
Research Article


Transition Firing Rules of Logic Petri Nets


Yu Yue Du,
Jing Wang
and
Yong Feng Zhang


ABSTRACT

Logical Petri nets (LPNs) can well describe and analyze batch processing functions
and passing value indeterminacy in cooperative systems. A new definition of
the LPNs is proposed based on our initial work in this study. The standard form
of logic expressions can be obtained and the logic input/output enabling vector
set is defined. How to determine the corresponding relationships between logic
input and output expressions is solved. In order to analyze their properties,
a vector matching method is given and a therom has been proved. Finally, the
feasibility of the proposed method is illustrated by an example.





Received:
November 11, 2013; Accepted: January 15, 2014;
Published: March 08, 2014 

INTRODUCTION
Petri nets (PNs) (Aziz et al., 2013) are the
mathematics representation of a parallel discrete system. PNs have a rigid mathematical
definition, with a welldeveloped mathematical theory for process analysis.
They are suitable for describing the concurrent asynchronous of computer system
models. With the continuous improvement of PN theory and the increasing popularity
of its application, some of their extensions have been defined, such as fuzzy
(Bayati and Dideban, 2012), colored (Barzegar
et al., 2011) and stochastic (Marin et al.,
2012) PNs.
Logic Petri nets (LPNs) (Du and Jiang 2002) are the
abstract and extension of inhibitor arcs PNs (IPNs) and highlevel PNs. The
inputs and outputs of a system can be described by logic transitions in LPNs,
respectively. The transitions restricted by logic input and output expressions
are called logic transitions. In our initial work, LPNs have been applied efficiently
to the modeling and analysis of trading systems (Du and
Jiang, 2002), cooperative systems (Du et al.,
2009) and electronic commerce (Du and Jiang, 2008).
Some weaknesses are found in the analysis of LPNs models. In this study, a new
definition of LPNs are proposed. The indeterminate data transmission caused
by logic output transitions is solved by introducing the matching expressions.
The standard form of logic expressions are defined and some indeterminate minterms
of the logic expression are deleted.
BASIC DEFINITIONS Three basic definitions are reviewed in this section before LPNs are introduced. Definition 1: N = (P, T, F) is a net, where:
• 
P is a finite set of places 
• 
T is a finite set of transitions with P∪T≠ø, P∩T=ø 
• 
F⊆(PxT)∪(TxP) is a set of arcs 
• 
dom(F)∪cod(F) = P∪T where: 
dom(F) = {x∈P∪T  ∃y∈P∪T: (x, y)∈F}
cod(F) = {x∈P∪T  ∃y∈P∪T: (y, x)∈F}

Definition 2: x∈P∪T is called a node in N:
^{•}x = {y (y, x)∈F} is called a preset of x
x^{•} = {y (x, y)∈F} is called a postset of x

If X∈P∪T, its preset and postset are as follows:
^{•}X = ∪_{x∈X}^{•}x
X^{•} = ∪_{x∈X}x^{•}

Definition 3: PN = (P, T, F, M_{0}) is a marked PN, where:
• 
N = (P, T, F) is a net 
• 
M:P→
is a marking function, where M_{0} is the initial marking and
= {1, 2,…} 
• 
Transition firing rules: 

• 
t is enabled at M if ∀p∈^{•}t: M(p) =
1, represented by M[t> 

• 
If t is enabled, it can fire and a new marking M' is generated from M,
represented by M[t >M', where: 
Based on the traditional definition PN, the LPN is put forward. Definition 4: Let LN = (P, T, F, I, O). LPN = (LN, M) is a logic Petri net where:
• 
P is a finite set of places 
• 
T = T_{D}∪T_{I }∪T_{O }is
a finite set of transitions, P∪T≠ø, P∩T = ø, ∀t∈T_{I}∪T_{O}:
^{•}t∩t^{•} = ø, where: 

• 
T_{D} denotes a set of traditional transitions 

• 
T_{I} denotes a set of logic input transitions, where ∀t∈T_{I},
the input places of t are restricted by a logic input expression f_{I}(t)
and 

• 
T_{O} denotes a set of logic output transitions, where ∀t∈T_{O},
the output places of t are restricted by a logic output expression f_{O}(t) 
• 
F⊆(P xT)∪(T xP) is a finite set
of directed arcs 
• 
I is a mapping from a logic input transition
to a logic input expression, i.e.,: 
∀t ∈T_{I}, I(t) = f_{I}(t)
= A_{1}∨A_{2}∨…∨A_{u} 
• 
O is a mapping from a logic output transition to a logic input
expression, i.e.,: 
∀t ∈T_{O}, O(t) = f_{O}(t)
= B_{1}∨B_{2}∨…∨B_{v} 
• 
M: P→{0,1} is a marking function, where ∀p∈P,
M(p) is the number of tokens in p 
• 
Transition firing rules: 

• 
∀t∈T_{D}, the firing rules of t are the
same as in PNs 

• 
∀t∈T_{I}, t is enabled only if ∃A_{i},
make f_{I}(t)_{M }= _{•}T_{•},
M[t>M', where ∀p∈^{•}t and p∈A_{i},
M'(p) = M(p)1; ∀p∈t^{•}, M'(p) = M(p)+1; ∀p∉^{•}t∪t^{•},
M'(p) = M(p);and ∀p∈^{•}t and p∉A_{i},
M'(p) = M(p) and 

• 
∀t∈T_{O}, t is enabled only if ∀p∈^{•}t
: M(p) = 1. M[t>M', where ∀p∈^{•}t M'(p) = M(p)1;
∀p∉^{•}t∪t^{•}: M'(p) = M(p); ∀p∈t^{•}
and ∀p∈B_{i }should satisfy f_{O}(t)_{M'}
= _{•}T_{•} and ∀p∈t^{•}
and p∉B_{i}, M'(p) = M(p) 
LPNs are the abstract and extension of inhibitor arcs PNs and highlevel PNs.
A logic input/output transition is restricted by the logic input/output expression
f_{I}(t)/f_{O}(t). All logic input/output transitions are called
logic transitions. The logic expressions can describe the indeterminacy of values
in input and output places. A_{i} and B_{i} represent input
and output ways of logic transitions, respectively. They are not the disjunctive
normal of f_{I}(t)/f_{O}(t), but the elements of A_{i}/B_{i}
are connected by the logic symbol “∧”.
Figure 1 shows a simple LPN model. t_{2} is a traditional transition and t_{1} and t_{3} are logic transitions restricted by the f_{I}/f_{O}, respectively, where f_{I} = (p_{1}∧p_{2})∨(p_{1}∧p_{2}∧p_{3}) and f_{O} = (p_{7}∧p_{8})∨(p_{7}∧p_{8}∧p_{9}). Each place of a logical expression has a logic value at marking M in an LPN and by substituting the values of all places into the logic expression, the expression corresponds to a logical value. In the LPN model of Fig. 1, M=(1, 1, 0, 1, 0, 0, 0, 0, 0)^{T}, I = {p_{1}, p_{2}, p_{3}}, having p_{1}_{M} = _{•}T_{•}, p_{2}_{M} = _{•}T_{•}, p_{3}_{M} = _{•}F_{•} and f_{I}_{M} = (_{•}T_{•}v_{C}T_{•})∨(_{•}T_{•}v_{C}T_{•}v_{C}F_{•}) = _{•}T_{•}w_{C}F_{•} = _{•}T_{•}.
FIRING RULES OF LPN TRANSITIONS From Definition 4, a logic transition is restricted by the logic expression f(t), next, the standard form of f(t) is putword. Definition 5: Suppose that a logic input/output transition t is restricted by f_{I}(t)/f_{O}(t) and the standard form is as follow: • 
For a logic input transition t, the standard form of f_{I}(t)
= A_{1}∨A_{2}∨…∨A_{m} can be obtained
by: 
For a logic input transition t, the standard form of f_{O}(t) = B_{1}∨B_{2}∨…∨B_{n }can be obtained by: A_{i} and B_{i }are called the standard minterms. Example 1: In Fig. 1, transitions t_{1} and t_{2} are restricted by f_{I}/f_{O}, respectively, where f_{I} = (p_{1}∧p_{2})∨(p_{1}∧p_{2}∧p_{3}) and f_{O} = (p_{7}∧p_{8})∨(p_{7}∧p_{8}∧p_{9}). From Definition 5, the standard form of f_{I} is f_{I} = (p_{1}∧p_{2}∧5p_{3})∨(p_{1}∧p_{2}∧p_{3}) and the standard form of f_{O }is f_{O} = (p_{7}∧p_{8}∧5p_{9})∨(p_{7}∧p_{8}∧p_{9}). A standard form of an expression consists of several standard minterms and each minterm can be represented by a vector. A standard minterm represented by a vector in this study and a standard form of an expression is described by a vector set. A logic input/output transition corresponds to an enabling vector set and a traditional transition corresponds a vector. Next, the enabling vectors are putford. Definition 6: Let LPN = (LN, M) be a logic Petri net, where P = {p_{1}, p_{2},…, p_{m}} and T = {t_{1}, t_{2},…, t_{n}}: • 
For t∈T_{I}, the standard form of f_{I}(t)
has a minterm A_{i} and A_{i} is a logic expression made
by ^{•}t. V_{i} = (v_{i1}, v_{i2},…,
v_{im})^{T} is called the enabling vector of t, where: 
• 
For t∈T_{O}, the standard form of f_{O}(t)
has a minterm B_{i} and B_{i} is a logic expression made
by t^{•}. V_{i} = (v_{i1}, v_{i2},…,
v_{im})^{T} is called the enabling vector of t, where: 
For t∈T_{D}, V_{i} = (v_{i1}, v_{i2},…, v_{im})^{T} is called the enabling vector of t, where:
In Eq. 3, 4 and 5, j
belongs to the set {1, 2,…, m}. Suppose that the standard form of f_{I}(t)
consists of u minterms, from Eq. 3, the corresponding vector
sets are V_{ii},_{1}, V_{ii},_{2},..., V_{Ii},_{u1}
and V_{Ii},_{u} and V_{ii} = {V_{Ii},_{1},
V_{Ii},_{2},…, V_{IiB},_{u}} is the logic
input enabling vector set of t. V_{I }is the logic input enabling set
of the LPN and has V_{Ii }∈V_{I}. Likewise, suppose that
the standard normal form of f_{O}(t) consists of v minterms, from Eq.
4, the corresponding minterm vector sets are V_{oi},_{1},
V_{oi},_{2},…, V_{oi},_{v} and V_{Oi
} = {V_{oi},_{1B}, V_{Oi},_{2},…,
V_{Oi},_{v}} is the logic input enabling vector set of t. V_{O
}is the logic output enabling set of the LPN and has V_{Oi }∈V_{O}.
The common enabling set V is composed by all enabling vectors of the traditional
transitions, i.e., V_{i }∈V.
The matching rule among vectors is proposed in the next.
Definition 7: Suppose that m∈^{+},
X = (x_{1}, x_{2},…, x_{m})^{T} is a 01vector
and Y = (y_{1}, y_{2},…, y_{m})^{T}, where
∀i∈_{m}
and _{m}
= {1, 2,…, m}, y_{i }∈{0, 1, *}. X matches Y donated by Y≅X,
if:
else Y≠X. Example 2: Let X = (1, 0, 1, 0, 1)^{T}, Y = (1, 0, 1, *, *)^{T} and Z = (1, 0, 0, *, *)^{T}. By Eq. 4, X ≅Y and X≠Z. Theorem 1: Let LPN = (LN, M) be a logic Petri net, where P = {p_{1}, p_{2},…, p_{m}} and T = {t_{1}, t_{2},…, t_{n}} and M_{0} is its inital marking, M∈R(M_{0}). t is enabled at M:
• 
For t∈T_{I}, suppose that V_{ii} = {V_{Ii},_{1},
V_{Ii},_{2},…, V_{Ii},_{u}} is its
logic input enabling vector set and V_{Ii}∈V_{I}. If
∃x∈{1, 2,…, u}, V_{Ii},_{x}∈V_{Ii}
and V_{Ii},_{x}≅ M 
• 
For t∈T_{O}, suppose that V_{oi} = {V_{Oi},_{1},
V_{Oi},_{2},…, V_{oi},_{v}} is its
logic output enabling vector set and V_{Oi }∈V_{O}.
If ∃x∈{1, 2,…, v}, V_{Oi},_{x }∈V_{Oi}
and V_{Oi},_{x}≅M; or 
• 
For t∈T_{D}, suppose V_{i} is its common enabling
vector and V_{i }≅M 
Proof: Suppose that t is enabled at M, i.e. M[t >:
• 
For t∈T_{I}, ∃V_{Ii},_{k}∈V_{Ii}
and V_{ii},_{k} = (v_{i1}, v_{i2}, …,
v_{im})^{T} is defined by Eq. 3 
If (t, p_{j})∈F, then v_{ij} = 0; if (p_{j}, t)
∈F and p_{j}_{Ai} = _{•}T_{•},
then v_{ij} = 1; the left v_{ij }are marked “*”.
If (t, p_{j})∈F, then M(j) = 0; if (p_{j}, t)∈F and p_{j}_{Ai} = _{•}T_{•}, then M(j) = 1; other elements of M are marked “1/0”. Therefore, if (p_{j}, t)∈F, M(j) = v_{ij}; if (t, p_{j})∈F and p_{j}_{Ai} = _{•}T_{•}, M(j) = v_{i}; the left v_{ij }are marked “*” and other elements of M are marked “1/0”. According to Eq. 6, V_{Ii},_{x }≅M: • 
For t∈T_{O}, ∃V_{Oi},_{k}∈V_{Oi}
and V_{oi},_{k} = (v_{i1}, v_{i2},…,
v_{im})^{T} is defined by Eq. 4 
If (p_{j}, t)∈F, then v_{ij} = 1; if (t, p_{j})∈F
and p_{j}_{Bi} = _{•}T_{•}, then
v_{ij} = 0; the left v_{ij} are marked “*”.
If (p_{j}, t)∈F, then M(j) = 1. If (t, p_{j})∈F and p_{j}_{Bi} = _{•}T_{•}, then M(j) = 0. other elements of M are marked “1/0”. Therefore, if (p_{j}, t)∈F, M(j) = v_{ij}; if (t, p_{j})∈F and p_{j}_{Bi} = _{•}T_{•}, M(j) = v_{ij}; the left v_{ij} are marked “*” and other elements of M are marked “1/0”. According to Eq. 6, V_{Oi},_{x }≅M: • 
For t∈T_{D}, V_{i} = (v_{i1},
v_{i2},…, v_{im})^{T} is defined by Eq.
5 
If (p_{j}, t)∈F, then v_{ij} = 1; if (t, p_{j})∈F,
then v_{ij} = 0; if p_{j}∉^{•}t∩t ^{•},
v_{ij} = *
If (p_{j}, t)∈F, then M(j) = 1; if (t, p_{j})∈F,
then M(j) = 0; if p_{j}∉^{•}t∩t ^{•},
M(j) = 1 or M(j) = 0.
Therefore, if (p_{j}, t)∈F, then M(j) = v_{ij}; if (t,
p_{j})∈F, then M(j) = v_{ij} and if p_{j}∉^{•}t∩t
^{•}, v_{ij} = * and M(j) = 1 or M(j) = 0. According to
Eq. 6, V_{i }≅M.
For logic transitions, Theorem 1 is the necessary condition of a transition to fire, not the necessary and sufficient condition. For all traditional transitons, Theorem is the necessary and sufficient condition. The firing of logic transitions are restricted by the enabling vectors and logic expressions. Next, the matching expression is put forward. Definition 8: Suppose that f_{1} = A_{1}∨A_{2}∨…∨A_{u} and f_{2} = B_{1}∨B_{2}∨…∨B_{u} are standard form. f_{1 }and f_{2 }are called matching expressions, if:
• 
The number of standard minterm between is the same, i.e.,
u = v 
• 
The standard minterm A_{i }of f_{1} corresponds to the
standard minterm B_{i} of f_{2} and the elements of A_{i
}corresponds to the elements of B_{i}, i.e., for ∀p_{m},
p_{n}∈A_{i}, their corresponding places p_{k},
p_{l}∈B_{i}, having km = ln 
Two matching expressions are causal relationship. Once a standard minterm A_{i
}of f_{1 }fired, the standard minterm B_{i }of its matching
expression could fire directionally in the running of the LPN.
Example 3: In Fig. 1, f_{I} = (p_{1}∧p_{2})∨(p_{1}∧p_{2}∧p_{3}) and f_{O} = (p_{7}∧p_{8})∨(p_{7}∧p_{8}∧p_{9}) are two matching expressions. f_{I} = (p_{1}∧p_{2}∧5p_{3})∨(p_{1}∧p_{2}∧p_{3}) and f_{O} = (p_{7}∧p_{8}∧p_{9})∨(p_{7}∧p_{8}∧5p_{9}) are their standard forms and they have satisfied the Definiton 8. They will be used in the section IV. Next, the equation vector used to calculate the marking M is put forward.
Definition 9: Let LPN = (LN, M) be a logic Petri net, where P = {p_{1},
p_{2},…, p_{m}} and T = {t_{1}, t_{2},…,
t_{n}}, For t∈T_{I}, the standard form of f_{I}(t)
has a minterm A_{i}. V' = (v_{i1}', v_{i2}',…,
v_{im}')^{T }is an equation vector of t, where:
•For t∈T_{O}, the standard form of f_{O}(t) has a minterm B_{i}. V' = (v_{i1}', v_{i2}',…, v_{im}')^{T} is an equation vector of t, where: •For t ∈T_{D}, V_{i}' = (v_{i1}', v_{i2}',…, v_{im}')^{T} is a equation vector of t, where:
In Eq. 78 and 9, j belongs
to the set {1, 2,…, m}. From Definitions 6 and 8, enabling vectors and
equation vectors are obtained by the same expression and their number is the
same. Each enabling vector corresponds to an equation vector. So V_{ii},_{1}',
V_{Ii},_{2}',..., V_{Ii},_{u1}' and V_{Ii},_{u}'
are equation vectors of t belonging to T_{I} and V_{ii}' = {V_{Ii},_{1}',
V_{Ii},_{2}',…, V_{IiB},_{u}'} is the input
equation vector set, V_{ii}'∈V_{I}'; likewise, V_{oi},_{1}',
V_{oi},_{2}',…, V_{oi},_{v}' are the equation
vectors of t belonging to T_{O} and V_{oi}' = {V_{Oi},_{1}',
V_{Oi},_{2}',…, V_{Oi},_{v}'} is the output
equation vector set, V_{Oi}'∈V_{O}' and V_{i}'∈V
is the common equation of t.
Based on Definitions 6 and 9, a therom and an algorithm are introduced in the following. Definition 10: Let LPN = (LN, M) be a logic Petri net, where P = {p_{1}, p_{2},…, p_{m}} and T = {t_{1}, t_{2},…, t_{n}} and M_{0 }is the initial marking of the LPN, M∈R(M_{0}): • 
For t∈T_{I}, V_{ii}' = {V_{Ii},_{1}',
V_{Ii},_{2}',…, V_{Ii},_{u}'} is the
input equation vector set of t and V_{ii} = {V_{Ii},_{1},
V_{Ii},_{2},…, V_{Ii},_{v}} is its
input enabling vector set. If V_{Ii},_{x}≅M, then M[t
>M' and: 
• 
For t∈T_{O}, V_{oi}' = {V_{Oi},_{1}',
V_{Oi},_{2}',…, V_{Oi},_{v}'} is its
output equation vector set and V_{oi} = {V_{Oi},_{1},
V_{Oi},_{2},…, V_{Oi},_{v}} is its
output enabling vector set. If V_{Oi},_{y}≅M, then
M[t >M': 
• 
For t∈T_{D}, V_{i}' is the common equation
vector of t and V_{i }is its enabling vector. If V_{i}≅M,
then M[t >M': 
Equation 10, 11 and 12
are used to calculate the marking M after a transition has fired. Based on the
definitions and Theorem 1, an algorithm is proposed:
Algorithm 1: 
The firing of transitions in LPNs 

AN EXAMPLE
In the LPN model of Fig. 1, LPN = (P, T, F, M), where t_{1}∈T_{I},
t_{2}∈T_{D}, t_{3}∈T_{O}, M = (1,
1, 0, 1, 0, 0, 0, 0, 0)^{T} and t_{1} and t_{3} are
restricted by f_{I} = (p_{1}∧p_{2})∨(p_{1}∧p_{2}∧p_{3})
and f_{O} = (p_{7}∧p_{8})∨(p_{7}∧p_{8}∧p_{9}),
respectively. f_{I} and f_{O} are matching expressions.
From Example 1, get the standard form f_{I} = (p_{1}∧p_{2}∧5p_{3})∨(p_{1}∧p_{2}∧p_{3}),
f_{O} = (p_{7}∧p_{8}∧5p_{9})∨(p_{7}∧p_{8}∧p_{9}).
By Definition 6, the input, output and common enabling vector sets can be obtained,
where:
• 
V_{I1} = {V_{I1},_{1}, V_{I1},_{2}}
is the input enabling vector set of t_{1}, where V_{I1},_{1}
= (1, 1, *, *, 0, *, *, *, *)^{T}, V_{I1},_{2} =
(1, 1, 1, *, 0, *, *, *, *)^{T} 
• 
V_{2} is the common enabling vector of t_{2} and V_{2}
= (*, *, *, 1, 1, 0, *, *, *)^{T} 
• 
V_{O3} = {V_{O3},_{1}, V_{O3},_{2}}
is the output enabling vector set of t_{3}, where V_{O3},_{1}
= (*, *, *, *, *, 1, 0, 0, *)^{T}, V_{O3},_{2} =
(*, *, *, *, *, 1, 0, 0, 0)^{T} 
By Definition 9, the equation vector set can be obtained, where:
• 
V_{I1}' = {V_{I1},_{1}', V_{I1},_{2}'}
is the input equation vector set of t_{1}, where V_{I1},_{1}'
= (1, 1, 0, 0, 1, 0, 0, 0, 0)^{T}, V_{I1},_{2}'
= (1, 1, 1, 0, 1, 0, 0, 0, 0)^{T} 
• 
V_{2}' is the common equation vector of t_{2} and V_{2}'
= (0, 0, 0, 1, 1, 1, 0, 0, 0)^{T} 
• 
V_{O3}' = {V_{O3},_{1}', V_{O3},_{2}'}
is the output equation vector set of t_{3}, where V_{O3},_{1}'
= (0, 0, 0, 0, 0, 1, 1, 1, 0)^{T}, V_{O3},_{2}'
= (0, 0, 0, 0, 0, 1, 1, 1, 1)^{T} 
According to Algorithm 1:
• 
Let M match V_{I1},_{1}, V_{I1},_{2},
by Definition 7, have V_{I1},_{1}≅M; by Eq.
10, M_{1} = M+V_{I1},_{1}' = (0, 0, 0, 1, 1,
0, 0, 0, 0)^{T}, M[t_{1}>M_{1} 
• 
By Definition 7, have V_{2}≅M_{1}; by Eq.
12, M_{2} = M_{1}+V_{2}' = (0, 0, 0, 0, 0, 1,
0, 0, 0)^{T}, M_{1}[t_{2}>M_{2} 
• 
Let M_{2} match V_{O3},_{1}, V_{O3},_{2},
by Definition 7, have V_{O3},_{1}≅M and V_{O3},_{2}≅M.
f_{I} and f_{O} are matching expressions and in (a), V_{I1},_{1}≅M,
V_{O3},_{1 }is the matching vector. By Eq.
11, M_{3} = M_{2}+V_{O3},_{1}' = (0,
0, 0, 0, 0, 0, 1, 1, 0)^{T}, M_{2}[t_{3}>M_{3} 
At the marking M_{3}, all enabling vectors can not match M_{3}
and the running of the LPN stops.
CONCLUSION Based on our work, the definition of LPNs is redefined. By introducing the enabling vectors and matching expressions, the indeterminate data transmission caused by logic output transitions is solved. The standard form of a logic expression can delete useless minterms and the analysis of LPN models is more concisionly. Further work will investigate the fundamental properties of LPNs according to the results proposed in this study, such as state equivalency, liveness and reachability. The Logic Petri Net Workflow will be put forward and be used in progress mining. ACKNOWLEDGMENTS This study is supported by the National Natural Science Foundation of China under Grants No. 61170078 and 61173042; the National Basic Research Program of China under Grant No. 2010CB328101; the Doctoral Program of Higher Education of the Specialized Research Fund of China under Grant No. 20113718110004; Basic Research Program of Qingdao City of China under Grant No. 1314116jch; the SDUST Research Fund of China under Grant No. 2011KYTD102 and Graduate Innovation Foundation of Shaudong University of Science and Technology under Grant No. YC130107.

REFERENCES 
Aziz, M.H., E.L.J. Bohez, R. Pisuchpen and M. Parnichkun, 2013. Petri Net model of repetitive push manufacturing with Polca to minimise valueadded WIP. Int. J. Prod. Res., 51: 44644483. CrossRef  Direct Link 
Barzegar, B., M. Mehrabanian, S. Bandegan and S. Bandegan, 2011. Fuzzy logic for a traffic signal control with colored petri net. Aus. J. Basic. Applied Sci., 5: 29612964. Direct Link 
Bayati, M. and A. Dideban, 2012. Controller synthesis using the novel fuzzy petri net. Int. Rev. Autom. Control, 5: 839843. Direct Link 
Du, Y.Y. and C.J. Jiang, 2002. Formal representation and analysis of batch stock trading systems by logical Petri net workflows. Proceedings of the 4th International Conference on Formal Engineering Methods, October 2125, 2002, Shanghai, China, pp: 221225.
Du, Y.Y. and C.J. Jiang, 2008. On the design and temporal Petri net verification of grid commerce architecture. Chinese J. Electron., 17: 247251. Direct Link 
Du, Y.Y., C.J. Jiang and M. Zhou, 2009. A petri netbased model for verification of obligations and accountability in cooperative systems. IEEE Trans. Syst. Man Cybern. A, 39: 299308. CrossRef  Direct Link 
Marin, A., S. Balsamo and P.G. Harrison, 2012. Analysis of stochastic Petri nets with signals. Perform. Eval., 69: 551572. CrossRef  Direct Link 



