=Paper=
{{Paper
|id=None
|storemode=property
|title=Generalised Fuzzy Petri Nets for Approximate Reasoning in Decision Support Systems
|pdfUrl=https://ceur-ws.org/Vol-928/0370.pdf
|volume=Vol-928
|dblpUrl=https://dblp.org/rec/conf/csp/Suraj12
}}
==Generalised Fuzzy Petri Nets for Approximate Reasoning in Decision Support Systems==
https://ceur-ws.org/Vol-928/0370.pdf
Generalised fuzzy Petri nets for approximate
reasoning in decision support systems
Zbigniew Suraj
Institute of Computer Science, University of Rzeszów, Poland
{zsuraj}@univ.rzeszow.pl
Abstract. The aim of this paper is to present a new class of Petri
nets called generalised fuzzy Petri nets. The new class extends the ex-
isting fuzzy Petri nets by introducing two operators: triangular norms
(t-norms) and t-conorms (s-norms), which are supposed to function as
substitute for the min and max operators. To demonstrate the power and
the usefulness of this model, an application of the generalised fuzzy Petri
nets in the domain of train tra c control is provided. The new model is
more exible than the classical one as in the former class the user has the
chance to de ne the input/output operators. The proposed approach can
be used for knowledge representation and reasoning in decision support
systems.
Key words: fuzzy Petri nets, knowledge representation, approximate
reasoning, decision support systems
1 Introduction
Petri nets serve as a graphical and mathematical modelling tool applicable to
many systems. Their graphical aspect allows representation of various interac-
tions between discrete events more easily. However, the mathematical aspect
allows formal modelling of these interactions and analysis of the modelled sys-
tem properties [11],[17],[22].
The concept of a Petri net has its origin in C.A. Petri's dissertation [23].
In the last four decades, several extensions of Petri nets have been proposed
improving such aspects as hierarchical nets, high level nets or temporal nets
[11]. Currently, Petri nets are gaining a growing interest among people both
in Arti cial Intelligence due to its adequacy to represent the reasoning process
as a dynamic discrete event system [1]-[9],[15]-[16],[18]-[21],[25]-[26] as well as
in Molecular Biology with respect to their powerfulness and usefulness in mod-
elling biological systems [10],[12]-[13],[24]. In 1988, C.G. Looney proposed in [16]
so called "F uzzy P etri N ets (FPNs )". In his model logical propositions can be
associated with Petri nets allowing for logical reasoning about the modelled sys-
tem and its behaviour. The application of fuzzy Petri nets includes the design
and implementation of decision support systems. In particular, they can be used
for knowledge representation and modelling of reasoning processes in such sys-
tems. In this class of Petri net models not only well-known pieces of information
but also imprecise, vague and uncertain information is admissible and taken
into account. Several authors have proposed dierent classes of fuzzy Petri nets.
These models are based on dierent approaches combining Petri nets and fuzzy
sets introduced by L. Zadeh in 1965 [27].
The aim of this paper is to further improve the fuzzy Petri net model for
knowledge representation and reasoning and to overcome some de ciencies of
former FPN approaches [2].
In the paper, we propose a new class of Petri nets called "Generalised F uzzy
P etri N ets (GFPNs )" for knowledge representation and reasoning in decision
support systems. This model is a natural extension of fuzzy Petri nets. The t-
norms and s-norms are introduced to the model as substitutes of min and max
operators. The latter ones generalize naturally AND and OR logical operators
with the Boolean values 0 and 1. The proposed model is not only more comfort-
able in terms of knowledge representation, but most of all it is more eective
in the modelling process of approximate reasoning as in the new class of fuzzy
Petri nets the user has the chance to de ne the input/output operators. The
preliminary results of real-life data experiments using this model are promis-
ing. In order to demonstrate the modelling aspects of GFPNs, an application of
generalised fuzzy Petri nets in the domain of train tra c control is provided.
The structure of this paper is as follows. Sect. 2 gives a brief introduction to
fuzzy Petri nets. In Sect. 3 generalised fuzzy Petri nets formalism is presented.
Sect. 4 describes an application of this class of FPN in the domain of train tra c
control. In Sect. 5 conclusions are made.
2 Fuzzy Petri nets - basic de nitions
One of the most known and applicable class of Petri nets in the domain of
Arti cial Intelligence are fuzzy Petri nets [3],[16]. They are a modi cation of
classical Petri nets [17] relying on interpretation of net places as logical variables
with values belonging to the closed interval [0,1] of all real numbers from 0 to 1
(0 and 1 are included). The concrete values of such variables represent a truth
degree of statements assigned to the variables. Net transitions are interpreted as
logical implications in which input places of a transition represent premises of a
given implication corresponding to the transition whereas output places of the
transition represent its conclusions. The de nitions of input/output places of a
transition in a net are given below. Similarly, we assume that the truth degree
of a given implication belongs also to the closed interval of all real numbers from
0 to 1. Moreover, we assume that threshold values for all transitions in a given
fuzzy Petri net are de ned. The role of these values is to limit the possibility of
transition rings. More precisely, if a logical premise value of a given transition
is less than a threshold value of the transition then this transition is not possible
to be red.
In this paper we view a fuzzy Petri net as a decision support system based
on speci c rules of the form: IF condition THEN action, for which the condition
is consumed and the action is produced each time the rule is used. In a decision
support system, the knowledge representation is based on a collection of rules. In
a fuzzy Petri net, each transition may be seen as a rule which depicts a possible
state change.
De nition 1. A fuzzy Petri net (FP-net) is a tuple N = (P, T, S, I, O, α, β, γ,
M 0), where:
- P = {p1 , p2 , . . . , pn } is a nite set of places, n > 0;
- T = {t1 , t2 , . . . , tm } is a nite set of transitions, m > 0;
- S = {s1 , s2 , . . . , sn } is a nite set of statements;
- the sets P , T , S are pairwise disjoint, i.e., P ∩ T = P ∩ S = T ∩ S = ∅ and
card(P ) = card(S );
- I : T → 2P is the input function;
- O : T → 2P is the output function;
- α : P → S is the statement binding function;
- β : T → [0, 1] is the truth degree function;
- γ : T → [0, 1] is the threshold function;
- M 0 : P → [0, 1] is the initial marking,
and 2P denotes a family of all subsets of the set P .
As for the graphical interpretation, places are denoted by circles and tran-
sitions by rectangles. The places are the nodes describing states (a place is a
partial state) and the transitions depict the state changes. The function I de-
scribes the oriented arcs connecting places with transitions. It represents, for
each transition t, the fragments of the state in which the system has to be, be-
fore the state change corresponding to t can occur. The function O describes the
oriented arcs connecting transitions with places. It represents, for each transi-
tion t, fragments of the state in which the system will be after the occurrence
of the state change corresponding to t. If I(t) = {p} then a place p is called an
input place of a transition t. Moreover, if O(t) = {p0 }, then a place p0 is called
an output place of t. The initial marking M 0 is an initial distribution of tokens
in the places. It can be represented by a vector of dimension n of real numbers
from the closed interval [0, 1]. For p ∈ P , M 0(p) is the token load of place p and
represents a partial state of the system described by the fuzzy Petri net. This
value can be interpreted as a truth value of a statement s bound with a given
place p by means of the statement binding function α, i.e., α(p) = s. Pictorially,
the tokens are represented by means of grey "dots" together with the suitable
real numbers placed inside the circles corresponding to appropriate places. We
assume that if a truth value of a statement attached to a given place is equal
to 0 then the token does not exist in the place. The number β(t) is placed in a
net picture under a transition t. Usually, this number is interpreted as a truth
degree of an implication corresponding to a given transition t. The meaning of
the threshold function γ is explained below.
Let N be a FP -net. A marking of N is a function M : P → [0, 1].
The fuzzy Petri net dynamics de nes how new markings are computed from
the current marking when transitions are red (the corresponding state change
occurs). It describes the state changes of the decision support system modeled
by the fuzzy Petri net.
Let N be a FP -net, t ∈ T , I(t) = {pi1 , pi2 , . . . , pik } be a set of input places
for a transition t, and M - a marking of N .
A transition t is enabled for marking M if the minimum min for all input
places of the transition t by M is positive and greater than or equal to the value
of the threshold function γ to t, i.e.,
min(M (pi1 ), M (pi2 ), . . . , M (pik )) ≥ γ(t) > 0 for each pij ∈ I(t), j = 1, . . . , k.
Only enabled transitions can be red. Informally, ring the enabled transition
t consists of removing (or not) dependently on the ring mode (it will be further
discussed in detail) the token load of its input places by I(t) (as the transition
is enabled no negative token load will be obtained), and increasing the token
load of all its output places by O(t) without any alteration of the token loads of
other places.
This informal de nition points out the fact that ring the transition t is
local in the sense that it only involves the tokens captured by I(t) and O(t)
independently from the other remaining tokens of the current marking. Firing is
based on two invisible primitives: removal of the tokens from input places and
insertion of the tokens in output places. Transitions which do not share any places
can be red independently. This is why fuzzy Petri nets are a mathematical tool
that can capture true concurrency.
There are two operating modes of the fuzzy Petri nets. In the rst mode,
each ring of an enabled transition t removes tokens from its input places and
adds a token to each of its output place (if the token does not exist in the
place yet). If in a given output place the token already exists then the red
transition does not place a new token. In the rst case, a new value related to
the generated token is computed as follows. After computing the minimum value
from all values corresponding to input places of a red transition, the computed
value is timed by the value of function β corresponding to a red transition.
A new value is placed in all output places of a red transition. In the second
case, we assume that ring enabled transition does not remove tokens from its
input places (copies of the tokens are only transmitted to output places of a red
transition). A nal value of tokens is computed in an analogous way to the rst
case. In both cases, if in a given output place the token already exists then the
red transition computes a new value of the token as follows. The nal value of a
given output place of a red transition is computed as maximum value from the
one residing in the output place and the computed value of the red transition.
Let N = (P, T, S, I, O, α, β, γ, M 0) be a FP -net, t ∈ T , I(t) = {pi1 , pi2 , . . . ,
pik } be a set of input places for a transition t, β(t) be a value of the truth degree
function β corresponding to t and β(t) ∈ (0, 1] (0 is not included), γ(t) be a
value of threshold function γ corresponding to t, and M be a marking of N .
Moreover, let min, ∗, max denote the minimum, the algebraic product and the
maximum, respectively.
Mode 1. If M is a marking of N enabling transition t and M 0 the marking
derived from M by the ring transition t, then for each p ∈ P :
0 if p ∈ I(t),
M 0 (p) = max(min(M (pi1 ), M (pi2 ), . . . , M (pik )) ∗ β(t), M (p)) if p ∈ O(t),
M (p) otherwise.
In this mode, a procedure for computing the marking M 0 is as follows: (1)
Tokens from all input places of the transition t are removed (the rst condition
from M 0 de nition). (2) Tokens in all output places of t are modi ed in the
following way: at rst the value of minimum min for all input places of t is
computed, then the product of the computed minimum for all input places of t
and the value of truth degree function β(t) is determined, and nally, a value
corresponding to M 0 (p) for each p ∈ O(p) is obtained as a result of maximum for
the computed product value and the current marking M (p) (the second condition
from M 0 de nition). (3) Tokens in the remaining places of net N are not changed
(the third condition from M 0 de nition).
Mode 2. If M is a marking of N enabling transition t and M 0 the marking
derived from M by the ring transition t, then for each p ∈ P :
max(min(M (pi1 ), M (pi2 ), . . . , M (pik )) ∗ β(t), M (p)) if p ∈ O(t),
0
M (p) =
M (p) otherwise.
The dierence in the de nition of a marking M 0 presented above (Mode 2 )
concerns input places of the red transition t. In Mode 1 tokens from all input
places of the red transition t are removed (cf . the rst de nition condition of
Mode 1 ), whereas in Mode 2 all tokens from input places of the red transition
t are copied (the second de nition condition of Mode 2 ).
Fig. 1. A fuzzy Petri net with the initial marking before ring the enabled transitions
Example 1. Let us consider a fuzzy Petri net in Figure 1. For the net we have:
the set of places P = {p1, p2, p3, p4, p5}, the set of transitions T = {t1, t2},
the input function I and the output function O in the form: I(t1) = {p1, p2},
I(t2) = {p2, p3}, O(t1) = {p4}, O(t2) = {p5}. Moreover, there are: the set of
statements S = {s1, s2, s3, s4, s5}, the statement binding function α : α(p1) =
s1, α(p2) = s2, α(p3) = s3, α(p4) = s4, α(p5) = s5, the truth degree function
β : β(t1) = 0.7, β(t2) = 0.8, the threshold function γ : γ(t1) = 0.4, γ(t2) = 0.3
and the initial marking M 0 = (0.6, 0.4, 0.7, 0, 0).
(a) (b)
Fig. 2. An illustration of a ring rule: (a) the marking after ring t1, where t2 is
disabled (Mode 1 ), (b) the marking after ring t2, where t1 and t2 are enabled (Mode
2)
Transitions t1 and t2 are enabled by the initial marking M 0. Firing transition
t1 by the marking M 0 according to Mode 1 transforms M 0 to the marking M 1 =
(0, 0, 0.7, 0.28, 0) (Figure 2(a)), and ring transition t2 by the initial marking M 0
according to Mode 2 results in the marking M 2 = (0.6, 0.4, 0.7, 0, 0.32) (Figure
2(b)).
3 Generalised fuzzy Petri nets
Now we are ready to de ne a new class of Petri net model called a generalised
fuzzy Petri net. Before giving a formal de nition of this model we remind some
notions used later on. This section presents the main contribution to the paper.
A t-norm is de ned as t : [0, 1]×[0, 1] → [0, 1] such that, for each a, b, c ∈ [0, 1]:
(1) it has 1 as the unit element, i.e., t(a, 1) = a; (2) it is monotone, i.e., if
a ≤ b then t(a, c) ≤ t(b, c); (3) it is commutative, i.e., t(a, b) = t(b, a); (4) it is
associative, i.e., t(t(a, b), c) = t(a, t(b, c)).
More relevant examples of t-norms are: the minimum t(a, b) = min(a, b)
which is the most widely used, the algebraic product t(a, b) = a∗b, the ukasiewicz
t-norm t(a, b) = max(0, a + b − 1).
An s-norm (or a t-conorm) is de ned as s : [0, 1] × [0, 1] → [0, 1] such that,
for each a, b, c ∈ [0, 1]: (1) it has 0 as the unit element, i.e., s(a, 0) = a, (2)
it is monotone, i.e., if a ≤ b then s(a, c) ≤ s(b, c), (3) it is commutative, i.e.,
s(a, b) = s(b, a), and (4) it is associative, i.e., s(s(a, b), c) = s(a, s(b, c)).
More relevant examples of s-norms are: the maximum s(a, b) = max(a, b)
which is the most widely used, the probabilistic sum s(a, b) = a + b − a ∗ b, the
ukasiewicz s-norm s(a, b) = min(a + b, 1).
Using the notions of t- and s-norms we formulate the de nition a generalised
fuzzy Petri nets as follows:
De nition 2. A generalised fuzzy Petri net (GFP-net) is a tuple N 0 = (P, T, S,
I, O, α, β, γ, Op, δ, M 0), where:
- P, T, S, I, O, α, β, γ, M 0 have the same meaning as in De nition 1;
- Op is a nite set of t-norms and s-norms called the set of operators;
- δ : T → Op × Op × Op is the operator binding function.
The operator binding function δ connects transitions with triples of operators
(opIn , opOut1 , opOut2 ). The rst operator appearing in the triple is called the
input operator, and two remaining ones are the output operators. The input
operator opIn concerns the way in which all input places are connected with
a given transition t (more precisely, statements corresponding to those places).
However, the output operators opOut1 and opOut2 concern the way in which
the next marking is computed after ring the transition t. In the case of input
operator we assume that it can belong to one of two classes, i.e., t- or s-norm.
This issue is more precisely discussed further on. In fuzzy Petri nets the operators
minimum, algebraic product and maximum are usually used. As we know, the
rst two belong to the class of t-norms whereas the third belongs to the class
of s-norms. In the new net model the input/output operators can be de ned
by a user of the model dependently on her/his needs. In general, selecting rule
operators and parameters (if we consider so called parameterised families of
t- and s-norms [14],[25]) depends on the rule type appearing in a given rule
knowledge base as well as the quality of experimental data from which the rules
are extracted [8],[21]. The behaviour of generalised fuzzy Petri nets is de ned
in an analogous way to the case of fuzzy Petri nets. It is worth emphasising
that the de nition of net marking is analogous to fuzzy Petri nets, although the
de nitions of transition rule and next marking are substantially modi ed.
Let N 0 be a GFP -net. A marking of N 0 is a function M : P → [0, 1].
A transition t ∈ T is enabled for marking M if the value of input operator
opIn for all input places of the transition t by M is positive and greater than or
equal to the value of threshold function γ corresponding to the transition t, i.e.,
opIn (M (pi1 ), M (pi2 ), . . . , M (pik )) ≥ γ(t) > 0 for each pij ∈ I(t), j = 1, . . . , k.
Let N 0 = (P, T, S, I, O, α, β, γ, Op, δ, M 0) be a GFP -net, t ∈ T , I(t) =
{pi1 , pi2 , . . . , pik } be a set of input places for a transition t and β(t) ∈ (0, 1].
Moreover, let opIn be an input operator and opOut1 , opOut2 be output operators
for the transition t.
Mode 1. If M is a marking of N 0 enabling transition t and M 0 the marking
derived from M by ring transition t, then for each p ∈ P :
0 if p ∈ I(t),
opOut2 (opOut1 (opIn (M (pi1 ), M (pi2 ), . . . , M (pik )), β(t)), M (p))
M 0 (p) =
if p ∈ O(t),
M (p) otherwise.
In this mode, a procedure for computing the marking M is similar to appro-
0
priate procedure corresponding to fuzzy Petri nets and Mode 1 presented above.
The dierence is that present procedure needs to set operators: opIn , opOut1 ,
opOut2 at rst. Remaining stages of the procedure are analogous to the previous
procedure concerning Mode 1.
Mode 2. If M is a marking of N 0 enabling transition t and M 0 the marking
derived from M by ring transition t, then for each p ∈ P :
opOut2 (opOut1 (opIn (M (pi1 ), M (pi2 ), . . . , M (pik )), β(t)), M (p))
M 0 (p) = if p ∈ O(t),
M (p) otherwise.
The main dierence in the de nition of the marking M presented above
0
(Mode 2 ) and Mode 1 is analogous to the fuzzy Petri nets.
Fig. 3. A generalised fuzzy Petri net with the initial marking before ring the enabled
transitions t1 and t2
Example 2. Let us consider a generalised fuzzy Petri net in Figure 3. For the
net we have: the set of places P , the set of transitions T , the set of statements
S , the input function I , the output function O, the statement binding function
α, the truth degree function β , the threshold function γ , and and the initial
marking M 0 are described analogously to Example 1. Moreover, there are: the
set of operators Op = {max, min, ∗} and the operator binding function δ de ned
as follows: δ(t1) = (max, ∗, max), δ(t2) = (min, ∗, max).
(a) (b)
Fig. 4. An illustration of a ring rule: (a) the marking after ring t1, where t2 is
disabled (Mode 1 ), (b) the marking after ring t2, where t1 and t2 are enabled (Mode
2)
Transitions t1 and t2 are enabled by the initial marking M 0. Firing transition
t1 by the marking M 0 according to Mode 1 transforms M 0 to the marking M 1 =
(0, 0, 0.7, 0.42, 0) (Figure 4(a)), and ring transition t2 by the initial marking M 0
according to Mode 2 results in the marking M 2 = (0.6, 0.4, 0.7, 0, 0.32) (Figure
4(b)).
4 Illustrating example
This section presents an application of GFPN in the domain of train tra c
control [5]. The example is based on a simpli ed version of the real-life problem.
We assume the following situation: a train B waits at a certain station for a
train A to arrive in order to allow some passengers to change train A to train
B . Now a con ict arises when the train A is late.
In this situation, the following alternatives can be taken into consideration:
Train B waits for train A to arrive. In this case, train B will depart with
delay.
Train B departs in time. In this case, passengers disembarking train A have
to wait for a later train.
Train B departs in time, and an additional train is employed for late train
A0 s passengers.
To make a decision, several inner conditions have to be taken into account
such as the delay period, the number of passengers changing trains, etc. The
discussion regarding an optimal solution to the problem of divergent aims such
as: minimization of delays throughout the tra c network, warranty of connec-
tions for the customer satisfaction, e cient use of expensive resources, etc. is
disregarded at this point.
In order to describe the tra c con ict, we propose to consider the following
three rules:
IF s2 OR s3 THEN s6;
IF s1 AND s4 AND s6 THEN s7;
IF s4 AND s5 THEN s8,
where: s1 = "Train B is the last train in this direction today", s2 = "The delay
of train A is huge", s3 = "There is an urgent need for the track of train B ", s4
= "Many passengers would like to change for train B ", s5 = "The delay of train
A is short", s6 = "(Let) train B depart according to schedule", s7 = "Employ
an additional train C (in the same direction as train B )", and s8 = "Let train
B wait for train A".
Fig. 5. An example of GFPN model of train tra c control
In Figure 5 the GFPN model corresponding to these rules, where the logical
operators OR, AND are interpreted as max and min, respectively, is shown. Note
that the places p1, p2, p3 and p4 include the fuzzy values 0.6, 0.4, 0.7 and 0.5
corresponding to the statements s1, s2, s3 and s4, respectively. In this example,
the statement s5 attached to the place p5 is the only crisp and its value is equal to
1. By means of evaluation of the statements attached to the places from p1 up to
p5, we observe that the transitions t1 and t3 can be red. Firing these transitions
according to the ring rules for the GFPN model allows the computation of the
support for the alternatives in question. In this way, the possible alternatives
are ordered with regard to the preference they achieve from the knowledge base.
This order forms the basis for further examinations and simulations and, in the
end, for the dispatching proposal. If one chooses a sequence of transitions t1t2
then they obtain the nal value, corresponding to the statement s7, equal to
0.35. In the second possible case (i.e., for the transition t3 only), the nal value,
corresponding now to the statement s8, equals 0.45.
5 Conclusions
The GFPN model combining the graphical power of Petri nets, possibilities of
fuzzy sets and the theory of t -norms to model rule-based expert knowledge in a
decision support system have been described in the paper. Having discussed basic
notions from the fuzzy Petri net theory, GFPN formalism has been presented.
Using a simple real-life example the suitability and the usefulness of the proposed
approach for the design and implementation of decision support systems have
been proved. Success of the elaborated approach looks promising with regard to
alike application problems that could be solved similarly.
An experimental application called PNES on IBM PC, in Java, consisting of
an editor and a simulator have been developed. The editor allows the inputting
and editing of generalised fuzzy Petri nets, while the simulator that starts with a
given initial marking and executes enabled transitions visualising reached mark-
ings. All gures and simulation results presented in the paper were produced by
the application.
Acknowledgment. The author is grateful to anonymous referees for their help-
ful comments.
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