SEKE 2012 Proceedings - Knowledge Systems Institute

Decidability of Minimal Supports of S-invariants and the Computation of their Supported
S-invariantsof Petri Nets
Faming Lu, Qingtian Zeng * , Hao Zhang,Yunxia Bao, Jiufang An
Shandong University of Science and Technology
Qingdao, China
Abstract—S-invariants play an important role in the structural
property analysis of Petri nets and there is no algorithm that can
derive all the S-invariants of a Petri net in polynomial time.
Fortunately what needed to do in some practical applications is
only to decide whether or not a given place subset is a minimal
support of S-invariants or to compute one of its supported S-
invariants. For this reason, a sufficient and necessary condition
for a place subset to be a minimal support of S-invariants is
proved in this paper. After that, 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.
Keywords: Petri nets; S -invariants; minimal supports of S-
invariants
I. INTRODUCTION
Petri nets are widely applied in modeling and simulation of
flexible manufacturing system, workflow management,
discrete event systems and many other fields [1-4] . S-invariants
are one of the basic analysis tools of Petri nets from which we
can analyze such properties as co nservativeness, liveness and
other important properties of a system.
For the computation of S-invariants, reference [1] h as
already pointed out that there is no algorithm which can derive
all the S-invariants of Petri nets in polynomial time
complexity. Even so, t here is still a lot of work devoted to
deriving S-invariants. In [5], a linear programming based
method is presented to compute part of S-invariant’s supports,
but integer S-invariants can’t be obtained. In [6-7], a
Fourier_Motzkin method is presented to compute a basis of all
S-invariants, but its time complexity is exponential. Reference
[8-9] put forward a ST FM method which has a g reat
improvement in efficiency compared to the above methods,
but there are some kinds of Petri nets the S-invariants of which
can’t be obtained with STFM method and the STFM method
has an exponential time complexity too in the worst case.
So the difficulty of deriving S -invariants is high. What
needed to do in some practical applications is only to decide
whether or not a given place subset is a minimal support of S-
invariants, or to compute only one S-invariant for a g iven
place subset. Even so, there is no efficient algorithm by now
This work is supported partly by the NSFC (61170079); Sci. & Tech. Development Fund
of Shandong Province of China (2010GSF10811); Specialized Research Fund for the
Doctoral Program of Higher Education of China (20103718110007); Sci. & Tech.
Development Fund of Qingdao(10-3-3-32-nsh and 2011-2-47), Excellent Young Scientist
Foundation of Shandong Province (BS2009DX004 and BS2010DX009);Natural Science
Foundation for Distinguished Young Scholars of Shandong and SDUST (JQ200816 and
2010KYJQ101),Guiding project of Coal Ministry(MTKJ2011-370); Project of Shandong
Province Higher Educational Sci.&Tech. Program(J12LN11); Research Project of SUST
Spring Bud(2010AZZ177, 2010AZZ069).
* corresponding author: Q. Zeng, Email: qtzeng@sdust.edu.cn
which can decide t he minimal supports of S-invariants or
compute an S-invariant for a given place subset. In this paper,
two polynomial algorithms for th e decidability of a minimal
support of S-invariants and for the computation of an S-
invariant supported by a given minimal support are presented.
The rest of this paper is organized as follows. In Section 2,
we provide the basic concepts about Petri nets and S-
invariants. In Section 3, so me properties of S-invariants are
presented. With these properties, a pol ynomial algorithm is
provided to decide th e minimal supports of S-invariants in
Section 4. After that a polynomial algorithm to compute an S-
invariant for a giv en place s ubset is presented in Section 5.
Finally a case study and a conclusion are given in Section 6-7.
II.
BASIC CONCEPTS ABOUT PETRI NETS
A Petri net is a 5-tuple ( ST , ; FW , , M)
, where S is a
0
finite set of places, T is a finite set of transitions,
F S T ( T S)
is a set of flow relation, W : F 1, 2,3,... is a
weight function, M : S {01,2, }
is the initial marking,
0
and S T S T
. Usually, a P etri net can be
represented by a bipartite graph just like in Fig.1 where a place
is presented by a circle, a transition is presented by a rectangle,
a flow relation is presented by an arc and t he marking is
presented by those black-points in places.
2
t 1
s
2
1
t 2 t
t 4
3
s 2 s 3
s 4
s 5 s 6
Fig 1. A Petri net example 1
2 1 0 0 1 0
0 1 1 0 0 1
A
0 0 1 1 1 0
2 0 0 1 0 1
According to [3-4], the incidence matrix of is the above
1
matrix A with each row corresponding to a transition and each
column corresponding to a place. Place subsets
S
1 s , s , s , s
1 2 3 4
and S
2 s , s , s
1 4 5
are two minimal
supports of S-invariants of . It has been proved in [4] that
1
there is a un ique minimal S-invariant supported by a given
minimal support. And the minimal S-invariants supported by
S and S are Y
1
2
1 1 2 2 2 0 0
T
and
Y
2 1 0 0 2 2 0
T
respectively. S
3 s , s , s , s , s
1 2 3 4 5
is
a support of S-invariants too and Y
3 1 1 1 2 1 0
T
is
one of its supported S-invariants. But S is not a minimal
3
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