CC BY
17
II. IHOOPMATHKA
YAK 519.873 + 519.7.24/25
LINEAR ALGEBRA AND PETRI NETS
A.Bourjij, M.Boutayeb
This contribution is devoted to fast computation of a basis of invariants (P-invariants or T-invariants) in Petri Nets with extension to large-scale systems. Unlike the previous techniques, our approach consists first, in transforming the homogeneous equation into a reduced form and next, with the aid of performed simple rules, a basis of invariants is deduced. In the second part of this note, we propose a useful decentralized algorithm for computing invariants in large-scale interconnected Petri Nets. The main feature of the proposed technique lies in elaborating each subsystem's decision by using only the local incidence matrix and by the aid of an adjustment procedure we deduce the global solutions taking the constraints interconnection into account. Furthermore, it is shown that computational requirements are reduced considerably in comparison with the global approach. Accuracy and performances of the proposed approach are illustrated through numerical examples.
INTRODUCTION
Petri Net has been one of the most frequently used tools for modeling and analysis of discrete event systems since its origins, about thirty years ago. It has been particularly used for representing computer systems to describe concurrency, conflicts synchronization of processes etc. Furthermore, as a graphical tool, Petri Net is well adapted to supervise dynamical systems in real time and thus to improve performances, such as the reliability and productivity of processes. It is not intended here to give a total overview and summary of the theory and applications of Petri Nets; for more details the reader is referred to T. Murata's [1] paper and the references mentioned inside.
In this note, we are interested in the particular case of determining a basis of invariants, which constitutes a powerful tool for studying the structural properties of Petri Nets[2]-[5], [15]. Several methods have been developed in literature to deal with this problem, see for example [11]-[14]. The basic idea of these techniques is to generate the space solutions by several linear transformations of the incidence matrix [7].
The method we put forward in this paper has an important advantage in that computational requirements are reduced in comparison with the existing methods. This property is particularly relevant when large-scale Petri Nets, or when time varying incidence matrices, are considered. Indeed, many dynamical processes, such as distributed computer systems, are described by a time varying incidence matrix, which may be due to changes in the communication structure, or to changes in system configuration. Thus structural properties must be determined continually in order to enhance the per-
formances of the process. The proposed approach consists first, in transforming the homogeneous equation into a reduced form. We next establish simple rules to determine the whole space solutions. To show the efficiency of the proposed algorithm, a numerical example is provided and computational requirements are compared to those of the method in [13], which is based on the well-know Farkas algorithm^].
In the second part, we propose a decentralized algorithm for computing invariants in large-scale interconnected Petri Nets, the latter may be obtained from systems that are composed of geographically distributed systems. The main feature of the proposed architecture lies in elaborating each subsystem's decision using only the local incidence matrix and with an adjustment procedure ensures the global solution taking the constraints interconnection into account. In the proposed architecture, we may use any method to determine invariants of the subsystems.
In the last section, we discuss some aspects of the decentralized algorithm implementation and the computational requirements. One advantage of the above structure is the possibility to implement the proposed algorithm on a multiprocessor environment where each subsystem is treated by a local processor. What is more, all local solutions are transmitted to a co-ordination processor to deduce the global solution.
Furthermore, we give an idea of the computational savings in the global and decentralized implementations of the proposed method. To show performances of the proposed technique, a numerical example is provided.
BASIC TERMINOLOGY
A Generalized Petri Net is a 4-tuple (P, T, F, Mq ) where: P = {P1, P2, ..., Pn} is a finite set of places, T = {Tp T2, ..., Tm} is a finite set of transitions, P n T = 0 and P u T , F c (P X T) u (TX P) is a set of arcs, Mq : P ^ N the initial marking is a n-vector where the ith entry Mq (P i) represents the number of tokens in place Pi, N is the set of positive integer numbers (hereafter, Z is
used as the set of integer numbers, Q as the set of rational numbers and R as the set of real numbers).
The fundamental relation of Petri Nets is:
tioned into the following form:
M = M0 + C G
(1)
C. x = 0
C=
where C e Zn x m is the incidence matrix, only generalized and capacity PNs are considered.
M is the marking reachable from Mq through a firing sequence.
O e Nm is called the firing count vector where the ith entry of O denotes the number of times the transition Ti must fire.
T-invariants are positive integer solutions of the homogeneous equation:
C1 C2
! C3 C4 "
with rank ( C ) = mi
(3)
where block C^ e Zmi X mi is a square full rank matrix, Ci is then invertible, with rank(C) = rank(Ci). C2 is an mi by m2 block matrix; C3 and C4 are block matrices of appropriate dimensions.
As the submatrix % C3 C4 & is linearly dependent on
(2)
C1 C2 J , we have
It corresponds to the firing sequence which can leads back to the initial marking and can help to improve the liveness of a system. P-invariants are solutions of the transposed incidence matrix:
cT.y = 0 .
It proves that the token count in a directed circuit is invariant under any firing. The solution of this kind of systems give invariant relations that can be used for the diagnosis of the modeled system [6]. The invariants computing can be reduced to the resolution of a homogeneous system (2) and so, hereafter only T-invariants are considered. To obtain the P-invariants enough is to transpose the incidence matrix C.
In this case the space of the solutions can be extended to Zn or Qn (as it can be shown in section 3 the space dimension is the same in Qn and Rn ).
I. INVARIANT COMPUTING
Most of the classical techniques for P and P-invariants computing consist in starting with the matrix [ C|/] and by linear combinations of its columns transform it into an equivalent form [0 |D] where the rows of the matrix D are all the invariants. This is an attractive and simple method to deal with this problem, but for large scale Petri Nets computational requirements are fairly numerous.
In the sequel we present a simple algorithm to determine the invariants [20], which has the advantage of being easily implemented and at the same time going faster than the previous techniques.
1. Proposed method
We are going first to reduce the system (2) to an equivalent form and after we give theorems that permit the computation of the solution in different spaces.
Consider the incidence matrix C e Zn X m , by appropriate permutations of rows and columns, it can always be parti-
C3 C4 J. = I j C1 C2 J V
C3 C4 J and ! Ci C2 J denote the ith and the jth
rows of ! C3 C4 " and ! Cx " respectively.
These constraints are redundant for the system resolution and thus the system (2) is equivalent to:
C1 C2 JX = 0 or ! C1 C2
( \
x2
! 2 J
=0
(4)
( \
with x =
x
! 2 J
, x1 e N 1, x2 e N 2
and m = m 1 + m2 .
Thus, instead of resolving (2) in the full order, we are interested in finding vectors x so that equation (4) is fulfilled.
From (4) we obtain a reduced and equivalent form:
x 1 = Mx2
(5)
with M=-C-iC2 or equivalently M= .(com(Ci))T.C2
1 2 Det (Ci) 1 2
where com(Ci) is the cofactors matrix of Ci .
Our goal now is to find out conditions on the matrix M so as to construct xi and x2 . The vectors xi and x2 are
related by equation (5). Remark:
While the elements of the matrix Ci are integers, elements of com(Ci) are integers too. But if |Det(Ci)| ^ i ,
k.M M
k • Im2 1 m-2
then M e (Q)m1 x m2 and the solutions x can be rational vectors. Thus, the space of the solutions depends on the matrix M, and by an appropriate choice of X2 we obtain Xi .
The rest of this section is dispatched into four parts that give the solutions to system (2) in Rm , Qm, Zm, and finally Nm .
Solutions in Rm
Below, we recall two well-known theorems in system resolution [17].
Theorem 1: The system (2) of n equations and m unknowns has a solution if and only if:
Rank(C) < m and then Det(C) = 0 .
Theorem 2: Furthermore, if r = rank(C) < m , the system has exactly m-r linear independent solutions.
Thus, the solutions are computed in the same way as in
the next part of this section (see solutions in Qm ), but if C is full column rank( mi = m) the system has no other solution than the trivial one: X = 0 (see theorem 1 above). Solutions in Qm
Theorem 3: The space dimension of the solutions to system (2) is m2 . By choosing vectors X2 in the canonical basis or
equivalently the columns I. of an identity matrix, we obtain Xj = M.I. with 1 < j < m2 , then a basis of the space of canonical basis of dimension m2
Theorem 4: The integer solutions to the system (2) are obtained by multiplying all the elements of the matrix S by an integer k = Det( C^ ) :
S =
Proof: (the same as in the previous section) Solutions in Nm
Below, two theorems are given to determine the whole space solutions of equation (5):
Theorem 5: If there exists a column M. of M with only
positive components, then the system (2) has exactly m2 linearly independent solutions in Nm . Proof and solutions computation:
If M has a positive column j, we can always choose X2j large enough to verify:
j - 1 m2 x1j = x2j • Mj + Z x2k-Mik + Z x2k-Mik ^ 0
k = 1
k = j + 1
for 1 < i < m
To obtain a basis of the solutions we choose: X2 in the
and
set
solutions to the system (2) is given by the columns of the following matrix S
x2j = a = |Min(Mkh)\ 1 < k < 1 < h < m2 .
For example if m2 = 4 , the constructed vectors X2 are:
S=
M
where M is given by (5) and Im is the identity matrix of dimension m2 .
Proof: Applying theorem 1 and 2 we obtain: Rank ( C ) = m 1 m = m 1 + m2
then the maximum number of linearly independent solutions to this system is m2 .
The solutions are the columns of the matrix S and we have:
Rank( S) = Rank (I ) = m2
thus we obtain m2 linearly independent solutions n
Solutions in Zm
In the previous section we have shown that the matrix M can contain rational elements if |Det( Ci )| ^ 1 .
1 0 0 0
a a a a
0 0 1 0
! 0 " ! 0 " ! 0 " ! 1 "
^ row j
The chosen basis is canonical and all other solutions are linear combinations of this basis.
Then the columns of the following matrix S give a basis of the solutions:
r 1
M-R
S= with R =
R • Im2
(
\
10 ... 0
a ... a a
0 1 0
0 ... 0 1
R is an identity matrix with all components of row j set to value a (where j is the number of the positive column of M).
Axiom 1: If all the components of M verify:
Mj > 0 (and not necessarily M. > 0 ) then the solutions
are immediate and are given by theorem 4.
m
2
Lemma 1:
i - if there exists a row Mt of M with only strictly negative components:
M/Mij < 0, i < j < m2, My e Z
the unique and trivial solution of (5) in Nm is:
( \
x=
x2 ! 2
Lemma 2: If the theorem 5 and the lemma 1 cannot be applied, a necessary but not sufficient condition to obtain a positive column in matrix M is: each row of M must contain at least one strictly positive component. Proof:
We can easily find positive integers (bp ..., bm ) such that (5) is transformed into an equivalent form like:
x1 = MDD-1x2
X1 = Mx2
(8)
(9)
2 - if there exists a rowMk of M with negative or null components
Mk/Mj < 0, 1 < j < m2, Mij e Z
the space dimension of the solutions is ( m2 - s ) where s is the number of strictly negative components.
3 - if contains d (d>1) negative rows (like Mk ), we create a vector v:
d
v = I Mk
k = 1
where M is with a positive column Mj and the matrix D is in the form:
% 10 ... 0 b1 0 ... 0 A
D=
0 1 ... 0 b2 0 ... 0
! 00 .•• 0 bm2 0 - 1
(10)
% h & b1
then the space dimension of the solutions is (m2 - s )
where s is the number of strictly negative components of v. Proof:
i - From (5) the ith component of xi is given by:
The vector
is the jth column of D with b, > 0
! bm2 J
with M = MD and X2 = DX2
(11)
Xl 1 = I Mijxj2 j = 1
(6)
as Mij < 0 for j = i, ..., m2 and with the condition Xn > 0 , the vector X2 must be set to zero, so from (6) we
deduce that also x1 = 0
2- and 3- to obtain m2 linearly independent solutions, we choose vectors x2j satisfying:
Then we apply theorem 5 on the obtained system (8), which gives the solutions.
Corollary 1: If the theorem 5 and the lemma i cannot be applied and it is not possible to transform the matrix M by use of lemma 2, then this approach cannot be used directly.
But, we can transform the system into an equivalent form, like this:
x1 = M. x2
^ [ M - 1 ]
[x2j] is an m2 x m2 matrix
if 3*/Mlk < 0 and Mi(j < 0)
=0
(7)
with 1 < j < m2 and j ^ k, then to be sure that x is a positive integer vector we must choose X2 k = 0 and therefore rank([x2j]) = m2 - 1 .
If M has s rows verifying condition (7) then rank([x2j]) = m2 - s because [x2j] has s null rows.
rank([X2j]) = m2 .
We obtain a new reduced homogeneous system made of two blocks, where the first is compact and the second is sparse. The Farkas algorithm is very appropriate for this system because the columns of the block M contain positive and negative components that are necessary for the transformations of the system.
Finally, the method can be synthesized in the following algorithm:
or
m
2
x
2
Step i:
Compute permutation of rows and columns of C to obtain the four blocs matrix (3) and memorize these permutations. Step 2:
Compute M = |Det( Ci )|. C-i.C2 . Step 3:
If (M has a row < 0) Then
The system has no other solution than 0 in N
Else
Find or construct a positive column of M Compute the solutions
Endif Step 4:
With use of memorized permutations made in step i rearrange the solutions x. Step 5:
Extract the minimal support of invariants from the computed basis. To do this we have: first to divide each column by the highest common factor of this column, and after to subtract from each column, which is linear combination of other columns, these latter.
2. Numerical example
Let C be the incidence matrix of a Petri Net with nine places and ten transitions [i8]:
% -i0 i i i 0000 & i -i -i 0 0 0 0 0 0 0 i 0 -i -i 0 0 0 0 0 0 -i 0 -i i 0 0 0 000 -i00i00 -i 0 0 0 0 0 i i i i 0 0 0 0 -i -i 0 0 0 0 0 0 0 i 0 -i -i 0 i 0 0 0 0 -i 0 -i 00 i 0000 -i 0 y
The step i of algorithm 2 gives the two blocks Ci and
then the reduced form (5) can be computed by use of Ci
C =
C2 , see (4)
C1 =
% -10 1 1 10 &
1 -1 -10 0 0
0 0 -10 -11
0 0 0 -1 0 0
-1 0 0 0 0 0
! 0 1 0 0 0 0 "
C1 is always a square full rank matrix (then invertible).
f
C2 =
A
0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 ! -10 -1 "
and C
2
Det( C1 ) = 1 ^ M = M =
% 111 & 1 0 1 0 1 0 1 0 0 0 0 1 ! 0 11 "
While all components of M are positive or null we can apply the axiom 1, then the space dimension of the solutions is m2 = 3 , and the solutions are given by the columns of the matrix S:
S =
M
After permutations memorized in step i and the execution of step 5, T-semiflows are:
Si = [i i 0 i 0 0 i 0 0 , S2 = [i 0 i 0 0 i 0 i 0 ,
and S3 = [i i 0 0 i i 0 0 i] .
To compute P-semiflows we use C', the transpose of the incidence matrix C. The main blocks Ci and C2 are:
C\ =
% -1 1 0 0 -1 0 $ % 0 1 0 0"
0 -1 0 0 0 1 1 0 0 0
1 -1 -1 0 0 0 , C'2 = 0 0 0 1
1 0 0 -1 0 0 -1 0 0 0
1 0 -1 0 0 0 -1 0 0 0
! 0 0 1 0 0 0 " ! 0 -1 1 0^
' 1 1 -1 2 $
1 0 0 2
0 1 -1 1
0 1 -1 1
0 0 1 1
! 0 0 0 1 "
then M =
M has a positive column, then by application of theorem 5, the space dimension of the solution is 4, and a basis of the solutions is the column of S:
% 3 3 i 2 &
3 2 2 2
3 i i i
i 2 0 i
i 2 0 i
i i 2 i
i 2 i i
0 0 i 0
i i i i
v i i i i "
S=
m
2
the step 5 of the algorithm gives the P-invariants:
F1 = [1 1 1 0 0 0 0 0 0 0 ,
f2 = [1 0 0 1 1 0 1 0 0 0 ,
F3 = [0 1 0 0 0 2 1 1 1 1] , F4 = [1 1 0 1 1 1 1 0 1 1] .
3. Computational requirements
In this paragraph, we compare execution time of two kinds of algorithms for invariants computing. The algorithm 1 is the Farkas algorithm and the algorithm 2 is our approach. The two algorithms have been implemented as Matlab procedures in a Macintosh centris 650 and tested for different incidence matrices. The obtained results are given in the table 1.
Table 1
Algorithm Size of C Execution time in Sec
1 10 x 9 0.75
2 10 x 9 0.95
1 24 x 22 14.5
2 24 x 22 6.48
1 48 x 44 89.22
2 48 x 44 32.28
group technology techniques [9], [10], or other mathematical tools to transform a sparse matrix into an equivalent diagonal block matrix as presented in the fig.1.
sparse incidence matrix
residual ties
-7TV
diagonal blocks
Figure 1
But the different blocks of the obtained matrix cannot be used separately to compute the invariants of the Petri Net due to the presence of residual ties. In order to do that, we apply a second transformation to the incidence matrix.
The columns containing residual ties are moved to the right of the matrix after the last column, as it can be shown in the fig. 2.
residual ties column
The incidence matrices are sparse and the use of the reduction procedure (steps 1 and 2) of algorithm 2 takes advantage of this property. Therefore, as can be seen from the previous table, the computational requirements of the proposed procedure are less numerous than for the first algorithm as the size of C (incidence matrix of the PN) increases.
12 3 4
1234
II. DECENTRALIZED APPROACH
1. Problem formulation
For large-scale systems, incidence matrices are huge, thus invariants time computing can be excessive, and analyzing this kind of Petri Net is not easy [16]. To optimize the analysis of these Petri Net models, we propose to decompose huge incidence matrices into much smaller blocks using the following technique[8],[19].
2. Proposed approach
To perform an algebraic analyze of Petri Nets we can compute invariants by solving these two homogeneous systems:
C. x = 0 and C'.x = 0
(12)
Figure 2
We may recall that all these transformations are regular and do not affect the solutions of the homogeneous system: C.X = 0 .
Finally, the incidence matrix of large scale Petri Nets is in the form:
(
\
C1 0 . 0 Cc 1
! 0 . 0 CpCcp "
(13)
An important remark is that C incidence matrices of large-scale systems are sparse. Thus it is always possible, by use of
In order to analyze one huge system we can analyze p smaller subsystems represented by their incidence matrices:
CC J for i = 1.....p
where the block Cci represents the coupling constraints.
Thus invariants are the solutions of the following homogeneous system:
1 < i <p of the global solution.
A simple procedure to obtain a basis of the solutions of the system (4), is to hold solutions from
n Sc/{0 }
i = i
(19)
Ci 0 . 0 Cd
( \
0 . 0 CpCcp J
=0
c
«
, C1 Xi + °x2 + ... + 0Xp + CciXc = 0 °*1 + C2X2 + ^ + 0Xp + Cc 2Xc = 0
0 X1 + 0x2 +
+ CpXp + CcpXc = 0
«
C1X1 + Cc 1Xc 0 C2 X2 + Cc 2Xc = 0
CpXp + CcpXc = 0
« ! CiCci
% & X
Xc V c
= 0 for i = 1, ...,p
(14)
where Sci is the set made by the coupling part of the basis
of the solution of the subsystem (i).
In other words, the local solutions that have the same coupling part are concatenated to give the global solution.
Corollary 1 : A solution
% & X
V Xci J
of the subsystem (i) and a
(15)
( \
solution
Xcj
V cJ J
of the subsystem (j) are the same part of a
(16)
solution of the global system if and only it have the same coupling components X ,. If X , = 0 the local solution
f \ X
V Xc J
is also a global solution
(17)
0 0
Xi Xi
0 0
V Xci V 0 J
Proof: For a system made up by two subsystems, let
With this decomposition we can notice that the invariants (also called "local invariants" in the following text) of each subsystem (7) can be computed simultaneously in a parallel or distributed architecture computer. Let Si be a generatrix
basis of the solution space of the subsystem (i), 1 < i <p :
CC
i ci
f \ X
V Xc J
=0
c1
x-2
XXc 2
be the ith solution of the subsystem (1) and
the jth solution of the subsystem (2). These two
solutions are each a part of the same solution of the global system if and only if:
and Sg a generatrix basis of the solution space of the global system (4), we can deduce from the previous equation that:
% X1 &
C.
= 0 and C.
c2
V J
=0
£ U Si -
(18)
Remark 1: By the proposed decomposition (7), local invariants computation can only gives a part of the global solution. To obtain the global solution we must concatenate the xt 1 < i < p components that have the same coupling
part (see corollary below and its proof).
The question is now: what local solutions must be concatenated to made a global solution?
The resolution of each subsystem gives a part
«
C1 0 Cc1
0 C2 Cc 2
C1 0 Cc1
0 C2 Cc 2
V Xc 1 J
=0
c2
=0
C1 x1 + Cc 1xc 1 0
C2 x2 + Cc 2xC 1 = 0
^ +
C1 x1 + Cc 1xc 2 = 0
C2x2 + Cc2xcj1 = 0
We can deduce: CciXCi = CcixJc2 and Cc2Xci = Cc2xc2 . For a system made up by p subsystems the proof is the same.
If a solution
xi x
C1 ••• Cc 1
• C •
0 • Ccp ,
cP
\ (
0
xi
0
! 0
= 0 if and only if C,x, = 0
interconnected subsystems, the time needed by the step i can be significant.
4. Numerical example
The example proposed below is not a huge model, but the method is just the same. This Petri Net model consist of two subsystems coupled by two places P4 and P8 (see Fig.3):
of the subsystem (i) verify:
xci = 0 , (solution with no coupling part)
this local solution is also a global solution. Due to the block diagonal form of the incidence matrix C:
Here is the approach summed up in a six-step algorithm: Step i:
Break up the incidence matrix of the Petri Net model into a diagonal block matrix (but residual ties appear). Step 2:
Rearrange the incidence matrix to isolate residual ties. Thus we obtain separate subsystems. Step 3:
Compute invariants of each subsystem by appropriate method (the proposed one or the Farkas approach). Step 4:
By the help of invariants retrieve the structural properties of each subsystem. If the model is inaccurate, it must be modified then repeat from step i. Step 5:
By means of corollary i assemble local invariants to obtain the invariants of the global system. Step 6:
Deduce the main properties of the global solution. 3. Computational requirements
Due to the block diagonal decomposition the invariants of each subsystem can be computed separately. Thus, the required computing time T will be the greatest computing time of each subsystem. If is the computing time of the
subsystem (i), and by using a parallel or distributed computer:
T = max (t;-)
ii
we can neglect the necessary time to assemble the local solutions (step 5), but for huge Petri Nets, which are not
Figure 3 - Petri Net model
Step 1:
The associated incidence matrix is:
CT =
-1 1 0 1 0 0 0 0
0 -1 11 0 0 0 0 0
1 0 -1 0 0 0 0 -1
0 0 0 -1 -1 1 0 0
0 0 0 0 0 -1 0
0 0 0 0 1 0 -1 1
C1
12345678 ^
column numbers
Step 2:
This matrix is sparse and thus if we move columns 4 and 8 (these columns corresponds to the coupling part between the two subsystems) to the end of C, we obtain an equivalent matrix with two interconnected subsystems like the diagonal blocks matrix presented in (3):
CT =
-1 1 0 0 -11 1 0 -1
0 0 0 0 0 0 ! 0 0 0
0 0 0
0 0 0 0 0 0
-1 1 0 ■M
0 -1 1
1 0 -1
1 0
0 0
0 -1
-1 0
0 0 0 1
C1
Cc1
C2
Cc2
P
8
Step 3:
Now we can solve the system by the decentralized method. The resolution of the subsystem (1):
C1 Cc 1
gives two solutions:
f \
! Xc 1 J
= 0
f \
and X42 =
! 8 J
Step 4:
With the computed local P-invariants we can proof that the subsystems (1) and (2) are bounded. Step 5:
The common coupling blocks are given by:
X1 =
% 1 & 1 1 0 0
( \
( \
where Xj =
and X\x =
Xl.
! 8
X1 % & 1
x2 = 1
x3 ! 3 " V 1 "
P 2
O Sc 1 = O Sc 1 = i = 1 i = 1
Thus the global solutions are obtained by concatenation of
' 1 &
the local solutions which have the coupling part
1
where local solutions Xi and X4 which have no coupling part are then global solutions:
X2 =
( \
1 1 1 0
! 0 J
( \
where X12 =
í \
X1 1
x2 = 0
X CO v 0 "
and Xc21 =
X, ! 8 J
The resolution of the subsystem (2):
C2 Cc 2
f \
! xc2 j
=0
gives two solutions:
X3 =
% 1 & 1 1 0 0
f \
where X23 =
í \
X5 1
X6 = 1
X7 ! ' ! 1 )
and Xc32 =
X
! 8 J
X4 =
% 1 & 0 0 1 1
where X24 =
í \
X5 1
X6 = 0
X7 ! ' ! 0 J
S = {(1, 1, 1, 0, 0, 0, 0, 0)T, (0, 0, 0, 0, 1, 1, 1, 0)T (1, 0, 0, 1, 0, 1, 1, 1 )T}. Step 6:
These three P-invariants can now be used to prove that this Petri Net is bounded because all the paces are in a P-invariant relation:
M( P1) + M( P 2 ) + M( P3 ) = 1, M( P 5 ) + M( P 6 ) + M( P7 ) = 1, M (P1) + M (P 4) + M (P 6) + M (Pg) = 1.
With the same method we can compute T-invariants and verify the reversibility of the Petri Net for the given initial marking of the figure 3.
III. CONCLUSION
In this paper, a new approach to determine invariants in Petri Nets has been presented. We have shown that the invariants may be obtained directly from the reduced form of the homogeneous equation. One major advantage of the approach presented above is the low computational requirements in the case of large scale Petri Nets. Besides the space dimension of the solution can be given directly (without major computations) and that space can be reduced from Qm to Zm and
finally to Nm . This has been shown by means of a numerical examples.
A decentralized method to analyze huge systems modeled by Petri Nets is developed. The proposed algorithm consists first in partitioning the incidence matrix and determining independently the invariants of each subsystem; this may be achieved by classical methods. We give a corollary for deducing the invariants of the global system. This approach has the
main advantage that it permits a structural analysis of all the subsystems that compose the system and a structural analysis of the global system. We have shown that the architecture presented reduces significantly the computational requirements, especially for a high number of interconnected subsystems and by using a multi-processor environment. We are now trying to evaluate the performances of our algorithms in terms of elementary operations (like addition, multiplication,...).
REFERENCES
[1] T. Murata, "Petri Nets: properties, analysis and applications," Proceedings of IEEE Vol. 77, N° 4, pp. 541-580, April 1989.
[2] R. David, H. Alla "Petri Nets for modeling of dynamic systems". Automatica, 2:175-202,1994.
[3] G. Berthelot "Verification des Reseaux de Petri" These de doctorat de troisieme cycle, universite P. et M. Curie (PARIS IV), 12 January 1978.
[4] K. Lauterbach, "Linear algebraic calculation of deadlocks and traps" {\it Concurrency and nets}, ed. K. Voss and H. J. Gen-rich, Springer-Verlag, 1987.
[5] E. R. Boer and T. Murata, "Generating basis siphons and traps of Petri nets using sign incidence matrix" {\it IEEE Trans. Circuits and Systems}, Vol. 41, No 4, pp 266-271, April 1994.
[6] A. Bourjij & al. "Algebraic Analysis of Discrete Event Systems Modelled by Generalized Petri Nets by use of Invariants" Mathematical Theory of Networks and Systems MTNS'96, Saint-Louis Missouri, June 1996.
[7] A. Bourjij "Contribution a la surete de fonctionnment des processus industriels par les reseaux de Petri", PhD Thesis, University Henri Poincare of Nancy I, France, 5 Dec, 1994.
0ДК 681.32
В статье представлены результаты сотрудничества между Германией и Румынией в рамках проекта EU-TEMPUS INCOT по разработке Web-центра. Этот центр поддерживает управление знаниями и европейскую сеть экспертов.
In this paper some results of a co-operation between the Germany and Romania within the EU-TEMPUS-project INCOT about the development of a Web-based competence centre are presented. This centre supports processes of knowledge management and telelearning and an European network of experts.
1. INTRODUCTION
The Internet with different forms of supported networking and knowledge management processes play an important role within the transition to the information society and knowledge-based economy.
On one hand, new information technologies (IT) e.g. the Internet and multimedia support knowledge management through services like information-transfer, navigation and access to different applications; on the other hand, the Internet causes an increasing flood of information and a decreasing validity period of it. So (potential) users of information
[8] M. Boutayeb, A. Bourjij, M. Darouach, "A decentralized algorithm to determine invariants in Petri Nets" MATHMOD Vienna, IMACS, 5-7 February, 1997.
[9] A. Kusiak, S. C. Wing, "Decomposition of manufacturing systems" IEEE Journal of Robotics and Automation, Vol. 4, N°5, Oct. 1988.
[10] N. Dridi, J. M. Proth "Ordonnancement des taches: une methode basee sur la technologie de groupes" Second International Conference on Production Systems, INRIA, tome 1, pp. 67-74, April 1987.
[11] J. Farkas, "Theorie der einfachen Ungleichungen" J.F.d. reine Angew Math., 124, 1902, 1-27.
[12] J. Martinez and M. Silva, "A simple and fast algorithm to obtain all invariants of a generalized Petri Net," 2nd European workshop on application and theory of Petri Net, Bod-honneff, 1981.
[13] H. Alaiwan and J.M. Toudic, "Recherche de semi-flots, de verrous et de trappes dans les reseaux de Petri," TSI Vol. 4, N° 1, pp. 104-112, 1985
[14] M. Silva and J.M. Colom, "On the computation of structural synchronic invariants in P/T nets," Lecture Notes in Computer Science Vol. 340, pp. 386-417, Springer-Verleg, 1988.
[15] T. Murata, B. Shenker and S.M. Shatz, "Detection of Ada deadlocks using Petri Net invariants" IEEE Trans. on Software Engineering Vol. 15, N° 3, pp. 314-326, march 1989.
[16] D.C. Marinescu, M. Beaven and R. Stansifer "A parallel algorithm for computing invariants of Petri Net models" Proceedings of the fourth international workshop on PNs and performance models, Melbourne, IEEE Computer Soc., pp. 136-43, 1991.
[17] P. Naudin and C.Quitte "Algorithme Algebrique" Masson 1992.
[18] K. Garg "An approach to performance specification of communication protocols using timed Petri Nets" IEEE Trans. on Software Engineering Vol. SE11, N° 10, pp. 1216-1224, 1985.
[19] A. Bourjij & al. "A decentralized approach for computing invariants in Petri Nets". IEEE SMC'97 orlando pp 1741-1746.
[20] A. Bourjij & al. "On generating a basis of invariants in Petri Nets". IEEE SMC'97 orlando pp 2228-2233.
PROCESSES
should be able to identify relevant one, to transform it in goal-oriented skills (knowledge generating), to process this knowledge according to the target group and so to contribute to the development of the organization they work with.
Innovative forms of IT-based learning (e-learning) like open distance learning (ODL) by using the Internet - tele-learning - are required in order to achieve such key competencies for the staff of the companies.
In part 2 of this paper we present some aspects of knowledge management (KM), particularly knowledge distribution (KD) which are important for manufacturing companies and some possibilities to support it by using IT -[i].
In part 3, telelearning methods and virtual communities which are based on the World Wide Web (WWW) are briefly described.
WWW is a global information system with a hypermedia structure; it is independent on the operating system and facilities the searching, retrieval and browsing of information via the Internet.
Virtual Competence centres which can be developed as suitable forums for KM, communication and experience
BUILDING EUROPEAN COMPETENCE CENTRES TO SUPPORT KNOWLEDGE DISTRIBUTION AND LEARNING
I.Hamburg, S.Balanica
44
ISSN 1607-3274 "Радтелектронжа, шформатика, управл1ння" № 1, 2001
