:Abstract formal model of information flow

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Petri Net

• Abstract formal model of information flow
• Major use
• Modeling of systems of events in which it is
  possible for some events to occur concurrently,
  but there are constraints on the occurrences,
  precedence, or frequency of these occurrences.

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Petri Net as a Graph

• Models static properties of a system
• Graph contains 2 types of nodes
• Circles (Places)
• Bars (Transitions)
• Petri net has dynamic properties that result from
  its execution
• Markers (Tokens)
• Tokens are moved by the firing of transitions of
  the net.

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Petri Net as a Graph (cont.)
(Figure 1) A simple graph representation of a
Petri net.
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Petri Net as a Graph (cont.)
(Figure 2) A marked Petri net.
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Petri Net as a Graph (cont.)
(Figure 3) The marking resulting from firing
transition t2 in Figure 2. Note that the token
in p1 was removed and tokens were added to p2
and p3
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Petri Net as a Graph (cont.)
(Figure 4) Markings resulting from the firing
of different transitions in the net of Figure
3.
(a) Result of firing transition t1
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Petri Net as a Graph (cont.)
(Figure 4) Markings resulting from the firing
of different transitions in the net of Figure
3.
(b) Result of firing transition t3
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Petri Net as a Graph (cont.)
(Figure 4) Markings resulting from the firing
of different transitions in the net of Figure
3.
(c) Result of firing transition t5
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Petri Net as a Graph (cont.)
(Figure 5) A simple model of three conditions and
an event
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(Figure 6) Modeling of a simple computer system
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Petri Net as a Graph (cont.)
(Figure 7) Modeling of a nonprimitive event
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Petri Net as a Graph (cont.)
(Figure 8) Modeling of simultaneous which
may occur in either order
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Petri Net as a Graph (cont.)
(Figure 9) Illustration of conflicting transition
s. Transitions tj and tk conflict since
the firing of one will disable the other
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Petri Net as a Graph (cont.)
(Figure 10) An uninterpreted Petri net.
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(Figure 11) Hierarchical modeling in Petri nets
by replacing places or transitions by subnets (or
vice versa).
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(Figure 12) A portion of a Petri net modeling
a control unit for a computer with multiple
registers and multiple functional units
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(Figure 13) Representation of an
asynchronous pipelined control unit. The
block diagram on the left is modeled by the
Petri net on the right
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Petri Net as a Graph (cont.)
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(Figure 15) A Petri net model of a P/V
solution to the mutual exclusion problem
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(Figure 16) Example of a Petri net used to
represent the flow of control in programs
containing certain kind of constructs
L S0 Do while P0 if P2 then S1
else S2 endif parbegin
S3,S4,S5, parend enddo goto L
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(Figure 17) A Petri net model for protocol 3
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Other properties for analysis

• Boundeness
• Safe net (bound 1)
• K-bounded net
• Conservation gt conservative net
• Live transition
• Dead transition

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State of a Petri net

• State - defined by its marking, m
• State space - set of all markings (m0, m1, m2,
  ...)
• Change in state - caused by firing a transition,
  defined by partial Fn, d
• (example) m1 d (m0 , tj)
• Note marking --
• For a marking m , m(Pi) mi
• A marked Petri net m (P, T, I, O, m)

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(Figure 19) A Petri net with a nonfirable transit
ion. Transition t3 is dead in this marking
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Petri Net as a Graph (cont.)
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Petri Net as a Graph (cont.)
(Figure 21) The reachability tree of the Petri
net of Figure 19
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Unsolvable Problems

• Subset problem - given 2 marked Petri nets, is
  the reachability of one net a subset of the
  reachability of the other net undecidable (Hack)
  ......
• Complexity
• reachability problem is exponential time-hard
  and exponential space-hard.