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Petri Net1 :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 Net2 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 Net3 Petri Net as a Graph (cont.) (Figure 1) A simple graph representation of a Petri net.

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Petri Net4 Petri Net as a Graph (cont.) (Figure 2) A marked Petri net.

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Petri Net5 Petri Net as a Graph (cont.) (Figure 3) The marking resulting from firing transition t 2 in Figure 2. Note that the token in p 1 was removed and tokens were added to p 2 and p 3

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Petri Net6 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 t 1

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Petri Net7 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 t 3

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Petri Net8 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 t 5

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Petri Net9 Petri Net as a Graph (cont.) (Figure 5) A simple model of three conditions and an event

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Petri Net10 (Figure 6) Modeling of a simple computer system

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Petri Net11 Petri Net as a Graph (cont.) (Figure 7) Modeling of a nonprimitive event

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Petri Net12 Petri Net as a Graph (cont.) (Figure 8) Modeling of “simultaneous” which may occur in either order

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Petri Net13 Petri Net as a Graph (cont.) (Figure 9) Illustration of conflicting transitions. Transitions t j and t k conflict since the firing of one will disable the other

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Petri Net14 Petri Net as a Graph (cont.) (Figure 10) An uninterpreted Petri net.

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Petri Net15 (Figure 11) Hierarchical modeling in Petri nets by replacing places or transitions by subnets (or vice versa).

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Petri Net16 (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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Petri Net17 (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 Net18 Petri Net as a Graph (cont.)

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Petri Net19 (Figure 15) A Petri net model of a P/V solution to the mutual exclusion problem

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Petri Net20 (Figure 16) Example of a Petri net used to represent the flow of control in programs containing certain kind of constructs L: S 0 Do while P 0 if P 2 then S 1 else S 2 endif parbegin S 3,S 4,S 5, parend enddo goto L

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Petri Net21 (Figure 17) A Petri net model for protocol 3

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Petri Net22 Other properties for analysis Boundeness –Safe net (bound = 1) –K-bounded net Conservation ==> conservative net Live transition Dead transition

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Petri Net23 State of a Petri net State - defined by its marking, State space - set of all markings: ( , , ,...) Change in state - caused by firing a transition, defined by partial F n, (example) = ( , t j ) Note: marking -- For a marking , (P i ) = i A marked Petri net: m = (P, T, I, O, )

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Petri Net24 = (1, 0, 1, 0, 2) ( , t 3 ) = (1, 0, 0, 1, 2) = ( , t 4 ) = (1, 1, 1, 0, 2) = etc.

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Petri Net25 (Figure 19) A Petri net with a nonfirable transition. Transition t 3 is dead in this marking

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Petri Net26 Petri Net as a Graph (cont.)

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Petri Net27 Petri Net as a Graph (cont.) (Figure 21) The reachability tree of the Petri net of Figure 19 (1, 0, 1, 0) (1, 0, 0, 1) (1, , 1, 0) (1, , 0, 0)(1, , 0, 1) (1, , 1, 0) t3t3 t2t2 t1t1 t3t3 t2t2

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Petri Net28 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.

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