# Petri nets boundedness

I want to ask about Petri net (PN) boundedness. When I have a state s1 = (2 0 0), then I find state s2 = (2 0 1), so since s1 < s2 can I declare the net as not bounded ? Because when I have this PN:

the PN is bounded but you can find there (2 0 0) < (2 0 1). Am I wrong about boundedness of Petri nets or is something wrong with the PT on the picture ?

• Can you identify what is the source of your uncertainty? What is the definition of "bounded" for Petri nets in the textbook/resource you're using? Have you tried plugging into the definition? If yes, where do you get stuck? How do you know that the Petri net in the picture is bounded?
– D.W.
Nov 8, 2015 at 22:40

Unboundedness can be studied from a fixed initial marking (=state). Then you want to find a state that is reachable, by firing transitions, but larger than the initial one.

You mention both states s1=(2,0,0) and s2=(2,0,1), which is in the picture. The problem is that I do not see how either one can be reached from the other.

So, start in the initial state, and analyse which transitions can be fired.

For the given Petri Net, the number of tokens for P_1,P_2 and P_3 is represented by m_1,m_2 and m_3 respectively. Based on the definition of boundedness of a Petri Net by Popova-Zeugmann (2013), the given Petri Net is bounded because m_1 ≤ 2,m_2 ≤ 2 and m_3 ≤ 2. In other words, there is a natural number (2 in this case) such that the number of tokens in P_1,P_2 and P_3 never exceeds the number.

I am also providing an example of an unbounded Petri Net and an example of a bounded Petri Net. Regardless of the initial marking, the Petri Net in Figure 1 is not bounded (or it is unbounded) because m_0 → ∞ when T_0 keeps firing.

For the given initial marking, the Petri Net in Figure 2 is bounded because m_0 ≤ 1 and m_1 ≤ 2. In general, the Petri Net in Figure 2 is bounded: let the initial value of m_0 = q_0 and the initial value of m_1 = q_1; m_0 ≤ q_0 and m_1 ≤ 2q_0+q_1

## References

Popova-Zeugmann, L. (2013). Chapter 2 The Classic Petri Net [Electronic version]. Time and Petri Nets. Berlin; Heidelerg: Springer-Verlag.

• Should that first reference be a link?
– Raphael
Nov 10, 2015 at 7:11
• Thank you for the helpful answer! A tip: You don't need to add a reference to the question that you are answering. The title of the question you're answering is right there at the top of the page, so it's just extraneous verbiage. Incidentally, it's possible to use LaTeX here to typeset mathematics, if you wish. See here for a short introduction.
– D.W.
Nov 10, 2015 at 7:33
• When I replied to this question, I did not have enough "points" to include more than one "link" in my reply. It is the reason why I only have "one image" -- considered a link by the online editor. Nov 11, 2015 at 4:20