Petri Nets – Lecture 1 - CAPSL - Research Laboratories€¦ · · 2014-02-06Petri Nets – Lecture 1 Chen Chen ... Problems of FSM Modeling of the Pipeline • Number of states

Petri Nets – Lecture 1

Chen ChenSep. 28th, 2011

2011/9/28 \course\867-11F\Topic-2.ppt 1

Outline

• History of Petri Nets• Basic Terminologies• Comparing with FSM• Petri Net Properties


Invention of Petri Nets

• C.A. Petri (1962). Kommunikation mitAutomaten. Ph. D. Thesis. University of Bonn.

• IEEE Computer Pioneer Award (2008)


Carl Adam Petri1926 –2010

“For establishing Petri net theory in 1962, which not only was cited by hundreds of thousands of scientific publications but also significantly advanced the fields of parallel and distributed computing”

http://awards.computer.org/ana/award/viewPastRecipients.action?id=13�

http://en.wikipedia.org/wiki/File:Carl_adam_petri.jpg�

Introduction to Petri Nets Theory

• A theory of systems for modeling concurrency and synchronization

• Feature Graphical representation Simplicity Expressiveness for concurrency and

asynchronous operations


ISO/IEC 15909 (2011)Petri Nets Standard

Rudiger Valk (1998)Elementary Object Nets

M.K. Molly (1982)Stochastic Petri Nets

Chiola, Dutheillet, Franceschinis, S. Haddad. (1991)

Well-Formed Coloured Nets

History of Petri Nets


1960 1970 1980 1990 2000 2010C.A. Petri (1962)

Invention of Petri Nets

Holt and Commoner (1970)Graphical expression of PN

Marked Graph

Ramchandani (1973)Timed Petri Nets

Kurt Jensen (1981)Colored Petri Nets

E. P. Dawis, J. F. Dawis, Wei-Pin Koo(2001)Dualistic Petri Nets

Bail, Alla, David (1991)Hybrid Petri Nets

R. David & H. Alla (1987)Continuous Petri Nets

Hack (1972)Free-choice Petri Nets

Simple Petri Nets

Buchs and Guelfi (1991)Object Oriented Petri Net

Haddad & Poitrenaud (2007)Recursive Petri Nets

First PN

Sub-class of PN

Extension of PN

Normalization of PN

Outline

• History• Basic Terminologies• Comparing with FSM• Petri Net Properties


Example: Critical Section

• 3 users try to access the same CS• Only one user can access CS each time


Critical Section

3 tokens = 3 users 1 token = lock place

transition

token

t2

p2

p3

p1

t1

arc


• Multiple users try to access the same CS• Only one user can access CS each time


One user entered CSThe others have to wait

t2

p2

p3

p1

t1

t2

p2

p3

p1

t1


• Multiple users try to access the same CS• Only one user can access CS each time


One user completes access.Another one can enter now.

t2

p2

p3

p1

t1

t2

p2

p3

p1

t1

t2

p2

p3

p1

t1

Example: Producer-Consumer


Producer

Consumer A Consumer B

Task Buffer

Producer produces tasks and put the tasks in the task buffer.Consumers take tasks from the task buffer and execute them.One task can only be executed by one consumer – either A or B.

1 2 3



Producer


Task Buffer

1 2 3

Producer produced one task.Either A or B can be fired. But only one will be fired!



Producer


Task Buffer

1 2 3

After firing A or B, the task is executed.



Producer


Task Buffer

What’s the difference between Petri Nets and Dataflow?

Petri NetsDataflow (similar structure

but different semantics)

What will happen after token pushing?

Definition of a Petri Net (Self-reading)

• A Petri Net is a bipartite graph (P,T,A) that comprises of A set of transitions: T A set of places: P A set of directed arcs: A


a transition a place

Firing Rules (Self-reading)

•


Petri Net Marking (Self-reading)

•


Outline



Example: Two’s Complement

• What is two’s complement?• How to compute two's complement?


Example: Two's Complement (cont.)

• What is two's complement?• How to compute two's complement? flip all bits, and then plus a carry bit Assume B = 110, then -B = 001 + 1 = 010



• What is two's complement?• How to compute two's complement? flip all bits, and then plus a carry bit Assume B = 110, then -B = 001 + 1 = 010 Compute two's complement of 110 from low bit flip(0) + carry bit 1 + 1 = 0 & carry bit flip(1) + carry bit 0 + 1 = 1 flip(1) 0




q1 q2

1/1

R/R

0/0

R/R

0/1

1/0State q1: need to add carry bitflip(0) + carry bit 1 + 1 = 0 + carry bitflip(1) + carry bit 0 + 1 = 1

State q2: no need to add carry bitflip(0) 1 flip(1) 0

R means reset of computation

Example: FSM (cont.)


q1 q2

1/1

R/R

0/0

R/R

0/1

1/0

Now try to compute two's complement of 110



q1 q2

1/1

R/R

0/0

R/R

0/1

1/0

1 2 3 4 5

Two's complement of 110 is 010



q1 q2

1/1

R/R

0/0

R/R

0/1

1/0

1 2

Input sequence : R11Output sequence: 0

Computing from low bitR indicates reset

3 4 5




q1 q2

1/1

R/R

0/0

R/R

0/1

1/0

1 2

Input sequence : R1Output sequence: 10


3 4 5




q1 q2

1/1

R/R

0/0

R/R

0/1

1/0

1 2

Input sequence : ROutput sequence: 010


3 4 5




q1 q2

1/1

R/R

0/0

R/R

0/1

1/0

1 2 3 4 5


Finite State Machine (FSM)Self-reading

•


Important Features of FSM


What are the features?




FSM is always at a single active state!

A single input event triggers state transition!




FSM is always at a single active state!How to represent multiple active states

as a single active state?A single input event triggers state transition!

How to represent synchronization between multiple states?

Another Example: 2-Stage Pipeline Modeling


Input sequence Output sequence

Unit1 Unit2

Each unit can be modeled by a three-state FSM

ready for input

ready to outputbusy

Unit I (I = 1,2)

a

b c

Another Example: 2-Stage Pipeline Modeling (cont.)



Unit1 Unit2

How to model the two units together in FSM?




Unit1 Unit2

How to model the two units together in FSM?– cross-functional states

State aa’: Unit1 is ready for input & Unit2 is ready for inputState ba’: Unit1 is busy & Unit2 is ready for inputState ca’: Unit 1 is ready to output & Unit2 is ready for input

…State cc’: Unit 1 is ready to output & Unit 2 is ready to output



ab’

aa’ac’

ba’

bb’

bc’

ca’

cc’

cb’The structure of the FSM for 2-stage pipeline modeling. Inputs and outputs are omitted.

Problems of FSM Modeling of the Pipeline

• Number of states grows exponentially with the number of stages

• The identity of the individual stage lost • The structure information is obscured• Concurrency / synchronization information

cannot be represented


Try Again: 2-Stage Pipeline Modeling by PN


Each unit can be modeled by 3 places and 3 transitions

ready for input

ready to outputbusy

Pickup input data

Done with processing

Output result


Unit1 Unit2 Unit I (I = 1,2)

Try Again: 2-Stage Pipeline Modeling by PN (cont.)


ready for input

ready to output

busy


Output result

ready for input

ready to output

busy

Pickup input data


Unit1 transfers intermediate result to Unit2

Input sequence Output sequenceUnit1 Unit2

Pipeline

PN



ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Scenario: D1 in the entrance of Unit1D2 in the entrance of Unit1

Unit1 Unit2




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Scenario: D1 being processed by Unit1D2 in the entrance of Unit1

Unit1 Unit2




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2

Scenario: D1 is done by Unit1D2 in the entrance of Unit1




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2

Scenario: D1 being processed by Unit2D2 in the entrance of Unit1




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2

Scenario: D1 is done by Unit2D2 being processed by Unit1




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2

Scenario: Result of D1 is outputted by Unit2D2 is done by Unit1




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2

Scenario: Result of D1 is outputted by Unit2D2 being processed by Unit2




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2

Scenario: Result of D1 is outputted by Unit2D2 is done by Unit2




ready for input

ready to output

busy


Output result

ready for input

ready to outputbusy

Pickup input data


Unit1 Unit2


Scenario: Result of D1 is outputted by Unit2Result of D2 is outputted by Unit2

Conclusion: Compare PN with FSM

• PN is powerful in constructing composite machines. FSM must do cross product.

• PN is powerful in expressing concurrency. (FSM cannot express concurrency).

• PN is powerful in expressing synchronization.


Outline



Petri Net Properties

• Reachability• Safeness and Boundness• Conservation• Liveness• Persistency• Consistency


Documents

Petri Nets – Lecture 1 - CAPSL - Research Laboratories€¦ · · 2014-02-06Petri Nets – Lecture 1 Chen Chen ... Problems of FSM Modeling of the Pipeline • Number of states