, 0)={2, 1, 0,0, 0, 1}

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Example

p1

p2

p3

t1

t3 t2

Post(1, 1, 0)={2, 1, 0,0, 0, 1}

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Example

p1

p2

p3

t1

t3 t2

Post(1, 1, 0)={2, 1, 0,0, 0, 1}

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Example

p1

p2

p3

t1

t3 t2

Post(1, 1, 0)={2, 1, 0,0, 0, 1}

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I l k


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Initial marking

Reachable markin s All PN are equipped with aninitial markingm0

If two markings mand mare s.t.: m!m1!m2!!mThenm is reachablefrom m

Let N be a PN with initial marking m0:Reach(N) = {m reachablefrom m0}

is the set of reachable markingsof N.

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Example

p1

p2

p3

t1

t3 t2

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Example

p1

p2

p3

t1

t3 t2

Reach(N) ={i,1,0| i N}

{i,0,1| i N}

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Example

p1

p2

p3

t1

t3 t2

Reach(N) ={i,1,0| i N}

{i,0,1| i N}

This set allows us toprove that the mutual

exclusion is indeedenforced

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Ordering on markings

Markings can be comparedthanks to 4: m4miff for any place p: m(p)6m(p)

mpmiff m4m andmm

Examples:

1, 0, 0p1, 1, 041, 1, 045, 7, 2 1, 0, 0is not comparableto 0, 1, 0

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Questions on PN Meaningful questionsabout PN include: Boundedness: is the number of reachable

markings bounded?

Place boundedness: is there a boundonthe maximal numberof tokens that can becreated in a given place ?

Semi-liveness: is there a reachable markingfrom which a given transitioncan fire? Coverability

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Exampleread

write

idle

write

read

idle

BoundedPN All the places are bounded

All the transitions are semi-live

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Some tools for theanalysis of PN

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Reachability Tree

Idea: the rootis labeled by m0 for each node labeled by m, create one

childfor each marking of Post(m)

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

W1, I2

I1,W2

M, R1, I2

M, I1, R2

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

W1, I2

I1,W2

M, R1, I2

M, I1, R2

M, I1, I2

R1,W2

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

W1, I2

I1,W2

M, R1, I2

M, I1, R2

M, I1, I2

R1,W2

W1, I2I1,W2 M, R1, I2

M, I1, R2

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

W1, I2

I1,W2

M, R1, I2

M, I1, R2

M, I1, I2

R1,W2

M, I1, I2W1, R2

W1, I2I1,W2 M, R1, I2

M, I1, R2

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

W1, I2

I1,W2

M, R1, I2

M, I1, R2

M, I1, I2

R1,W2

M, I1, I2W1, R2

W1, I2I1,W2 M, R1, I2

M, I1, R2

Reachability Tree

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Reachability Tree


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I1

R1W1

W2 R2

I2

M

M, I1, I2

W1, I2

I1,W2

M, R1, I2

M, I1, R2

M, I1, I2

R1,W2

M, I1, I2W1, R2

W1, I2I1,W2 M, R1, I2

M, I1, R2

Reachability Tree

Reachability trees canbe infinite

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Reachability graph

Idea: build a nodefor each reachablemarkingand add an edgefrom m to m ifsome transition transforms m into m

remark: now, if we meet the same markingtwice, we do not createa new node, but

re-use the previously created node.

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Reachability graph

I1

R1W1

W2 R2

I2

MM, I1, I2

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Reachability graph

I1

R1W1

W2R2

I2

MM, I1, I2

M, R1, I2 M, I1, R2

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Reachability graph

I1

R1W1

W2R2

I2

MM, I1, I2

M, R1, I2 M, I1, R2

M, R1, R2

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Reachability graph

I1

R1W1

W2R2

I2

MM, I1, I2

W1, I2I1,W2

M, R1, I2 M, I1, R2

M, R1, R2

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Reachability graph

I1

R1W1

W2R2

I2

MM, I1, I2

W1, I2I1,W2

W1, R2R1,W2

M, R1, I2 M, I1, R2

M, R1, R2

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Reachability graph

I1

R1W1

W2R2

I2

MM, I1, I2

W1, I2I1,W2

W1, R2R1,W2

M, R1, I2 M, I1, R2

M, R1, R2

The reachability graphallows us to prove thatthe mutual exclusion is

indeed enforced

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Reachability graph

The reachability graph of a PN contains allthe necessary information to decide:

boundedness place boundedness

semi-liveness ...

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Reachability graph

Unfortunately...

p1

p2

p3

t1

t3 t2

p2

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Reachability graph

Unfortunately...

p1

p2

p3

t1

t3 t2

p2

p1,p2

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Reachability graph

Unfortunately...

p1

p2

p3

t1

t3 t2

p2

p1,p2

2p1,p2 p3

3p1,p2 p1,p3

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Reachability graph

Unfortunately...

p1

p2

p3

t1

t3 t2

p2

p1,p2

2p1,p2 p3

3p1,p2 p1,p3

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Reachability graph

Unfortunately...

p1

p2

p3

t1

t3 t2

p2

p1,p2

2p1,p2 p3

3p1,p2 p1,p3

Reachability graphs canbe infinite

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The hard stuff...

The main difficulty in analysing Petri nets is

due to the possibly infinitenumber ofreachable markings.

We have to find techniquesto deal withthis infiniteset.

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The hard stuff...

Remark: finitedoesnt mean easy

The set of reachable markingsof aboundednet can be huge! Efficient techniquesto deal with bounded

nets have been developped.

e.g.: net unfoldings

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Place invariants

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Place Invariantsread

write

idle

write

read

idle

R1 R2

I2

m(R1) + m(R2) + m(I2) =1

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write

idle

write

read

idle

R1 R2

I2

m(R1) + m(R2) + m(I2) =1

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write

idle

write

read

idle

R1 R2

I2

m(R1) + m(R2) + m(I2) = 2

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write

idle

write

read

idle

R1 R2

I2

m(R1) + m(R2) + m(I2) = 0

The total number oftokens in these placesis not constant

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write

idle

write

read

idle

R1

I1

m(R1) + m(W1) + m(I1) = 1

W1

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Pl I


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write

idle

write

read

idle

R1

I1

m(R1) + m(W1) + m(I1) = 1

W1

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Pl I


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write

idle

write

read

idle

R1

I1

m(R1) + m(W1) + m(I1) = 1

W1

The total number oftokens in these places

is constant

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Pl I i


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write

idle

write

read

idle

R1

I1

m(R1) + m(W1) + m(I1) = 1

W1


is constant

This providesmeaningful information

about the system: a

process is either idle,or reading or writing

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Pl I i


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p1

p2

p3

p42

2

Place Invariants

m(p1) + m(p2) + m(p3) + m(p4) = 3

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Pl I i


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p1

p2

p3

p42

2

Place Invariants

m(p1) + m(p2) + m(p3) + m(p4) = 2

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Pl I i


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p1

p2

p3

p42

2

Place Invariants

m(p1) + m(p2) + m(p3) + m(p4) = 1

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Pl I i


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p1

p2

p3

p42

2

Place Invariants

m(p1) + m(p2) + m(p3) + m(p4) = 1


is not constant

60

Pl I i


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p1

p2

p3

p42

2

Place Invariants

m(p1) + m(p2) + m(p3) + m(p4) = 1


is not constant

In some sense, tokensin p1are heavierthan

those in p2

60

Pl I i


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p1

p2

p3

p42

2

Place Invariants

m(p1) + m(p2) + m(p3) + m(p4) = 1


is not constant

In some sense, tokensin p1are heavierthan

those in p2

Lets add weightstothe places !

60

Pl I i t


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p1

p2

p3

p42

2

Place Invariants

3 m(p1) + m(p2) + m(p3) + 2 m(p4) = 3

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Pl I i t


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p1

p2

p3

p42

2

Place Invariants

3 m(p1) + m(p2) + m(p3) + 2 m(p4) = 3

62

Pl I i t


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p1

p2

p3

p42

2

Place Invariants

3 m(p1) + m(p2) + m(p3) + 2 m(p4) = 3

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Example: other


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p1

p2

p3

p42

2

p

invariants

m(p1) + m(p3) = 1

2 m(p1) + m(p2) + 2 m(p4) = 2

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Invariants as over-


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a roximations A place-invariant expresses a constrainton

the reachable markings.

Ifm is reachableand i is an invariant, then:

The reverseis not true!

pP

i(p)m(p) = pP

i(p)m0(p)

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E l


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p1

p2

p3

p42

2

Example

m(p1) + m(p3) = 1is an invariant

but1, 25, 0, 234isnot reachable

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Invariants as over-


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Theorem: For any Petri net N:

Reach(N)

{m| mrespects some invariant of N}

approximations

68

Invariants as over-


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Reach(N)


approximations

This setoverapproximatesthe

reachable markings

68

Invariants as over-


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Reach(N)


approximations

This setoverapproximatesthe

reachable markings

Place invariants arethus usefulto finitely

approximatethe set of

reachable markings68

Place invariant and


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boundedness Theorem: Ifthere exists a place invariant i

and a place p s.t. i(p)>0 thenp is bounded.

Remark: the reverseis not true. One can find a boundednet that doesnt

have a place invariant i with i(p)>0 foreach place.

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Place invariant


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Place invariant

Question: how do we computethem ?

70

Matrix characterisation


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The negative effect(consumption) of allthe transitions on all the places can besummarisedin one matrix:

where, for any i: ti =Ii,Oi

W=

I1(p1)I2(p1) Ik(p1)I1(p2)I2(p2) Ik(p2)

.

.. .

.. .

. . ...

I1(pn)I2(pn) Ik(pn)

neg. eff. on p1

neg. eff. on p2

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The same can be done with the positiveeffects:

where, for any i: ti =Ii,Oi

W+ =

O1(p1)O2(p1)

Ok(p1)O1(p2)O2(p2) Ok(p2)

... ... . . . ...

O1(pn)O2(pn)

Ok(pn)

pos. eff. on p1

pos. eff. on p2


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Incidence Matrix


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Incidence Matrix

The global effectof every transition can besummarised as a single matrix:

W=W+W

W is called the incidence matrixof the net

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Example


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Example

W+=

1 0 1

0 0 1

0 1 0

W

=

0 1 0

0 1 0

0 0 1

W =

1 1 1

0

1 10 1 1

p1

p2

p3

t1

t3 t2

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Example


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Example

W+=

1 0 1

0 0 1

0 1 0

W

=

0 1 0

0 1 0

0 0 1

W =

1 1 1

0

1 10 1 1

p1

p2

p3

t1

t3 t2

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Computing place


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p g p

invariants Intuitively, if i is a place invariant it shouldassign weightsto the places such that thepositiveand negativeeffects of every

transition are balanced

Thus, for any transition t =I,Oweshould have:

pPI(p) i(p) = pPO(p)

i(p)

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Computing place


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i(p)

2

1 2

75


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Computing place


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i(p)

2

1 2

75

Computing place


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i(p)

2

1 2

75

Computing place


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invariantspP

I(p) i(p) = pP

O(p) i(p)

pP

O(p)I(p)

i(p) = 0

means

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Computing place


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invariantspP

I(p) i(p) = pP

O(p) i(p)

pP

O(p)I(p)

i(p) = 0

means

t=I,O

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Computing place


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invariantspP

I(p) i(p) = pP

O(p) i(p)

pP

O(p)I(p)

i(p) = 0

means

t=I,O W=

O(p1)I(p1)

O(p2)

I(p2)

... ... ...

O(pn)I(pn)

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Computing place


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invariantspP

O(p)I(p)

i(p) = 0

is thus the scalar productof i and the columnof W that corresponds to transitiont

77

Computing place


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invariantspP

O(p)I(p)

i(p) = 0


Since this must hold for any t, we obtain:

77

Computing place


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invariantspP

O(p)I(p)

i(p) = 0



Theorem: any solutioni to the following system ofequationsis a place-invariant:

77

Computing place


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invariantspP

O(p)I(p)

i(p) = 0



iW = 0

Theorem: any solutioni to the following system ofequationsis a place-invariant:

77

Example


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Example

p1

p2

p3

t1

t3 t2

W =

1

1 1

0 1 1

0 1 1

78

Example


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Example

p1

p2

p3

t1

t3 t2

i1, i2, i3W = 0

W =

1

1 1

0 1 1

0 1 1

78

Example


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Example

p1

p2

p3

t1

t3 t2

i1 = 0i1i2+i3 = 0

i1+i2i3 = 0

i1, i2, i3W = 0

W =

1

1 1

0 1 1

0 1 1

78

Example


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Example

p1

p2

p3

t1

t3 t2

i1 = 0i1i2+i3 = 0

i1+i2i3 = 0

i1, i2, i3W = 0

W =

1

1 1

0 1 1

0 1 1

i1 = 0i2+i3 = 0

+i2i3 = 0

78

Example


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Example

p1

p2

p3

t1

t3 t2

i1 = 0i1i2+i3 = 0

i1+i2i3 = 0

i1, i2, i3W = 0

W =

1

1 1

0 1 1

0 1 1

i1 = 0i2+i3 = 0

+i2i3 = 0

Any vector of the form

0, i, iis a place invariant

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Proving properties


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p1

p2

p3

t1

t3 t2

Let us choose 0, 1, 1as place-invariant

This means that p2and p3arebounded!

79

Proving properties


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p1

p2

p3

t1

t3 t2

For any reachable marking m:

0m(p1) +1m(p2) + 1m(p3) = 0m0(p1) + 1m0(p2) + 1m0(p3)

m(p2) + m(p3) = 1



79

Proving properties


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p1

p2

p3

t1

t3 t2

For any reachable marking m:

0m(p1) +1m(p2) + 1m(p3) = 0m0(p1) + 1m0(p2) + 1m0(p3)

m(p2) + m(p3) = 1


Hence, mutual exclusionis enforced !


79

Proving properties


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read

write

idle

write

read

idle

M

W1 W2

i(M) = i(W1) = i(W2) = 1 and i(p) = 0 otherwiseis a place invariant

80

Proving properties


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read

write

idle

write

read

idle

M

W1 W2

i(M) = i(W1) = i(W2) = 1 and i(p) = 0 otherwiseis a place invariant

Hence, mutual exclusionis enforced !80


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Karp & Miller

andthe coverability set

81

The reachability tree


146/341

revisited Reminder: reachability trees can be infinite

p1

p2

p3

t1

t3 t2

0p1,p2

1p1,p2

2p1,p2 p3

3p1,p2 p1,p3

82



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p1

p2

p3

t1

t3 t2

0p1,p2

1p1,p2

2p1,p2 p3

3p1,p2 p1,p3

82



148/341


p1

p2

p3

t1

t3 t2

0p1,p2

1p1,p2

2p1,p2 p3

3p1,p2 p1,p3

Increasing sequencesof markings appear

on unbounded

places

82



149/341

revisited Let us summarisethis infinite sequence

0p1,p2

1p1,p2

2p1,p2

3p1,p2

83



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0p1,p2

1p1,p2

2p1,p2

3p1,p2

limit

83

The reachability treei i d


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0p1,p2

1p1,p2

2p1,p2

3p1,p2

"p1,p2limit

83

The reachability treei i d


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0p1,p2

1p1,p2

2p1,p2

3p1,p2

"p1,p2limit

"must be regarded as:

any number of tokens

83

The reachability treei it d


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0p1,p2

1p1,p2

2p1,p2

3p1,p2

"p1,p2limit

"must be regarded as:

any number of tokens

Main idea of theKarp andMiller algorithm

83


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Monotonicity


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y

Petri nets induce (strongly) monotonictransition systems:

In articular:m1 m2

m3

t

if

85

Monotonicity


156/341

y


In articular:

m4

m1 m2

m3

t

if

85

Monotonicity


157/341

y


In articular:

m4

m1 m2

m3

t

t

if

85

Monotonicity


158/341

y


In articular:

m4

m1 m2

m3

t

t

if

85


159/341

Monotonicity


160/341

y


In articular:

m4

m1 m2

m3

t

t

if

i1, i2, i3 i1, i2, i3

85

Monotonicity


161/341

y


In articular:

m4

m1 m2

m3

t

t

if

i1, i2, i3 i1, i2, i3

85

Monotonicity


162/341

y


In articular:

m4

m1 m2

m3

t

t

if

i1, i2, i3 i1, i2, i3

85

Monotonicity


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y


In articular:

m4

m1 m2

m3

t

t

Documents

Tutorial Perti Nets Geeraerts