An equivalent generating algorithm to model fuzzy Petri net for knowledge-based system

Abstract

The simulation of knowledge-based systems (KBSs) has become a significant challenge owing to the rapid increase in the scale of accumulated data. The extended formalisms that are widely used to test, model, and analyze such systems include the fuzzy production rule (FPR) and fuzzy Petri net (FPN). However, with the growth in magnitude of KBSs, it has become difficult to manually generate an FPN. Hence, the authors propose an equivalent transformation algorithm that automatically models an FPN for a sizeable KBS. The proposed method produces a final FPR by initially investigating the inner-inference path(s) between FPRs, followed by a four-phase transformation algorithm that automatically generates an equivalent FPN model for the corresponding KBS rooted in the inner-inference path(s) obtained. A KBS with 13 FPRs is used to demonstrate both the validity and feasibly of the proposed transformation algorithm. The results validate the capability of the generated FPN to fully represent the complete information base contained in the corresponding KBS.

Full text

J Intell Manuf
DOI 10.1007/s10845-017-1355-x
An equivalent generating algorithm to model fuzzy Petri net
for knowledge-based system
Kai-Qing Zhou1,2·Li-Ping Mo1·Jie Jin1·Azlan Mohd Zain3
Received: 29 April 2017 / Accepted: 4 August 2017
© Springer Science+Business Media, LLC 2017
Abstract The simulation of knowledge-based systems
(KBSs) has become a significant challenge owing to the rapid
increase in the scale of accumulated data. The extended for-
malisms that are widely used to test, model, and analyze
such systems include the fuzzy production rule (FPR) and
fuzzy Petri net (FPN). However, with the growth in magni-
tude of KBSs, it has become difficult to manually generate
an FPN. Hence, the authors propose an equivalent transfor-
mation algorithm that automatically models an FPN for a
sizeable KBS. The proposed method produces a final FPR
by initially investigating the inner-inference path(s) between
FPRs, followed by a four-phase transformation algorithm
that automatically generates an equivalent FPN model for
the corresponding KBS rooted in the inner-inference path(s)
obtained. A KBS with 13 FPRs is used to demonstrate both
the validity and feasibly of the proposed transformation algo-
rithm. The results validate the capability of the generated
FPN to fully represent the complete information base con-
tained in the corresponding KBS.
Keywords Fuzzy production rule ·Fuzzy Petri net ·
Modeling ·Equivalent ·Inner-inference path(s)
BKai-Qing Zhou
jsu_computer@163.com
1College of Information Science and Engineering,
Jishou University, Jishou 416000, Hunan,
People’s Republic of China
2College of Information Science and Engineering, Central
South University, Changsha 410083, Hunan,
People’s Republic of China
3Soft Computing Research Group, Faculty of Computing,
Universiti Teknologi Malaysia, UTM, 81310 Skudai, Johor,
Malaysia
Introduction
Knowledge-based systems (KBSs), which are considered
expert systems, are a type of computerized artificial intel-
ligence applied in various industrial fields for capturing and
utilizing knowledge to solve complex problems, such as fault
diagnosis or knowledge reasoning, using computer systems
(García-Crespo et al. 2011;Paredes-Frigolett and Gomes
2016;Nasiri et al. 2017). Iqbal (2014) employed compu-
tational intelligence and a KBS to predict the flow stress
of an AISI 4340 based on different settings, such as the
microstructure, applied temperature, strain, and strain rate
of the material. Law et al. (2016) proposed a modified KBS
to realize the design and selection of equipment based on
achieving low-grade waste heat recovery in process indus-
tries. Maciol (2017) proposed a method for verifying the
hypothesis of estimating the variable costs of metal casts
based on knowledge-based systems. Meanwhile, a foundry
production program for manufacturing water-system fittings
has been used to demonstrate the feasibly of the proposed
method through three classical reasoning techniques.
The last few decades have witnessed a series of new meth-
ods for representing knowledge and automatic reasoning
implementation. Two major formalisms have been applied to
fulfill the KBS requirements: fuzzy production rules (FPRs)
(Awan and Awais 2011;Peng et al. 2013) and a fuzzy Petri
net (FPN) (Liu et al. 2013;Wai and Liu 2009).
FPRs have been widely manipulated for data representa-
tion in a KBS using an ‘IF–THEN’ configuration to conduct
reasoning tasks for capturing data (Balazinski et al. 2002;
Novák and Lehmke 2006;Ting et al. 2008). In a correspond-
ing manner, an FPN is also broadly employed to implement
approximate inferences owing to its superb application to
descriptive graphics and parallel mathematical analysis. To
explore hidden associations between an FPN and FPRs,
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Chen et al. (1990) employed an equivalent FPN for the cor-
responding basic FPR types: ‘Simple’, ‘OR’, ‘AND’, and
‘Multi-condition’ rules. Gao et al. (2003) proposed a new
FPN model to represent FPRs that contain negative literals.
In addition, Shen (2006) utilized multiple-output places to
describe ‘IF–THEN’ and ‘IF–THEN–ELSE’ rules in a KBS.
Based on the corresponding FPNs, a KBS can be gen-
erally transformed manually into an FPN. However, the
transformation process of generating an FPN for a large-
size KBS is complex and difficult owing to the increasing
numbers of FPRs. Furthermore, such a huge KBS transfor-
mation also hinders further exploration of the output FPN
in areas representing knowledge and reasoning. Our pre-
vious studies (Zhou et al. 2015a,b) on the inner reasoning
relationships of FPNs used a large-scale FPN divided into a
series of sub-FPNs by utilizing both an index function and
an incidence matrix. These studies proposed a decomposition
algorithm, the results of which indicated that a correspond-
ing FPN for a large-size KBS can be automatically generated
and enabled by searching for inner reasoning paths between
FPRs. Accordingly, the present study proposes an equivalent
transformation algorithm used to model an FPN for a corre-
sponding large-size KBS. To achieve this, the following two
steps are applied.
1. An innovative method is used to represent an FPR that
reveals the inner inference relationships between FPRs.
2. A four-phase transformation algorithm is utilized that
automatically generates FPNs for a large-size KBS,
including separation, simplification, generation, and
merging.
The remainder of this paper is organized as follows. A review
of the FPN, FPRs, and the corresponding inner relations
between both formalisms is described in “Fuzzy Petri net
and fuzzy production rules” section. “Novel representation
method of fuzzy production rule” section presents a novel
method for representing an FPR from a reasoning perspec-
tive, and introduces a transformation algorithm that enables
the generation of an FPN model equivalent to the given FPRs.
Based on a case study, “Proposed transformation algorithm”
section discusses the entire transformation process, from the
FPRs to the FPN model. “Conclusion” section concludes
with a discussion of the results.
Fuzzy Petri net and fuzzy production rules
This section reviews the fundamental definitions of an FPN
and FPR. Moreover, a systemic comparison is presented by
relating the differences and similarities between both formal-
ities.
Formalism of fuzzy Petri net
The general formalism of an FPN is as follows.
Definition 1 Fuzzy Petri net (FPN)
An FPN is represented as {P,T,M,I,O,W,μ,CF},
where the following hold.
1. The term P={p1,p2,...,pn}is a finite set of places.
2. The term T={t1,t2,...,tm}is a finite set of transitions.
3. The term M={m1,m2,...,mn}Tis a fuzzy marking
vector, where mi∈[0,1]indicates the truth degree of
pi(i=1,2,...,n). The initial truth degree vector is
denoted as M0.
4. The term I:P×T→{0,1}is an n×minput matrix.
Here, I(pi,tj)(i=1,2,...n;j=1,2,...,m)records
whether a directed arc from pito tjexists, where
Ipi,tj=1 if there is an arc from pito tj;
0 if there is no an arc from pito tj.
5. The term O:P×T→{0,1}is the n×moutput matrix.
Here, O(pi,tj)(i=1,2,...n;j=1,2,...,m)
records whether a directed arc from tjto piexists, where
Opi,tj=1 if there is an arc from tjto pi;
0 if there is no an arc from tjto pi.
6. The term μ:t→(0,1], where μjis the threshold of tj.
7. The term W(i,j)is the weight on the arc from pito
tj. Here, W(i,j)∈[0,1]indicates how much place pi
impacts its following transition tj.
8. The term CF is the belief strength. Here, CF ∈[0,1]
indicates how much transition tjimpacts its output place
pi.
Fuzzy production rule
An FPR is used to illustrate the inner relationship between
prepositions and conclusions with fuzzy parameters. A for-
mal definition of an FPR is given in Definition 2.
Definition 2 Fuzzy production rule
A general FPR formalism is described as follows:
if D(λ) then Q (CF ,μ,w) , where the following
hold:
1. Dis a finite set of preconditions, D={D1,D2,...,Dn};
2. Qis a finite set of conclusions, Q={Q1,Q2,...,Qm};
3. λis the truth degree of each precondition, λ∈[0,1];
4. CF is the belief strength of this rule, where CF ∈[0,1]
is the credibility after the rule is implemented;
5. μis the threshold of the rule, μ∈[0,1]; and
6. wis the weight of each precondition, w∈[0,1].
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Correspondence between FPR and FPN
Table 1demonstrates the correspondence between an FPR
and FPN.
Table 1circumscribes a theoretical path used to generate
an FPN model from the FPRs. For example, an FPR with two
inputs and two outputs is given below.
if D1(λ1)and D2(λ2)then Q1and Q 2
(w1,w
2;μ;CF1,CF 2)
According to the correspondence between the FPR and FPN,
an equivalent FPN for this FPR can be generated, as shown
in Fig. 1.
Novel representation method of fuzzy production
rule
In a real KBS, preconditions for the rule implementation uti-
lize conclusions from other rules. Based on an intricate set
of connections between rules, inferences can be executed
for the entire KBS. Although the traditional representation
of the FPR method clearly illustrates the inner relationships
between the preconditions and conclusions for each rule, the
difficulty in the manual recognition of the inner inference
relationships between FPRs increases with the growth in the
size of the KBS. Hence, it is logically necessary to find a
novel FPR representation method. Compared to a conven-
tional representation method, the proposed method is capable
of revealing inner reasoning relations between FPRs in a
large-size KBS. From a reasoning perspective, each FPR
is separated into inference and parameter chains. Details of
the proposed novel FPR representation method are given in
Table 2.
According to Table 2, each FPR type is divided into two
sub-parts as inference and parameter chains, and the conclu-
sion of an FPR inference chain is marked as a head, whereas
the prepositions for the conclusions are listed in parentheses.
Meanwhile, the order of each parameter chain type is set as
the degree of truth (λ), weight (w), threshold (μ), and belief
strength (CF). In addition, if the precondition is a conclu-
sion of other rules within a KBS, the certainty factor for the
precondition is marked as null.
Table 3presents three FPRs for a KBS using a tradi-
tional representation method. Table 4illustrates the same
FPRs for a KBS using the proposed novel representation
method.
Tabl e 1 The corresponding
relationship between an FPR
and FPN
Fuzzy production rule (FPR) Fuzzy Petri net (FPN)
FPRs FPN model
FPR Transition
Precondition and conclusion Place
Range of application of rule Extension of transition
Weight of rule (w) Input weight from place to transition (w)
Certainty factor of the precondition (λ) Value of token (M(pi))
Threshold of rule (μ) Threshold of transition (μ)
Belief strength of rule (CF) Credibility from transition to place (CF)
Fig. 1 Corresponding FPN of
two-input and two-output FPR Q1
D1
w
1
D2
w
2
Q2
CF
1
CF
2
M(D1)= 1
M(D2)= 2
μ
Tabl e 2 Novel representation of FPR
Type of rule Inference chain Parameter chain
‘Simple’ rule Q[D][(λ, w =1),μ,CF]
‘AND’ rule Q[D1,D2,...,Dn][
((λ1,w
1),(λ2,w
2),...,(λn,w
n)) ,μ,CF]
‘OR’ rule Q{D1,D2,...,Dn}{
[(λ1,w
1=1),μ
1,CF1],[(λ2,w
2=1),μ
2,CF2],...,[(λn,w
n=1),μ
n,CFn]}
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Tabl e 3 KBS using traditional
representation method Rule Content
1if R1(λ1)and R2(λ2)then Q1(w11 ,w
12,μ
1,CF1)
2if R3(λ3)then Q2(w2=1,μ
2,CF2)
3if Q1(null)or Q 2(null)then Q3(w31 =1,w
32 =1,μ
31,μ
32,CF 31,CF32)
Tabl e 4 KBS using novel
representation method Rule Inference chain Parameter chain
1Q1[R1,R2][
((λ1,w
11),(λ2,w
12)) ,μ
1,CF1]
2Q2[R3][(λ3,w
2=1),μ
2,CF2]
3Q3{Q1,Q2}{[(null,w
31 =1),μ
31,CF 31],[(null,w
32 =1),μ
32,CF 32]}
Tabl e 5 Reasoning path of Table 3
Inference chain Parameter chain
Q3{Q1[R1,R2],Q2[R3]}{
[([((λ1,w
11),(λ2,w
12)) ,μ
1,CF1],w
31 =1),μ
31,CF 31],[([(λ3,w
2=1),μ
2,CF2],w
32 =1),μ
32,CF 32]}
From Table 4, it is easy to find the inner-reasoning rela-
tion of each rule (Q1, Q2, and Q3 are the conclusions for
Rules 1, 2, and 3, respectively). Meanwhile, some conclu-
sions are the parts of other rules (e.g., Q1 and Q2 are the
preconditions of Rule 3). Utilizing the proposed method, the
inner-inference relationships (reasoning paths) among FPRs
can be recognized and generated automatically. Based on the
novel representation, the inner-reasoning path of the FPRs is
carried out as shown in Table 5. Accordingly, Fig. 2illus-
trates the search process used for the final reasoning path of
the novel representation method.
Table 4allows easy recognition of the inner reasoning
relation of each rule, where Q1, Q2, and Q3 are conclusions
for Rules 1, 2, and 3, respectively. Moreover, some conclu-
sions are components of other rules (e.g., Q1 and Q2 are pre-
conditions for Rule 3). Utilizing the proposed method, inner-
inference-relationships (reasoning paths) between FPRs are
recognized and automatically generated. The novel repre-
sentation method completes the inner reasoning paths of the
FPR, as shown in Table 5. Accordingly, Fig. 2illustrates the
process of searching for the final reasoning paths utilized in
the proposed novel representation method.
Q
3
{Q
1
,Q
2
}
Q
2
[R
3
]
Q
1
[R
1
,R
2
]
Search process of inference chain
Q
3
{Q
1
[R
1
,R
2
] ,Q
2
[R
3
]}
Search process of parameter chain
31 31 31 32 3 2 32
{[( , 1), , ],[( , 1), , ]}null w CF null w CF
μμ
==
32 2 2
[( , 1,), , ]wCF
λμ
=
111 2 12 1 1
[(( , ), ( , )), , ]wwCF
λλ μ
Fig. 2 Search process for reasoning path
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In Fig. 2, the conclusive reasoning path is easily gener-
ated by executing repetitive iterate operations. According to
such repetitive iterate operations, the reasoning path is grad-
ually generated by different conclusions of the FPRs. The
proposed representation method has the ability to reveal the
inner-inference relationship(s) among FPRs in a large-size
KBS.
As shown in Fig. 2, executing reiterate operations eas-
ily generates a conclusive reasoning path. Accordingly, the
reasoning path is gradually generated by different FPR con-
clusions. The proposed representation method can thus reveal
inner inference relationships between FPRs in a large-size
KBS.
Proposed transformation algorithm
The novel representation method includes a transformation
algorithm that automatically generates an equivalent FPN
model from a corresponding large-size KBS. Figure 3shows
a flowchart of the transformation algorithm. The steps for the
transformation process are as follows:
Step 1 Simplify each FPR.
Step 2 Merge repeated FPR items.
Step 3 Represent each simplified FPR using the proposed
novel method.
Step 4 Generate reasoning path(s) between FPRs through
reiteration.
Step 5 Generate corresponding FPN(s) for the current rea-
soning path(s).
Step 6 Store the generated FPN(s).
Step 7 If a processed reasoning path is the last reasoning
path in the KBS, move to Step 11. Otherwise, move
to Step 8.
Step 8 Appoint another unprocessed reasoning path as the
current reasoning path and then move to Step 5.
Step 9 If sharing relationships exist among the generated
FPNs, move to Step 10. Otherwise, move to Step 11.
Step 10 Compose FPN models that have sharing relation-
ships.
Step 11 Output the entire FPN model.
Design of transformation algorithm
According to the transformation steps, the algorithm design
has four phases: separation, simplification, generation, and
merging.
Separation
The separation process of the FPR has two steps. First, the
respective meaning and location of the preconditions and
conclusions are extracted and stored using a two-column
table. The first column records the original locations of the
preconditions using labels, and the second column stores the
meanings of the corresponding preconditions and/or conclu-
sions. The rules for labeling are as follows.
1. The term Pij is used to record the location of the precon-
dition (where i is the ith rule, and j is the jth precondition
in the ith rule).
2. The term Cij is used to record the location of the con-
clusion (where i is the ith rule, and j is the jth conclusion
in the ith rule). In addition, if a conclusion only exists in
the ith FPR, then Cij is indicated as Ci.
Accordingly, labels are used to replace the previous precondi-
tions and conclusions in the FPRs. For example, the following
FPR illustrates the separation process. The separation process
for the FPR is demonstrated in Fig. 4.
IF Compressor is noisy (0.5) OR temperature ob bump
is high (0.5)
Then blade of compressor is broken
(W1=1,μ1=0.3,CF1=0.9;W2=1,
μ2=0.2,CF2=0.95)
Labels are used to replace the preconditions and con-
clusions after the separation process is completed. For
example, the condition ‘compressor is noisy’, is replaced
by P11. Meanwhile, the meanings of the preconditions and
conclusion are stored in the corresponding locations in a two-
column table.
Simplification
In a KBS, some rules share the same preconditions or conclu-
sions. Hence, it is natural to assume there are some repeated
items in a two-column table. The function of the simpli-
fication phase is to check for repeated items and delete
unnecessary items.
A four-rule KBS is utilized to illustrate the simplification
process. The four rules are without parameters, and are listed
in Table 6. Table 7shows the corresponding two-column table
for the four-rule KBS.
The repeated items indicated in Table 7are deleted. The
simplification process is then conducted as follows.
1. Appoint the first row as the current row.
2. Check if the same meaning in the second column of the
appointed row is stored by other rows.
3. Delete the repeated rows instead of replacing the label
of the repeated rows in the FPRs with the label of the
appointed row.
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Yes
Begin
Separate every FPR in KBS
Merge repeated items for KBS
Represent every FPR by the proposed method
Obtain reasoning paths among FPRs in KBS
Generate the FPN model for the current reasoning path
Are there unprocessed reasoning paths?
Store the generated FPN model
Select next unprocessed
reasoning path as the
current reasoning path
Are there sharing relationships among the
stored FPN models?
Compose the FPNs who
owns sharing relationship
Output the final FPN model(s)
End
No
No
Yes
Separation
Simplification
Generation
Merging
Fig. 3 Flowchart of the proposed transformation algorithm
(0.65)(0.5)IF compressor
is noisy OR temperature of
bump is high THEN blade of compressor is
broken
(W1=1,µ1=0.3,CF1=0.9;
W2=1,µ2=0.2,CF2=0.95)
P11 P12 C1
P11 compressor is noisy
C1 blade of compressor is broken
P12 temperature of bump is high
(0.65)(0.5)P11 OR P12 THEN C1 (W1=1,µ1=0.3,CF1=0.9;W2=1,µ2=0.2,CF2=0.95)IF
Fig. 4 Separation process of FPR
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Tabl e 6 Four-rule KBS
Rule Content
1IFR1THENC1
2IFR2THENC2
3 IF R31 AND R32 THEN C3
4IFR4THENC4
Tabl e 7 Two-column table of KBS
Label of precondition and conclusion Meaning
R1 Hardware failure
C1 Measuring resolution decrease
R2 Hardware failure
C2 Control unit failure
R31 Hardware failure
R32 Software failure
C3 Servo system failure
R4 Control unit failure
C4 Tool breakage
hardware fails
R1
tools are brokenC4
control unit failsR4
servo systems failC3
software failsR32
control unit failsC2
hardware failsR31
hardware failsR2
measuring resolution decreasesC1
A
B
Fig. 5 Simplification process for repeated items
Tabl e 8 Modified four-rule KBS
Rule Content
1IFR1THENC1
2IFR1THENC2
3 IF R1 AND R32 THEN C3
4IFC2THENC4
Tabl e 9 The modified two-column table of KBS
Number Meaning
R1 Hardware failure
C1 Measuring resolution decrease
C2 Control unit failure
R32 Software failure
C3 Servo system failure
C4 Tools breakage
4. Repeat this operation until all rows are checked.
Figure 5demonstrates the simplification process for repeated
items.
Arrow A in Fig. 5indicates the three repeated items (R1,
R2, R31) in Table 7. Hence, R2, and R31 are deleted. All
repeated items in the KBS are replaced by R1. Arrow B shows
the same operation for another repeated item. Tables 8and 9
are appropriately modified versions of Tables 6and 7.
Generation
The generated reasoning paths described make it easier to
model a corresponding FPN with the following advantages.
1. The head of the inference chain corresponds to the output
place of the FPN model.
2. According to the rule types, [] indicates ‘Simple’ and
‘AND’ rules, and {} indicates the ‘OR’ rule; it therefore
becomes easier to recognize and generate the places, tran-
sitions, and arcs for FPN models.
The reasoning path Q3{Q1[R1,R2],Q2[R3]} gained from
Table 5is an eligible case for demonstrating the modeling
process from the reasoning path to the FPN.
1. From the inference path in Table 5, it was found that
Q3is the output place. The FPR is classified as an ‘OR’
rule because of symbol {}. In addition, two branches of
the generated transition of the FPN model are identified
because two components of {} exist.
2. Table 5demonstrates the corresponding parameter chain
that is conversely marked in the FPN model. For example,
the belief strength (CF) is marked along the arc between
the transition and the output place. The threshold (μ)is
marked on the transition.
Figure 6demonstrates the process used for generating the
output places and related transitions for the reasoning path
showninTable5. Figure 7models the corresponding ‘AND’
FPN according to the bracket types identified in the first
branch with two preconditions. Figure 8indicates the final
‘Simple’ FPR generated according to the bracket types in the
second branch.
Merging
The complexity of a large-size KBS requires merging some of
the components of the inference path. Figure 9demonstrates
the implementation of merging three FPN reasoning paths
that are then combined into one FPN, as shown in Fig. 10.
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Q3
3112 23
{[, ], [ ]}QQRR QR
32
μ
31
μ
Fig. 6 Initial converse generation of corresponding FPN reasoning path
11 2 2 3
[, ], [ ]QRR Q R
32
μ
31
μ
1
μ
1
λ
2
λ
1 11 2 12 1 1 31 3 2 2 2 32
{([(( , ), ( , )), , ], 1), ([( , 1, ), , ], 1)}wwCFw w CFw
λλ μ λ μ
== =
Fig. 7 Generation of corresponding FPN for the first branch of reasoning path
32
μ
31
μ
1
μ
1
λ
2
λ
32 2 2 32
{([( , 1,), , ], 1)}wCFw
λμ
==
2
μ
3
λ
Fig. 8 Generation of corresponding FPN for the second branch of reasoning path
R1
R2
C1CF1
W11
W12
µ1
R3
R4
C2CF3
W31
W32
R1
R2
C2CF2
W11
W12
µ1 µ3
Fig. 9 FPNs for each of the three inference paths
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R1
R2
W11
C2
C1CF 1
CF2
W12
R3
R4
W31
W32
CF 3
µ1
µ2
Fig. 10 Merged FPN
Validation of the transformation algorithm
We now present a case study conducted by Huang (2000)
demonstrating how the process of the proposed transfor-
mation algorithm generates an FPN model from a KBS.
After conducting the separation and simplification opera-
tions, the KBS is divided into two components: simplified
FPRs (Table 10) and a two-column table (Table 11).
Based on the meanings of the preconditions and con-
clusions shown in Table 11, simplified FPRs are des-
cribed (Table 10) using the proposed representation method
(Table 12). Based on the FPRs listed in Table 12,twodif-
ferent reasoning paths are generated by conducting reiterate
operations.
Reasoning path 1:
Inference chain for reasoning path 1:
C12[C8{C4[C1[R1], R42], C5{C2[R2], C3[R2, R32, R33]}},
C10[C7[C3[R2, R32, R33]]]]
Parameter chain for reasoning path 1:
[(({[([(([(λ1, w=1), 0.3, 0.9], 0.5), (λ42, 0.5)), 0.3, 0.8], 1),
0.3, 0.9], [({[([(λ2, w=1), 0.2, 0.75], 1), 0.1, 0.95], [([((λ2,
Tabl e 11 Two-column table
Location Meaning
R1 Molecular pump is not in proper position
C1 Pressure exerted is too high
R2 Temperature of cooling water is high
C2 Cooling system failure
R32 Pump is insufficiently dry
R33 Air exhaust is insufficient
C3 Compressor operates in magnetic field
R42 Roller bearing is worn
C4 Compressor is noisy
C5 Pump temperature is high
C7 Turbine blade is worn
C8 Compressor blade is broken
C10 Pressurization ratio of compressor is low
C11 Turbine blade is scaled
C12 Compressor is experiencing turbulence
C13 Turbine blade is broken
0.3), (λ32; 0.5), (λ33; 0.2)), 0.3, 0.9], 1), 0.2, 0.9]}, 1), 0.2,
0.9]}, 0.7), ([([([((λ2, 0.3), (λ32; 0.5), (λ33; 0.2)), 0.3, 0.9],
1), 0.2, 0.8], 1), 0.2, 0.9], 0.3)), 0.4, 0.95]
Reasoning path 2:
Inference chain for reasoning path 2:
C13[C11[C7[C3[R2, R32, R33]]]]
Parameter chain for reasoning path 2:
[([([([((λ2, 0.3), (λ32; 0.5), (λ33; 0.2)), 0.3, 0.9], 1), 0.2,
0.8], 1), 0.2, 0.8], 1), 0.5, 0.9]
Based on the two reasoning paths above, two correspond-
ing FPNs are built by executing a generation phase (Figs. 11,
12). After merging, the completed FPN is generated (Fig. 13).
Table 11 shows the meaning of each olace.
Tabl e 10 Simplified FPRs No. Content
1IFR1THENC1 [(λ1,w=1),0.3,0.9]
2IFR2THENC2 [(λ2,w=1),0.2,0.8]
3 IF R2 AND R32 AND R33 THEN C3 [((λ2,0.3),(λ32;0.5),(λ33;0.2)) ,0.3,0.9]
4 IF C1 AND R42 THEN C4 [((null,0.5),(λ42,0.5)) ,0.3,0.8]
5 IF C2 THEN C5 [(null,1), 0.1, 0.95]
6 IF C3 THEN C5 [(null, 1), 0.2, 0.9]
7 IF C3 THEN C7 [(null, 1), 0.2, 0.75]
8 IF C4 THEN C8 [(null, 1), 0.3, 0.9]
9 IF C5 THEN C8 [(null, 1), 0.2, 0.9]
10 IF C7 THEN C10 [(null, 1), 0.2, 0.9]
11 IF C7 THEN C11 [(null, 1), 0.2, 0.8]
12 IF C8 AND C10 THEN C12 [((null, 1), (null, 0.3)), 0.4, 0.95]
13 IF C11 THEN C13 [(null, 1), 0.5, 0.9]
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Tabl e 12 Simplified FPRs
represented using the proposed
method
No. Content
1C1[R1] [(λ1,w=1),0.3,0.9]
2C2[R2] [(λ2,w =1),0.2,0.8]
3 C3[R2, R32, R33] [((λ2,0.3),(λ32;0.5),(λ33;0.2)) ,0.3,0.9]
4 C4[C1, R42] [((null,0.5),(λ42,0.5)) ,0.3,0.8]
5 C5[C2] [(null,1), 0.1, 0.95]
6 C5[C3] [(null, 1), 0.2, 0.9]
7 C7[C3] [(null, 1), 0.2, 0.75]
8 C8[C4] [(null, 1), 0.3, 0.9]
9 C8[C5] [(null, 1), 0.2, 0.9]
10 C10[C7] [(null, 1), 0.2, 0.9]
11 C11[C7] [(null, 1), 0.2, 0.8]
12 C12[C8, C10] [((null, 1), (null, 0.3)), 0.4, 0.95]
13 C13[C11] [(null, 1), 0.5, 0.9]
Figure 11 shows the corresponding FPN model of rea-
soning path 1. Compared with the obtained FPN model and
reasoning path 1, it easy to see that the output place, C12,
is the head of inference chain 1. The remaining parts of the
obtained FPN are reverse generated from the output place to
the input places based on the inference chain in a step-by-
step manner. Meanwhile, each parameter of the gained FPN
model is also gradually obtained using the parameter chain
of reasoning path 1.
Figure 12 shows the corresponding FPN model of rea-
soning path 2. Similar to the generation process of the FPN
model for reasoning path 1, the output place, C13, of the
FPN model shown in Fig. 13 represents the head of infer-
ence chain 2, and each parameter of the obtained FPN model
is also gradually acquired based on the parameter chain of
reasoning path 2.
According to the obtained FPN models shown in Figs. 11
and 12, it can be seen that some sharing components exist,
such as certain places (R2, R32, R33, C3, C7, and C11),
transitions, arcs, and parameters. Hence, a merging operation
will be further applied to combine the common parts of these
two FPN models. After completing all four implementation
0.2
0.3
0.2
0.3
0.4
0.2
0.3
0.2
R1
R2
R32
R33
R42
C1
C2
C3
1
1
0.5
0.3
0.2
0.9
0.75
0.9
0.5
1
1
0.5
C10
C12
C8
C7
C5
C40.8
0.95
0.8
1
1
1
0.95
0.9
0.9
0.7
0.9
0.3
0.3
0.1
0.9
Fig. 11 Corresponding FPN of reasoning path 1
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J Intell Manuf
phases of the proposed transformation algorithm, an entire
FPN is generated for the KBS, as shown in Fig. 13.
Through an analysis of the transformation procedure from
the KBS to the final obtained FPN model, it can be seen that
the completed FPN is capable of representing all information
within the KBS based on the following validated qualifica-
tions.
1. The method described for representing an FPR reveals
all inner-reasoning relations between the FPRs of the
KBS, and denotes a rigorous one-to-one correspondence
between the generated reasoning path(s) and FPRs.
2. Because of these structural one-to-one correspondences
between a reasoning path and an FPN, the converse gen-
eration of the final FPN model, as per the appointed
reasoning paths, represents all information of the KBS.
The paths of both these carefully derived one-to-one cor-
respondences pave a theoretic passageway for the automatic
generation of an equivalent FPN from a corresponding large-
size KBS. The case study of the implementation process
for the corresponding FPN model clearly validates the abil-
ity of the proposed algorithm to automatically generate an
equivalent large-scale FPN from a corresponding large-size
KBS.
0.20.3 0.2
R2
R32
R33
C30.5
0.3
0.2
0.9 1 C13
C7 C11
0.8 1 0.9
1
0.8
0.5
Fig. 12 Corresponding FPN model of reasoning path 2
0.20.3
0.2
0.3
0.4
0.2
0.3
0.2
R1
R2
R32
R33
R42
C1
C2
C3
1
1
0.5
0.3
0.2
0.9
0.75
0.9
0.5
1
1
0.5
C10
C13
C12
C8
C7
C5
C4
C11
0.8
0.95
0.8
1
1
1
0.95
0.91
0.9
0.9
0.7
0.9
0.3
0.8
0.3
0.1
0.5
0.9
Fig. 13 Completed FPN of KBS
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J Intell Manuf
Conclusion
This study proposed an automatic transformation process
for building an equivalent FPN model from the correspond-
ing FPRs. The transformation process implements a novel
method for describing the FPRs through reasoning. As a
bridge between the FPRs and the FPN model, the proposed
method enables the collection of all information in a KBS and
classifies the FPRs as a single or few inference paths, includ-
ing the reasoning and parameter chains. Moreover, the FPN
model was conversely generated in a step-by-step manner,
based on rigorously derived inference paths:
1. Automatic generation of an FPN model from other for-
malisms.
2. Automatic calculation of a corresponding incidence
matrix for an FPN model.
3. Automatic implementation of a decomposition function
for a large-size FPN based on different inference paths.
4. Automatic application of relevant reasoning algorithms
based on appointed output places.
Acknowledgements This paper was supported through the National
Natural Science Foundation of China (NCFC) (No. 61462029), the
Research Foundation of the Education Bureau of Hunan Province,
China (Nos. 16C1314 and 16B212), and the Research University Grant
(RUG) UTM (No. Q. J13000. 2528. 11H72).
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