Decidability of Minimal Supports of S-invariants and the Computation of their Supported S-...
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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
Decidability of Minimal Supports of S-invariants and the Computation of their Supported S- invariants of Petri Nets Faming Lu Shandong university of Science
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
Slide 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
Slide 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
Slide 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 ||.
Slide 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-invariants supports, but
integer S- invariants cant 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 cant 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.
Slide 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.
Slide 7
Main Conclusions Decidability algorithm of a minimal support of
S- invariants
Slide 8
Main Conclusions Construction of a non-trivial integer solution
for Examples:
Slide 9
Main Conclusions Computation of a minimal-supported
S-invariant
Slide 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
Slide 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
