Petri Nets and Their Extensions - Part 1

Petri Nets: Expressibility and Ease of Modeling
Part 1 - Introduction and Extensions
Irina A. Lomazova
[email protected]
National Research University Higher School of Economics,
Laboratory of Process-Aware Information Systems (PAIS Lab)
Moscow, Russia
International Conference on Software Testing, Machine Learning
and Complex Process Analysis (TMPA 2019)
Tbilisi, Georgia (7-9 November 2019)

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Petri Nets - Agenda
Part 1 - Introduction and Extensions
on Thursday 07.11
Introduction and Initial Examples
Notion of Runs
Petri Net Extensions
Part 2 - Formal Analysis
on Saturday 09.11
Behavioral Properties
Structural Analysis
Irina A. Lomazova Petri Nets - Part 1: Introduction and Extensions 1 / 46

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Petri Nets - An Introduction
and some Initial Modeling Examples

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Communicating Finite-State Automata
a
2
a
1
b
2
b
1
sender receiver
+ack -req +req -ack

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From Automata to Petri Nets
Drawing Automata as Petri Nets
a
2
a
1
b
2
b
1
sender receiver
+ack -req +req -ack
t
1
t
2
t
3
t
4

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Petri Nets - Synchronous Communication
a
2
a
1
b
2
b
1
sender receiver
ack ack
t
1
t
2
t
3
req

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Petri Nets - Asynchronous Communication
a
2
a
1
b
2
b
1
sender receiver
ch
1
ch
2
send receive
t
2
t
3
t
4
t
1

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Carl Adam Petri (1926 - 2010)
Petri nets were invented in 1939 by C. A. Petri for describing chemical
processes. He documented Petri nets as part of his dissertation
Kommunikation mit Automaten (Communication with Automata) in 1962.

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What are Petri Nets?
A Formalism for modelling and analysis of concurrent,
asynchronous, distributed, parallel, non-deterministic and/or
stochastic systems.
A Graphical tool for constructing and visualizing models of
distributed systems.
A Mathematical tool to conduct a formal analysis of
distributed systems.

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Design Principles
Places (store tokens)
(Pre- Post-) Conditions
Resources needed and released
Input/Output Data or Signals
Buﬀers
Transitions
Events or activities
Tasks or jobs
Signal processors
Arcs link places to transitions, and transitions to places. They may denote
some abstract relation, logical connection, physical proximity, etc.

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Design Principles
t
2
Never
Never t
1
Wrong
Correct!
t
2
t
1
computer 1 computer 2
computer 1 t
1
communication
channel
t
2
computer 2

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Formal Definition
A Petri net is a tuple PN = (P, T, F, W , M0) where:
P = {p1, p2, ..., pn} is a finite set of places.
T = {t1, t2, ..., tn} is a finite set of transitions. P ∩ T = ∅.
F ⊆ (P × T) ∪ (T × P) is a set of directed arcs (flow
relation).
W : F → {1, 2, 3, ...} is a weight function.
M0 : P → {0, 1, 2, 3, ...} is the initial marking.
A state of a Petri net is defined by a marking M : P → {0, 1, 2, 3, ...}

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Transition Firing Rule
1 A transition t is enabled in a marking M iﬀ each input place
p of t has at least w(p, t) tokens.
∀p ∈ •t M(p) ≥ w(p, t) •t is the set of input places of t.
2 The ﬁring of an enabled transition t removes w(p, t) tokens
from each input place p of t, and adds w(t, p ) tokens to each
output place p of t.
Firing of t produces a new marking M , such that
M (p) = M(p) − W (p, t) + W (t, p )

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Transition Firing Rule - An Example
A Petri net N in a marking M with transition t enabled.
H
2
t
O
2
2
2 H
2
O
N with a new marking M after ﬁring transition t.
H
2
t
O
2
2
2 H
2
O

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Modeling Examples of Petri Nets
Finite-State Machines (FSM) - a sub-class of Petri nets
A vending machine with 15¢ and 20¢ candies.
Two types of transitions: a user may insert 5¢ or 10¢ coins.
Places are states 0¢, 5¢, 10¢, 15¢, 20¢.
0¢
10¢
5¢
10¢
5¢
5¢ 15¢
20 ¢
5¢
10¢
10¢
5¢
c
1
get 15¢ candy
c
2
get 20¢ candy
start

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Modeling Choice (Conﬂict, Decision)
At place start two transitions are enabled and may ﬁre,
i.e, to deposit either a 5¢ or a 10¢ coin.
0¢
10¢
5¢
10¢
5¢
5¢ 15¢
20 ¢
5¢
10¢
10¢
5¢
c
1
get 15¢ candy
c
2
get 20¢ candy
start

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Modeling Concurrent Activities
Transitions t2
and t3
represent concurrent activities.
Par
End
t
4
t
3
t
2
t
1
Par
Begin
p
1
p
2
p
3
p
4
p
5

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Communication Protocols
buffer
1
t
1
t
1
t
1 buffer
2
t
2
t
3
t
4
t
5
t
6
ready to send ready to receive
send receive
wait ack. message
received
receive
ack.
send
ack.
process
1
process
2
ack
received.
ack
sent.
sender receiver
channels

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Dataﬂow Computation
t
1
a
copy
a
t
2
b
b
copy
t
3
add
t
4
substract
a+b
a−b
t
6
divide
t
5
if a−b≠0
t
7
if a−b=0
undefined
Compute x =
a+b
a−b
x =
a+b
a−b

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Petri Nets - Runs
Sequential Run
A Petri net N
Its sequential run...
A a
−
→ B b
−
→ C c
−
→ A a
−
→ ...
short: abca

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Petri Nets - Runs
Sequential Run with an External Resource
Sequential run...
AD a
−
→ BD b
−
→ C c
−
→ AD a
−
→ ...
short: abca

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Petri Nets - Runs
One more Process sharing the Resource
Many sequential runs! Two of them...
ADE a
−
→ BDE b
−
→ CE c
−
→ ADE a
−
→ ...
ADE a
−
→ BDE d
−
→ BDF e
−
→ BG f
−
→ BDE...

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Petri Nets - Runs
State Transition Diagram (Reachability Graph)
Each path in the graph is a sequential run in the Petri net

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Petri Nets - Runs
Concurrent Runs
Transition ocurrences are partially ordered (ordering based on causality).

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Petri Nets - Runs
Many Concurrent Runs!

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Petri Nets - Runs
A Run Consists of Scenarios...

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Petri Nets - Runs
A Run Consists of Scenarios...
Two scenarios in the net above corresponding to red and blue processes.

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Petri Nets - Runs
Petri Nets Semantics
Sequential runs - interleaving semantics
Concurrent runs - true concurrency semantics
Parallel (simultaneous) ﬁring of diﬀerent enabled transitions -
step semantics.

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Augmenting the Expressibility of Petri Nets
for Modeling Complex Systems

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Basic Extensions
Petri Nets with Inhibitor Arcs
Petri Nets with Reset Arcs
Petri Nets with Time
Time Petri Nets (TPN)
Stochastic Petri Nets (SPN)
Petri Nets with Data Types
Colored Petri Nets (CPN)
Nets-within-Nets
Nested Petri Nets (local semantics)
Valk’s Object Petri Nets (reference semantics)

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Petri nets with Inhibitor Arcs
An inhbitor arc tests the absence of tokens (instead of testing the
presence of a minimum number of tokens).
t
1
p
1
p
2
2
p
3
p
4
p
5
2 3
inhibitor arc

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Petri nets with Inhibitor Arcs
Producer-Consumer System with Priority
t
1
p
1
Producer A
t
2
p
2
t
3
p
4
t
4
p
5
Consumer A
Buffer A
p
3
t
5
p
6
Producer B
t
6
p
7
t
7
p
9
t
8
p
10
Consumer B
Buffer B
p
3
Consumer B can consume tokens from buﬀer B only if buﬀer A is empty

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Petri nets with Reset Arcs
A reset arc does not impose a precondition for firing,
and it empties the place when the transition fires.
Before firing of transition t1
t
1
p
1
p
2
p
3
Reset arc
After firing of transition t1
t
1
p
1
p
2
p
3
Reset arc

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Time and Petri Nets
Petri nets can be extended with time, introducing time delays
on transitions.
Useful for analyzing time-dependent distributed systems, i.e,
for performance evaluation, scheduling problems, etc.
We consider two kind of Petri nets with time:
(Deterministic) Time Petri Nets (TPN) → delay intervals
are statically assigned.
Stochastic Petri Nets (SPN) → delays are randomly
generated from a probabilistic distribution.

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Each transition t has an internal clock with value v(t), and a
delay interval [e(t), l(t)].
An enabled transition may ﬁre just if its clock value v(t) lies within
its interval [e(t), l(t)].
t
1
p
2
p
1
t
2
0
0
[ 1 , 4 ]
[ 2 , 2 ]

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In TPN, a state is deﬁned as a pair (M, v) where M is a net
marking, and v is an array of clock values.
A state S = (M, v) is transformed into a new state S :
By some time elapsing d:
(M, v) d
−
→ (M, v + d)
By the ﬁring of a transition t:
(M, v) t
−
→ (M − Pre(t) + Post(t), v)

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Time Petri Nets (TPN) - Sequence Example (0/4)
Initial state
(p1
+ 2p2, (0, 0))
t
1
p
2
p
1
t
2
0
0
[ 1 , 4 ]
[ 2 , 2 ]
Clock values are set to 0.
Transitions enabled, but cannot ﬁre.
(clock values not within their delay range)

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Elapsing time d = 1.5 secs.
(p1
+ 2p2, (0, 0)) d=1.5
−
−
−
−
→ (p1
+ 2p2, (1.5, 1.5))
t
1
p
2
p
1
t
2
1.5
1.5
[ 1 , 4 ]
[ 2 , 2 ]
Clock values updated.
Enabled transition t2
can ﬁre. v(t2
) ∈ [1, 4]

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Firing of transition t2
.
(p1
+ 2p2, (0, 0)) d=1.5
−
−
−
−
→ (p1
+ 2p2, (1.5, 1.5)) t2
−
→ (p1
+ p2, (1.5, 0))
t
1
p
2
p
1
t
2
0
1.5
[ 1 , 4 ]
[ 2 , 2 ]
Clock value v(t2
) is reset.

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Elapsing time d = 0.5 secs.
(p1
+ 2p2, (0, 0)) d=1.5
−
−
−
−
→ (p1
+ 2p2, (1.5, 1.5)) t2
−
→ (p1
+ p2, (1.5, 0))
d=0.5
−
−
−
−
→ (p1
+ p2, (2, 0.5))
t
1
p
2
p
1
t
2
0.5
2
[ 1 , 4 ]
[ 2 , 2 ]
Enabled transition t1
is allowed to ﬁre. v(t1
) ∈ [2, 2]

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Firing of transition t1
.
(p1
+ 2p2, (0, 0)) d=1.5
−
−
−
−
→ (p1
+ 2p2, (1.5, 1.5)) t2
−
→ (p1
+ p2, (1.5, 0))
d=0.5
−
−
−
−
→ (p1
+ p2, (2, 0.5)) t1
−
→ (p2, (0, 0.5))
t
1
p
2
p
1
t
2
0.5
0
[ 1 , 4 ]
[ 2 , 2 ]

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Stochastic Petri Nets (SPN)
Each enabled transition ti has a delay for firing di , which is
a positive random variable with the exponential distribution.
di ∼ Exponential(λi )
λi is the firing rate of transition ti .
The average delay of transition ti is 1
λi
If several transitions are enabled, the transition with shortest
sampled delay fires first.

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Stochastic Petri Nets (SPN) - Example
t
1
p
1
t
2
p
2
t
3
λ
1
=7
d
1
~ exp(7)
E[d
1
]=
1
7
λ
2
=2
d
2
~ exp(2)
E[d
2
]=
1
2
λ
3
=5
d
3
~ exp(5)
E[d
3
]=
1
5
All transitions are enabled. Generate random delays...

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t
1
p
1
t
2
p
2
t
3
λ
1
=7
d
1
~ exp(7)
λ
2
=2
d
2
~ exp(2)
λ
3
=5
d
3
~ exp(5)
d
1
=0.04 d
2
=0.53 d
3
=0.08
Generate random delays: di
= ln(1−u)
−λi
u ∈ (0, 1)
Transition t1
with shorest delay d1
ﬁres...

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t
1
p
1
t
2
p
2
t
3
λ
1
=7
d
1
~ exp(7)
λ
2
=2
d
2
~ exp(2)
λ
3
=5
d
3
~ exp(5)
New marking after ﬁring transition t1

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Stochastic Petri Nets (SPN) - Markov Chain
Consider the following Petri net N and its reachability graph R(N, M0
)
t
1
p
1
t
2
t
3
p
3
t
4
p
2
λ
1
=7 λ
3
=4
λ
4
=2
λ
2
=5
(0, 2, 0)
M
0
(1, 1, 0)
(2, 0, 0)
t
1
M
1
M
3
t
1
t
2
t
2
(0, 1, 1)
M
2
t
3
t
4 (0, 0, 1)
M
5
t
3
t
4
(1, 0, 1)
M
4
t
1
t
2
t
3
t
4

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Stochastic Petri Nets (SPN) - Markov Chain
The Markov Chain of N can be constructed from R(N, M0
)
(0, 2, 0)
M
0
(1, 1, 0)
(2, 0, 0)
t
1
M
1
M
3
t
1
t
2
t
2
(0, 1, 1)
M
2
t
3
t
4 (0, 0, 1)
M
5
t
3
t
4
(1, 0, 1)
M
4
t
1
t
2
t
3
t
4
M
0
M
1
M
3
M
2
M
5
M
4
7/11
5/16
7/16
1
4/11
2/13
7/13
4/13
1
5/7
2/7
4/16
For instance, being at marking M0
the probability of passing to M1
is equal to
the probability of ﬁring t1
.

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Stochastic Petri Nets (SPN) - Marking-dependent delays
Predator-Prey Model
p
1
t
2
t
3
antelopes
t
1
p
2
cheetahs
predation death
birth
m
1
λ
1
m
1
m
2
λ
2
λ
3
Predator-Prey Model
Antelopes reproduce (by ﬁring birth) with rate m1λ1
, where m1
is the
number of tokens at p1
. (more antelopes, faster reproduction).
Cheetahs reproduce (by ﬁring predation) with rate m1
m2λ2
, where m2
is
the number of tokens at p2
. (more antelopes and cheetahs, more chance
to predate and reproduce new cheetahs).

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Towards Petri Net Extensions with Objects
Consider the following Ordinary Petri Net
t
1
p
1
at home
p
2
t
3
t
2
at work projects
p
3
discussion

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A Colored Petri Net (CPN)
t
1
p
1
at home
p
2
t
3
t
2
at work projects
p
3
discussion
x x
x
x
x
y
z
A
B C

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A Nested Petri Net (NP-net)
t
1
p
1
at home
p
2
t
3
t
2
at work projects
p
3
produce
x x
x
x
x
y
System Net (SN)
Element Net (EN)
u
1
u
2
passive
active
q
1
q
2
work with a
colleague
λh
u
3
produce ready
q
3
λv
λv

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A Nested Petri Net (NP-net) - Marking Example
t
1
p
1
at home
p
2
t
3
t
2
at work projects
p
3
produce
x x
x
x
x
y
u
1
u
2
passive
active
q
1
q
2
work with a
colleague
λh
u
3
produce ready
q
3
λv
λv
u
1
u
2
active
q
1
q
2
λh
u
3
produce ready
q
3
λv
passive work with a
colleague
Two agents in place “at work” in the system net.

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A Nested Petri Net (NP) - Horizontal Synchronization
t
1
p
1
at home
p
2
t
3
t
2
at work projects
p
3
produce
x x
x
x
x
y
u
1
u
2
passive
active
q
1
q
2
work with a
colleague
λh
u
3
produce ready
q
3
λv
λv
u
1
u
2
active
q
1
q
2
λh
u
3
produce ready
q
3
λv
passive work with a
colleague
Simultanenous ﬁring of transition “work with a colleague” in each element net.

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A Nested Petri Net (NP) - Vertical Synchronization
t
1
p
1
at home
p
2
t
3
t
2
at work projects
p
3
produce
x x
x
x
x
y
u
1
u
2
passive
active
q
1
q
2
work with a
colleague
λh
u
3
produce ready
q
3
λv
λv
u
1
u
2
active
q
1
q
2
λh
u
3
produce ready
q
3
λv
passive work with a
colleague
Simultanenous ﬁring of transition “produce” in the system and in the element
nets.

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Reference vs Local Semantics
In Nested Petri Nets (NP-nets) each agent resides locally in
one place of the system net (local semantics).
In R. Valk’s Object Petri Nets (OPN) each agent may be in
some sense distributed over a system net: different tokens
(acting as pointers) in different places may refer to the same
object net (reference semantics).
In the following, we illustrate this difference with an example
in fork-join situations.

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Object Petri Nets (with reference semantics)
t
1
t
2
t
3
t
4
t
2
p
1
p
6
p
2
p
4
p
3
p
5
fork
join
task 1 task 2
⟨i
4
⟩
⟨i
3
⟩
⟨i
2
⟩
l
1
b
1
l
2
l
3
l
4
b
2
b
4
b
5
b
3
b
6
⟨i
2
⟩
⟨i
3
⟩
⟨i
4
⟩

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t
1
t
2
t
3
t
4
t
2
p
1
p
6
p
2
p
4
p
3
p
5
fork
join
task 1 task 2
⟨i
4
⟩
⟨i
3
⟩
⟨i
2
⟩
l
1
b
1
l
2
l
3
l
4
b
2
b
4
b
5
b
3
b
6
⟨i
2
⟩
⟨i
3
⟩
⟨i
4
⟩

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t
1
t
2
t
3
t
4
t
2
p
1
p
6
p
2
p
4
p
3
p
5
fork
join
task 1 task 2
⟨i
4
⟩
⟨i
3
⟩
⟨i
2
⟩
l
1
b
1
l
2
l
3
l
4
b
2
b
4
b
5
b
3
b
6
⟨i
2
⟩
⟨i
3
⟩
⟨i
4
⟩
Two tokens in the system net point to the same agent.
The tokens denote some progress of the agent on each task.

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Nested Petri Nets (with local semantics)
t
1
t
2
t
3
t
4
t
2
p
1
p
6
p
2
p
4
p
3
p
5
fork
join
task 1 task 2
λ
3
l
1
b
1
l
2
b
3
x x
x y
x y
x y
x
x
x
b
2
l
3
l
4
b
4
l
5 λ
3
λ
2
λ
4
λ
2
λ
4

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t
1
t
2
t
3
t
4
t
2
p
1
p
6
p
2
p
4
p
3
p
5
fork
join
task 1 task 2
λ
3
l
1
b
1
l
2
b
3
x x
x y
x y
x y
x
x
x
b
2
l
3
l
4
b
4
l
5
λ
3
λ
2
λ
4
λ
2
λ
4

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t
1
t
2
t
3
t
4
t
2
p
1
p
6
p
2
p
4
p
3
p
5
fork
join
task 1 task 2
λ
3
l
1
b
1
l
2
b
3
x x
x y
x y
x y
x
x
x
b
2
l
3
l
4
b
4
l
5
λ
3
λ
2
λ
4
λ
2
λ
4
λ
3
l
1
b
1
l
2
b
3
b
2
l
3
l
4
b
4
l
5
λ
2
λ
4
Two copies of the same agent have been produced to perform, each of them, a
single task.

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Petri Nets and Their Extensions - Part 1

Petri Nets and Their Extensions - Part 1

Exactpro
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Petri Nets and Their Extensions - Part 1

Petri Nets and Their Extensions - Part 1

Exactpro PRO

More Decks by Exactpro

Other Decks in Technology

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Transcript

Exactpro
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