1. Petri Nets Section 2
Roohollah Abdipur

2. Properties Reachability Boundedness Safeness
“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?”

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

4. Properties Reachability : Vending machine example p1 t8 t1 t2 p2 t3 p3
Initial marking:M0
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) M0 M1 M2 M3 M4 t3 t1 t5 t8 t2 t6
“M2 is reachable from M1 and M4 is reachable from M0.”
In fact, all markings are reachable from every marking.

5. Properties Boundedness The PN for vending machine is 1-bounded.
“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.

6. 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

7. 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

8. Properties Liveness: An Example A bounded but non-live Petri net t1 p3

9. Properties Liveness: Another Example An unbounded but live Petri net

10. 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

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

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

13. Analysis Methods
Reachability Tree: An exmple

14. Analysis Methods
Reachability Tree: Another exmple

15. Analysis Methods
Linear Algebraic Technique

16. Analysis Methods
Linear Algebraic Technique

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

18. THE END