1. Decidability of Minimal Supports of S-invariants and the Computation of their Supported S- invariants of Petri Nets Faming Lu Shandong university of Science and Technology Qingdao, China

2. Outline Basic concepts about Petri nets and S-invariants Review about the computation of S-invariants Main conclusions of this paper Outlook on the future work Q&A

3. Basic concepts Petri net A Petri net is a 5-tuple, where S is a finite set of places, T is a finite set of transitions, is a set of flow relation, is a weight function, is the initial marking, and       Graph representation & incidence matrix

4. Basic concepts S-Invariants & supports of S-invariants An S-invariant is a non-trivial integral vector Y which satisfies, where A is an incidence matrix of a Petri net An support of S-invariant Y is the place subset generated by, where S is the place set of a Petri net. Examples:Y 1, Y 2 and Y 3 are all S-invariants. ||Y 1 || and ||Y 2 || are two minimal supports while ||Y 3 || is a support but not a minimal support because ||Y 1 ||=||Y 1 || ∪ ||Y 2 ||.

5. Review about S-invariants Computation reference [1]: no algorithm can derive all the S-invariants in polynomial time complexity. Reference [5]: a linear programming based method is presented which can compute part of S-invariant’s supports, but integer S- invariants can’t be obtained References [6-7]: a Fourier_Motzkin method is presented to compute a basis of all S-invariants, but its time complexity is exponential. References [8-9]:a Siphon_Trap based Fourier_Motzikin method which has a great improvement in efficiency on average, but there are some Petri nets the S-invariants of which can’t be obtained with STFM method and the the time complexity is exponential in the worst case. This paper: two polynomial algorithms for the decidability of a minimal support of S-invariants and for the computation of an S- invariant supported by a given minimal support are presented.

6. Main Conclusions Judgment theorem of minimal supports of S-invariants Let be an arbitrary non-trivial solution of. Place subset S 1 is a minimal support of S-invariants if and only if and is positive or negative, where is the generated sub-matrix of A corresponding to S 1. Examples: considering and in Fig.1. After the following elementary row transformation, we can see that =[0.5 1] T is an positive solution and. According to the above theorem, S 2 is an minimal support, as is consistent with the facts.

7. Main Conclusions Decidability algorithm of a minimal support of S- invariants

8. Main Conclusions Construction of a non-trivial integer solution for Examples:

9. Main Conclusions Computation of a minimal-supported S-invariant

10. Outlook on the future work Based on the conclusion presented in this paper, we have realized the following algorithm with Matlab: –(1)An algorithm used to judge the existence of S-invariants and generate one S-invariant if it exist, which is a polynomial time algorithm on average. Running Time(Unit:100seconds) Number of Place s/Transitions Petri nets with (|T|*|S|)/3 flows on average Petri nets with 2*(|T|*|S|)/3 flows on average The running time statistics of the above algorithm

11. Outlook on the future work – (2)An algorithm used to judge the S-coverability of a Petri net and generate a group of corresponding S-invariants, which is a polynomial time algorithm on average too. Running Time(Unit:100seconds) Number of Place s/Transitions Petri nets with (|T|*|S|)/3 flows on average Petri nets with 2*(|T|*|S|)/3 flows on average The running time statistics of the above algorithm

12. Q&A Any questions, please contact fm_lu@163.com Thank you!