Mourjo Sen
October 17, 2014
16

# A short introduction to Petri Nets

Covers the basics of Petri Nets

October 17, 2014

## Transcript

2. ### What a Petri Net is ▶ A directed bi-partite graph

▶ With two types of nodes: ▶ Places ▶ Transitions ▶ Arcs from places to transitions and vice versa ▶ Tokens placed inside places, or a marking ▶ Optional weights in arcs Cl 2 + H 2 → 2 HCl 2
3. ### What a Petri Net is 3 ▶ Mathematically, a petri

net is a 5-tuple ▶ PN = (P, T, F, W, M 0 ) ▶ P = Finite set of places ▶ T = Finite set of transitions ▶ F = Finite set of arcs ▶ W: F → {1, 2, 3, …} is a weight function ▶ M 0 : P → {0, 1, 2, …} is the initial marking ▶ A petri net structure N =(P, T, F, W) is without any initial marking

data 5

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released 8

19. ### Reachability set 19 ▶ Let M be a marking of

a Petri net N = (P,T,F,W,M 0 ) ▶ The set of markings reachable from M is called the reachability set of M, written as reach(M) ▶ Reach(M) is the set of markings, such that: ▶ M ∈ reach(M), and ▶ If M’ → M’’ for some transition t ∈ T and M’ ∈ reach(M), then M’’ ∈ reach(M) ▶ The set of reachable markings of a petri net as a whole is defined to be reach(M 0 ) ▶ The “reachability problem” is to decide whether M ∈ reach(N), given a Petri net N and a marking M

21. ### Boundedness 21 ▶ A place is called k-bounded or bounded

if it does not contain more than k tokens in all reachable markings, including the initial marking ▶ It is said to be safe if it is one-bounded ▶ A (marked) Petri net is called (k-)bounded and/or safe if all of its places are (k-)bounded and/or safe.
22. ### Boundedness 22 If in the net N, both places are

assigned capacity 2, we obtain a Petri net with place capacities, say N2 An unbounded petri net N A 2-bounded version N2
23. ### Liveness 23 A transition t in a Petri net N

is said to be: ▶ L0-live or dead, if t can never be fired in any firing sequence ▶ L1-live or potentially fireable, if t can be fired at least once in some firing sequence of the Petri net ▶ L2-live, if for a given positive integer k, t can be fired at least k times in some firing sequence of the Petri net ▶ L3-live, if t can be ﬁred infinite number of times in some ﬁring sequence which can be obtained from M 0 of N ▶ L4-live, or live, if t is L1-live for every marking in reach(N)
24. ### Liveness 24 ▶ A Petri net is called Lk-live if

every transition in the petri net is k-live, for k = 0, 1, 2, 3, 4 ▶ L4-liveness is the strongest ▶ L4-liveness ⇒ L3-liveness ⇒ L2-liveness ⇒ L1-liveness
25. ### Reversibility 25 A Petri net (N, M 0 ) is

reversible if, in any marking M k reachable from initial marking M 0 , M 0 is reachable from Mk, that is, it is always possible to go back to the initial marking.
26. ### Persistence 26 ▶ A Petri net (N, M 0 )

is persistent if, for any two enabled transitions, the firing of one will not disable the other. ▶ An enabled transition, in such a petri net, stays enabled till it fires.

You!