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Petri Nets Section 2 Roohollah Abdipur 1

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Properties Reachability – “Can we reach one particular state from another?” Boundedness – “Will a storage place overflow?” – the number of tokens in a place is bounded Safeness - the number of tokens in a place never exceeds one Deadlock-free - none of markings in R(PN, M0) is a deadlock Liveness – “Will the system die in a particular state?” 2

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Properties Reachability – Question: “Can we reach one particular state from another?” – To answer: find R(PN, M0) 3

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Reachability : Vending machine example Properties t8t8 t1 p1 t2 p2 t3 p3 t4 t5 t6 p5 t7t7 p4 t9 Initial marking:M0 M0 M1M2M3M0M2M4 t3t1t5t8t2t6 M0 = (1,0,0,0,0) M1 = (0,1,0,0,0) M2 = (0,0,1,0,0) M3 = (0,0,0,1,0) M4 = (0,0,0,0,1) “M2 is reachable from M1 and M4 is reachable from M0.” In fact, all markings are reachable from every marking. 4

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Properties Boundedness – “Will a storage place overflow?” – A Petri net is said to be k -bounded or simply bounded if the number of tokens in each place does not exceed a finite number k for any marking reachable from M0. The PN for vending machine is 1-bounded. A 1-bounded Petri net is also safe. 5

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Properties Safeness – The number of tokens in a place never exceeds one – i.e., a 1-bounded PN is a safe PN – e.g., the PN for vending machine is safe 6

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Properties Liveness: A Petri net with initial marking M0 is live if, no matter what marking has been reached from M0, it is possible to ultimately fire any transition by progressing through some further firing sequence. A live Petri net guarantees deadlock-free operation, no matter what firing sequence is chosen. E.g., the vending machine is live and the producer-consumer system is also live. A transition is dead if it can never be fired in any firing sequence 7

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Properties A bounded but non-live Petri net p1 p2 p3 p4 t1 t2 t3t4 M0 = (1,0,0,1) M1 = (0,1,0,1) M2 = (0,0,1,0) M3 = (0,0,0,1) 8 Liveness: An Example

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Properties p1 t1 p2p3 t2t3 p4 p5 t4 An unbounded but live Petri net M0 = (1, 0, 0, 0, 0) M1 = (0, 1, 1, 0, 0) M2 = (0, 0, 0, 1, 1) M3 = (1, 1, 0, 0, 0) M4 = (0, 2, 1, 0, 0) 9 Liveness: Another Example

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Properties Liveness: – a PN is live if, no matter what marking has been reached, it is possible to fire any transition with an appropriate firing sequence – equivalent to deadlock-free 10

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Properties Other properties: – Reversibility – Coverability – Persistency – Synchronic distance – Fairness – …. 11

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Analysis Methods Enumeration – Reachability Tree – Coverability Tree Linear Algebraic Technique – State Matrix Equation – Invariant Analysis: P-Invariant and T-invariant Simulation 12

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Analysis Methods Reachability Tree: An exmple 13

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Analysis Methods Reachability Tree: Another exmple 14

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Analysis Methods Linear Algebraic Technique 15

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Analysis Methods Linear Algebraic Technique 16

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Other Types of Petri Nets Coloured Petri nets Timed Petri nets Object-Oriented Petri nets 17

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THE END 18

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