Petri%20Nets

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Petri Nets
Formal Methods for SoC Design
Sorin Manolache
sorma_at_ida.liu.se
Compiled from lecture slides by Petru Eles
petel_at_ida.liu.se
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Outline

• History and rationale
• Structure
• Behaviour
• Analysis
• Extensions
• Petri Nets resources

Petri Nets Sorin Manolache and Petru Eles
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History and Rationale

• Graphical and mathematical modelling language
• Applicable to a large variety of systems
• Able to model synchronisation, concurrency,
  non-determinism
• Amenable to analysis
• With extensions, able to model real-time
• With extensions, they have the expressive power
  of Turing Machines

• Carl Adam Petri, Kommunikation mit Automaten,
  PhD dissertation, 1962
• Kurt Jensen, Coloured Petri Nets and the
  Invariant Method, Theoretical Computer Science
  (14) 1981
• M.K. Molloy, Performance Analysis Using
  Stochastic Petri Nets, IEEE Trans. on Computers,
  C-31, 9, 1982

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Structure

• Directed, bipartite graph
• The two types of vertices are places and
  transitions

• Multisets of input arcs and multisets of output
  arcs (input and output relative to transitions)

• Places are like holders of something (defining
  state), while transitions are like activities
  (defining state transformations/behaviour)

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Behaviour

• Places hold tokens, the place is said to be
  marked
• The marking of a net indicates how many tokens
  there are in each place

• Tokens circulate among places through transitions

• We say that the transition fires
• A transition fires when it is enabled
• A transition is enabled when each of its input
  places contain more tokens than the multiplicity
  of the corresponding input arc
• Upon firing, Ip tokens are removed from each
  input place and Op tokens are placed in each
  output place, where Ip and Op are the
  corresponding input and output arc multiplicities

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Producer-Consumer Example
prod
cons
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Producer-Consumer Example
prod
cons
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Concurrency, Synchronisation
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Analysis

• We're interested in certain properties of the
  modelled system expressed in terms of properties
  of the Petri Net representation
• Reachability
• Liveness
• Boundedness
• Safety
• Computation of invariants

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Reachability
Marking M'
Marking M
Is there a sequence of transition firings such
that M M'?
NO
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Liveness

• A transition T is live if in any marking there
  exists a firing sequence such that T becomes
  enabled
• An entire net is live if all its transitions are
  live
• Important for checking deadlock

Live?
YES
NO
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Boudedness
YES
NO
Is there a limit of the number of tokens in any
place?
A net is safe if its bound is 1
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Place Invariants
P1
P2
P1 P2 2
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Extensions

• Sometimes non-determinism is not desirable ?
  Introduction of transition priorities
• We can have partial order on transition firings,
  but no explicit notion of time ? Introduction of
  transition firing delays (Timed Petri Nets) or
  transition firing delay probability distributions
  (Stochastic Petri Nets or Extended Stochastic
  Petri Nets) or a combination thereof
  (Deterministic and Stochastic Petri Nets)
• Some behaviours cannot be expressed with Basic
  Petri Nets ? Introduction of inhibitor arcs
  giving Petri Nets the same expressive power as
  Turing Machines

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Hierarchy
1
2
1
1
1
2
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Coloured Petri Nets

• Petri Nets quickly grow to large dimensions when
  modelling non-trivial systems
• In order to reduce modelling complexity, tokens
  have associated values of different types
  (colours) and transitions are functions defined
  on the colour domains of their input places with
  values in the colour domains of their output
  places

• The most useful application is associating time
  stamps to tokens, easily modelling real-time
  systems
• Danger the modelling language is so powerful
  that it easily leads to non-analysable models
  (real-valued time stamps lead to unbounded nets,
  as the set of real numbers is uncountable)

• It is just syntactic sugar, CPN are equivalent to
  Basic Petri Nets and therefore CPN models can be
  translated to BPN models and the other way around

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Worker-Resource Example
idle
update
use
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Multiprocessor System Example
A
B
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Application Modelling
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Reachability Graph
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Discussion

• Pros

• Powerful modelling language
• Amenable to analysis
• Models parallelism, synchronisation,
  non-determinism, real-time
• Extended to support hierarchy

• Cons

• Almost all analysis methods die because of
  complexity issues
• Unsatisfying composability when it comes to
  analysis because the modelling hierarchy is
  usually lost in a flat representation at analysis

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Petri Nets Resources

• http//www.daimi.au.dk/PetriNets/

• Books
• Tools
• Papers
• Conferences
• Success stories
• Mailing lists
• News

Petri Nets Sorin Manolache and Petru Eles