petri nets refresher

/faculteit technologie management

PN-1

Petri netsrefresher

Prof.dr.ir. Wil van der AalstEindhoven University of Technology, Faculty of Technology Management,

Department of Information and Technology, P.O.Box 513, NL-5600 MB,

Eindhoven, The Netherlands.


PN-2

Elements

(name)

(name)

place

transition

arc (directed connection)

token

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

place

transition

token


PN-3

Rules

• Connections are directed.

• No connections between two places or two transitions.

• Places may hold zero or more tokens.

• First, we consider the case of at most one arc between two nodes.

wait enter before make_picture after leave gone

free

occupied


PN-4


PN-5

Enabled

• A transition is enabled if each of its input places contains at least one token.


free

occupied

enabled Not enabled

Not enabled


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Firing

• An enabled transition can fire (i.e., it occurs).

• When it fires it consumes a token from each input place and produces a token for each output place.


free

occupied

fired


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PN-8

Play “Token Game”

• In the new state, make_picture is enabled. It will fire, etc.


free

occupied


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Remarks

• Firing is atomic.

• Multiple transitions may be enabled, but only one fires at a time, i.e., we assume interleaving semantics (cf. diamond rule).

• The number of tokens may vary if there are transitions for which the number of input places is not equal to the number of output places.

• The network is static.

• The state is represented by the distribution of tokens over places (also referred to as marking).


PN-14

Non-determinism

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

transition t23fires

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

Two transitions are enabled but only one can fire


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PN-16

Example: Single traffic light rg

go

or

red

green

orange


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PN-18

Two traffic lights

rg

go

or

red

green

orange

rg

go

or

red

green

orange

rg

go

or

red

green

orange

OR


PN-19

Problem


PN-20

Solution

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

How to make them alternate?


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PN-22

Playing the “Token Game” on the Internet

• Applet to build your own Petri nets and execute them: http://is.tm.tue.nl/staff/wvdaalst/workflowcourse/pn_applet/pn_applet.htm

• FLASH animations: www.workflowcourse.com


PN-23

Exercise: Train system (1)

• Consider a circular railroad system with 4 (one-way) tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we do not care about the identities of the two trains.


PN-24


• Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time and we want to distinguish the two trains.


PN-25


• Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains (A,B). No two trains should be at the same track at the same time. Moreover the next track should also be free to allow for a safe distance. (We do not care about train identities.)


PN-26


• Consider a railroad system with 4 tracks (1,2,3,4) and 2 trains. Tracks are free, busy or claimed. Trains need to claim the next track before entering.


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Modeling problem


PN-28

Modeling problem (1): Zero testing

• Transition t should fire if place p is empty.

t ?

p


PN-29

Solution

• Only works if place is N-bounded

tN input and output arcs

Initially there are N tokens

p

p’


PN-30

Modeling problem (2): Priority

• Transition t1 has priority over t2

t1

t2

?

Hint: similar to Zero testing!


PN-31

A bit of theory

• Extensions have been proposed to tackle these problems, e.g., inhibitor arcs.

• These extensions extend the modeling power (Turing completeness*).

• Without such an extension not Turing complete.• Still certain questions are difficult/expensive to

answer or even undecidable (e.g., equivalence of two nets).

* Turing completeness corresponds to the ability to execute any computation.

petri nets refresher

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