Faming Lu Shandong university of Science and Technology Qingdao, China

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 • AnS-invariantisa 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:Y1, Y2and Y3areall S-invariants. ||Y1|| and ||Y2|| aretwo minimal supports while ||Y3|| is a support but not a minimal support because||Y1||=||Y1||∪||Y2|| .

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 S1 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 S1 . • Examples: considering and in Fig.1. After the following elementary row transformation, we can see that =[0.5 1]Tis an positive solution and . According to the above theorem, S2 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 • Thankyou!