▶ 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
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
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
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.
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)
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
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.
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.