S S symmetry
Article An Automatic Algorithm to Generate a Reachability Tree for Large-Scale Fuzzy Petri Net by And/Or Graph
Kai-Qing Zhou 1,2,* , Li-Ping Mo 1, Lei Ding 1 and Wei-Hua Gui 2
1 College of Information Science and Engineering, Jishou University, Jishou 416000, Hunan, China; [email protected] (L.-P.M.); [email protected] (L.D.) 2 College of Information Science and Engineering, Central South University, Changsha 410083, Hunan, China; [email protected] * Correspondence: [email protected]; Tel.: +86-0743-856-3673
Received: 14 September 2018; Accepted: 25 September 2018; Published: 1 October 2018
Abstract: Fuzzy Petri net (FPN) is widely used to repre sent, model and analyse knowledge-based systems (KBSs). Meanwhile, a reachability tree is an important tool to fully represent the flow relationship of FPN and is widely applied to implement inference in industrial areas. However, the traditional reachability ignores recording the dependence relationships (‘and/or’ relationship) among the places in the neighbouring layers. This paper develops a modified reachability tree based on an and/or graph and presents a three-phase generation algorithm to model the reachability tree for the corresponding FPN automatically via fuzzy production rules (FPRs). Four cases are used to verify the correctness and feasibility of the proposed algorithm from different viewpoints, such as general FPRs, FPRs with a condition-sharing situation, FPRs with a conclusion-sharing situation, and FPRs with multi-conclusions. Simulation results reveal that the proposed approach has the ability to automatically generate the reachability tree for the corresponding FPN correctly.
Keywords: fuzzy Petri net (FPN), reachability tree; and/or graph; modeling; equivalent
1. Introduction The fuzzy Petri net (FPN), a kind of backward extension high-level Petri net (HLPN), has been widely applied to model and analyse uncertainty systems or knowledge-based systems (KBSs) [1]. Due to two unique features of FPN-structural store representation for fuzzy information and visualization description for uncertainty reasoning, FPN has been widely applied in various industrial areas, such as robotics [2,3], intelligent manufacturing systems [4,5], traffic control [6,7], semantic network [8,9], etc. According to the above two characteristics of FPN, the main mechanisms of a FPN model’s application could be classified into two types, a reasoning algorithm using the reachability tree and a reasoning algorithm using the algebra form [10,11]. The reachability tree is an equivalent model of FPN to record the reasoning path(s) among places intuitively. Chen et al. proposed various reasoning algorithms to implement fault diagnosis based on the corresponding reachability tree of a FPN model utilizing the immediate reachable set (IRS) and reachable set (RS), such as forward-reasoning algorithm [12], backward-reasoning method [13] and weighted fuzzy reasoning [14]. Manoj et al. extended Chen et al.’s research and proposed a hierarchical FPN for all types of data abstraction based on a reachability tree model [15]. In industrial applications, the reachability tree method is also widely applied to implement special tasks. Wu et al. proposed a hybrid method to analyze the reliability of the solar array by employing fault tree analysis and a fuzzy reasoning Petri net. The simulation results reveal that the top two factors to cause the solar array fault are harsh thermal environment and impact caused by particles in
Symmetry 2018, 10, 454; doi:10.3390/sym10100454 www.mdpi.com/journal/symmetry Symmetry 2018, 10, 454 2 of 9 space array [16]. Chiang et al. presented a model to evaluate sleep quality by combing the fast Fourier transform and fuzzy Petri net. Compared with other existing data mining approaches, the proposed electroencephalogram (EEG)-based sleep quality detection model is able to increase the assessment accuracy to 82.4% [17]. Reference [18–20] also demonstrate some typically applications of inference by using reachability tree in various industrial areas, such as micro distribution, risk evaluation, et al. Based on the literature mentioned above, it is easy to find that reachability tree is a powerful tool to fully represent the flow relationships among places and transitions of the entire FPN model and to implement inference operations based on a visible method. According to the above highlights of the reachability tree, various reasoning mechanisms based on the reachability tree mechanism were widely discussed in different industrial areas. However, with the rapidly increasing complexity of real industrial systems, the scale of the corresponding FPN model is also raised. The state-explosion issue hinders the further application of the inference mechanism using a reachability tree in industrial areas. The drawbacks of the reachability tree under the state-explosion issue could be summarized into two aspects:
• It difficult to directly describe the dependence relationships (‘and’ or ‘or’ relationship) among the places in the neighbouring layers; • It difficult to generate automatically a large reachability tree for a rather complex expert system.
To overcome the shortcomings mentioned above, the goal of this manuscript is to propose an algorithm to automatically generate a reachability tree for a corresponding large-scale FPN model with complex flow-relationships among places and transitions. To fulfil this goal, this paper includes three main phases. First, a modified reachability tree based on an and/or graph model [21], decision tree theory [22] and fuzzy relations [23] is presented to fully record the flow-relationship among places and describe the dependence relationships among places in the neighbouring layers. Furthermore, an automatic generation algorithm is proposed in this manuscript to the reachability tree for the corresponding large-scale FPN model for solving the state-explosion issue. Finally, four groups of cases are employed to test the feasibility of the proposed generation algorithm. The organization of the remaining parts is as follows: the modified reachability is illustrated in Section2 after discussing the general definition of FPN, fuzzy production rule (FPR) and and/or graph. Section3 demonstrates the entire flowchart and reveals the algorithm complexity of the presented generation algorithm of the modified reachability tree. A series of examples are given to prove the validity and feasibility of the presented algorithm in Section4. Section5 summarises and gives a brief conclusion.
2. Modified Reachability Tree for Different Fuzzy Petri Net (FPN) Models FPN and FPR are the two main formal ways to fulfil the KBS requirements. Meanwhile, the reachability tree is a stronger tool to fully represent flow-relationship among places of FPN. This section first introduces the basic concepts both of FPN, FPR. Then, the corresponding modified reachability trees of different types of FPN models are generated after introducing the improved thinking of the reachability tree utilizing the and/or graph.
2.1. FPN and Fuzzy Production Rule (FPR) FPN is generally defined as the following 8-tuple formalism.
FPN ∑ = (P, T, I, O, M, µ, W, CF) where, P = {p1, p2, ··· , pn} represents a finite set of places; n represents the number of places in the rule; T = {t1, t2, ··· , tm} represents a finite set of transitions; m represents the number of transitions in the rule;I(O) is the input (output) function, i.e., the mapping relation between places and transitions; T M = (m1, m2, ··· , mn) indicates the identity of places; wi indicates the weight of places pi, i.e., Symmetry 2018, 10, x 3 of 9
T transitions; M (,,,) m12 m mn indicates the identity of places; wi indicates the weight of places
pi , i.e., the support degree for the rule establishment by preconditions pi ; CFj indicates the Symmetry 2018, 10, 454 3 of 9 credibility, i.e., the true extent of the conclusion after transitions t j fired; and : (0,1] , i is the threshold of transitions t j . the support degree for the rule establishment by preconditions pi; CFj indicates the credibility, i.e., FPR is commonly used to represent the uncertainties in expert systems [20–22]. General FPRs the true extent of the conclusion after transitions tj fired; and µ : µ → (0, 1], µi is the threshold of are formalized and described as follows. transitions tj. FPR is commonly used to representif D the()(,,) uncertainties then Q CF in expert w systems [20–22]. General FPRs are formalized and described as follows. DDDD{,,} Q where, D is a limited set of preconditions; 12 n ; is a limited set of conclusions; QQQQ{,,} ( ) ( ) [0,1] 12 m ; is the true extenti f D ofλ eachthen precondition Q CF, ;µ , w ; CF is the credibility of the CF (0,1] rule; is the credibility of the conclusion obtained after the rule is executed; is the where, D is a limited set of preconditions; D = {D1, D2, ··· Dn}; Q is a limited set of conclusions; (0,1] w w(0,1] thresholdQ = {Q1, Qof2 the, ··· rule,Qm} ; λ is the true; and extent is ofthe each weight precondition; of each precondition,λ ∈ [0, 1]; CF is the credibility. of the rule; Moreover,CF ∈ (0, 1] FPRsis the can credibility be divided of the into conclusion three main obtained types, after which the ruleare ‘simple’, is executed; ‘or’,µ andis the ‘and’ threshold rules accordingof the rule, toµ the∈ ( 0,different 1]; and relationshipw is the weight among of each conditions. precondition, w ∈ (0, 1]. Moreover, FPRs can be divided into three main types, which are ‘simple’, ‘or’, and ‘and’ rules Type 1: Simple rule according to the different relationship among conditions. Type 1: Simple rule if D( ) then Q ( w 1, , CF )
i f D(λ) then Q (w = 1, µ, CF) Type 2: And rule Type 2: And rule n if D() and D () and and D () then Q (1,,) wn CF 1 1 2 2 n( n) ( i = ) i f D1(λ1) and D2(λ2) and ··· and Dn λn then Q i1∑ wi 1, µ, CF i=1
Type 3: Or rule if D() or D () or or D () then Q (1,,) w CF 1 1 2 2 n n i i i i f D1(λ1) or D2(λ2) or ··· or Dn(λn) then Q (wi = 1, µi, CFi)
2.22.2.. A Modified Modified Reachability Tree BasedBased onon And/OrAnd/Or GraphGraph The reachability tree was the main method to implement the inference process using FPN’s capability forfor graphicgraphic description description during during the the 20th 20th century. century. According According to [13 to], [13], the reachability the reachability tree could tree couldbe generated be generated by the immediateby the immediate reachability reachability set (IRS) s andet (IRS) reachability and reachability set (RS). Although set (RS). theAlthough traditional the treachabilityraditional reachability tree can fully tree represent can fully the flow represent relationship the flow among relationship places, the among logical places, relationship the logical such relationshipas ‘and’ and such ‘or’ among as ‘and’ the and connected ‘or’ among places the isconnected ignored in places the current is ignored version. in the Hence, current a modified version. Hence,reachability a modified tree is proposedreachability in tree this sectionis proposed based in on this the secti and/oron based graph on the and/or graph If the original problem can only be solved when solutions of multiple simple problems are are simultaneously established, the relationship among these these simple simple problems problems is is an an ‘ ‘and’and’ relation; if the original problem problem can can be be solved solved when when multiple multiple simple simple problems problems are established, are established, the relationship the relationship among amongthese simple these problemssimple problems is an ‘or’ is an relation; ‘or’ relation;if the original if the original problem problem can be solvedcan be whensolved any when simple any simpleproblem problem is established, is established, the relationship the relationship is called is ‘Simple’ called ‘Simple’ elation. Theelation. problem The thatproblem can nothat longer can no be longerresolved be is resolved called the is called original the problem original [ 21problem]. [21]. Based on the characteristics of the and/orand/or graph graph mentioned mentioned above, above, the the differences between the traditional reachability reachability tree tree and and the the modified modified reachability reachability tree could tree be could summarized be summarized into the following into the followingthree aspects: three aspects:
·• The symbol ‘︶’ is used to mark the ‘and’‘and’ relation for the ‘and’‘and’ type FPR in the modifiedmodified reachability tree.tree. ·• All parameters’ valuesvalues are markedmarked inin thethe modifiedmodified reachabilityreachability tree.tree. • The modified reachability reverse records the flow relationship between the input place(s) and · The modified reachability reverse records the flow relationship between the input place(s) and the output place(s) to aid in decomposition and reasoning operations. the output place(s) to aid in decomposition and reasoning operations. According to the logic relations among conditions of FPRs, there are three types of FPR, which are the ‘simple’, ‘or’, and ‘and’ rules. Furthermore, the corresponding FPN model and the equivalent modified reachability tree of each type of FPR are drawn in Figures1–3, respectively. Symmetry 2018, 10, x 4 of 9
Symmetry 2018, 10, x 4 of 9 According to the logic relations among conditions of FPRs, there are three types of FPR, which Symmetryare the 2018 ‘simple’,, 10, x ‘or’, and ‘and’ rules. Furthermore, the corresponding FPN model and the equivalent4 of 9 According to the logic relations among conditions of FPRs, there are three types of FPR, which modified reachability tree of each type of FPR are drawn in Figures 1—3, respectively. areAccording the ‘simple’, to the‘or’, logic and ‘and’relations rule amongs. Furthe conditionsrmore, the of corresponding FPRs, there are FPN three model types and of theFPR, equivalent which modified reachability tree of each type of FPR are drawn in Figures 1—3, respectively. are the ‘simple’, ‘or’, and ‘and’ rules. Furthermore, the correspondingpj FPN model and the equivalent Symmetry 2018, 10, 454 4 of 9 modified reachability tree of each type of FPR are drawn in Figures （1—μ,CF3, ） respectively. pj （μ,CF ） mi μ CF pj （μ,CF ） m μ CF p i i t pj mi μ CF pi t pj pi
pi t pj (a) p (b) i
Figure 1. (a) Corresponding fuzzy Petri net(a )(FPN ) and (b) modifiedpi reachability(b) tree of ‘simple’ rule.
Figure 1. (a) Corresponding fuzzy Petri(a net) (FPN) and (b) modified(b )reachability tree of ‘simple’ rule.
FigureFigure1. 1.( (aa)) CorrespondingCorresponding fuzzyfuzzy Petri Petri net net (FPN) (FPN) and and ( b(b)) modified modified reachability reachabilitypj tree tree of of ‘simple’ ‘simple’ rule. rule. M(p1) （μ,CF ）
pj p1 w 1 M(p1) （μ,CF ）
pj Mp(p12) w2 μ CF M(p1) 1 （μ,CF ） p 2 3 pj p1 M(p2) w w tμ CF 1 w2
M(p2i) p w1 w2 wi M(p2) w2 w 3 μ t CF j pi p1 p2 pi p2 M(pi) w 3 t pj w1 w2 wi
pi (a) (b) p1 p2 pi M(pi) w1 w2 wi
pi Figure 2. (a) Corresponding(a) FPN and (b) modifiedp1 reachabilityp2(b) pi tree of ‘and’ rule.
FigureFigure 2. ( a2.) Corresponding(a) Corresponding(a) FPN FPN and and (b) modified(b) modified reachability reachability(b) tree tree of ‘and’ of ‘a rule.nd’ rule.
Figure 2. (a) Corresponding FPN and (b) modified reachabilitypj tree of ‘and’ rule.
M(p1) W1=1 μ1
pj p C 1 t1 F M(p1) W1=1 μ1 1
pj p C M(p12) W2=1 μt21 CF2 M(p1) W1=1 μ1 1
p2 i p1 t2 C F pj （μ1,CF1 ）（μ2,CF2 ）（μi,CFi ） M(p2) W2=1 t1 μ2 F CCF2 1
M(p2i) W3=1 μt2 F i p （μ ,CF ）（μ ,CF ）（μ ,CF ） M(p2) W2=1 μ2 2 CF2 C j 1 1 2 2 i i
pi ti i p p p p 2 M(p ) W3=1 t μ F p （μ ,CF 1）（μ ,CF ）2 （μ ,CF ）i i 2 2 C j 1 1 2 2 i i pi (ati) (b) M(pi) W3=1 μ2 p1 p2 pi
pi ti FigureFigure 3. Corresponding 3. Corresponding(a) FPN FPN and and modified modified reachabilityp1 reachabilityp2(b tree) tree ofpi ‘or’ of ‘ rule.or’ rule.
Figure 3. Corresponding(a) FPN and modified reachability(b) tree of ‘or’ rule. 3. Generation3. Generation Algorithm Algorithm of Reachabilityof Reachability Tree Tree for for FPN FPN Figure 3. Corresponding FPN and modified reachability tree of ‘or’ rule. 3.1.3. Introduction Generation to A Breadth-Firstlgorithm ofSearch Reachability (BFS) Tree for FPN 3.1. Introduction to Breadth-First Search (BFS) 3. GenerationBreadth-first Algorithm search (BFS) of Reachability is one of the T classicalree for FPN algorithms to seek a tree or graph. The entire 3.1. IntroductionBreadth-first to searchBreadth (BFS)-First isSearch one of(BFS the) classical algorithms to seek a tree or graph. The entire implementation of BFS starts at the tree root or some arbitrary node of a graph. Then, it moves on the implementation of BFS starts at the tree root or some arbitrary node of a graph. Then, it moves on the nodes3.1. Introduction atBreadth the next- firstto depth Breadth search level-F irst(BFS) and S executesearch is one (BFS of the) the same classical operations algorithms once all tothe seek neighbouring a tree or graph. nodes The at theentire nodes at the next depth level and executes the same operations once all the neighbouring nodes at presentimplementation prior depth areof BFS analyzed. starts at The the queue tree root data or structure some arbitrary is often node used of because a graph. of Then, the non-recursive it moves on the theBreadth present-first prior search depth (BFS) are is analyzed. one of the The classical queue algorithms data structure to seek is a oftentree or used graph. because The entire of the mechanismnodes at ofthe BFS next [24 depth]. level and executes the same operations once all the neighbouring nodes at implementationnon-recursive ofmechanism BFS starts of at BFSthe tree[24]. root or some arbitrary node of a graph. Then, it moves on the nodesthe at present the next prior depth depth level ar ande analyzed. executes the The same queue operations data structure once all is the often neighbo usedu r becauseing nodes of atthe 3.2.non Algorithm-recursive to Generate mechanism the Reachability of BFS [24]. Tree for FPN the3.2. present Algorithm prior to Gdepthenerate ar thee analyzed.Reachability The Tree queuefor FPN data structure is often used because of the non-Therecursive proposed mechanism generation of BFS algorithm [24]. owns the capability to build the corresponding reachability 3.2. AlgorithmThe proposed to Generate generation the Reachability algorithm Tree for owns FPN the capability to build the corresponding tree for a large-scale FPN model. The main thinking of the presented algorithm includes three phases, reachability tree for a large-scale FPN model. The main thinking of the presented algorithm which3.2. Algorithm are:The proposed to Generate generation the Reachability algorithm Tree for FPN owns the capability to build the corresponding includes three phases, which are: reachability tree for a large-scale FPN model. The main thinking of the presented algorithm • PhaseThe 1proposed is used to generationredefine and algorithm optimize the owns FPRs, the which capability includes twoto build operations. the correspondingFirst, redefine reachabilityincludes three tree phases, for a largewhich-scale are: FPN model. The main thinking of the presented algorithm all FPRs based on the following formalism one by one: includes three phases, which are: < type(simple, or, and?), {condition(s)}, conclusion, {parameters} >
Symmetry 2018, 10, x 5 of 9
Phase 1 is used to redefine and optimize the FPRs, which includes two operations. First, redefine
Symmetryall FPRs2018, 10 based, 454 on the following formalism one by one: 5 of 9 type( simple ,, or and ?),{ condition ()}, s conclusion ,{ parameters } Then,Then, settle settlethe conclusion the conclusion-sharing-sharing case in caseKBS. in In KBS.KBS, Ina common KBS, a commonsituation that situation exists thatis that exists many is thatFPRs many could FPRs obtain could the obtainsame conclusion. the same conclusion. The conclusion The conclusion-sharing-sharing situation will situation lead to will chaotic lead toand chaotic complex and complexflow-relationships flow-relationships in the reachability in the reachability tree. Hence, tree. Hence, in this in algorithm, this algorithm, the theconclusion conclusion-sharing-sharing situations situations will will be bechecked checked first first and and the the FPRs FPRs with with the the conclusion conclusion-sharing-sharing situationssituations willwill bebe divideddivided intointo twotwo FPRsFPRs byby usingusing aa virtualvirtual conclusionconclusion asas follows.follows. type( simple ,,?),{ or and condition ()}, s virtual conclusion ,{ parameters } < type(simple, or, and?), {condition(s)}, virtual conclusion , {parameters } > (1(1))
type( simple ,,?),{ or and virtualn conclusion ()}, s previouo s conclusion ,{ parameters } (2) < type(simple, or, and?), virtual conclusion (s) , previous conclusion , {parameters} > (2) Phase 2 is used to generate all reachability trees for each FPR and to calculate the number of all • Phase 2 is used to generate all reachability trees for each FPR and to calculate the number of allroot root-nodes.-nodes. • PhasePhase 3 is 3 isto to expand expand the the reachability reachability tree tree of of an an any any given given root root-node-node from from the the root root-nodes’-nodes’ set by using thethe BFSBFS mechanismmechanism repeatedlyrepeatedly until until the the final final completed completed reachability reachability tree tree is gained,is gained and, and to executeto execute the the BFS BFS operation operation for for each each root-node. root-node. The flowchartThe flowchart of the of generation the generation algorithm algorithm is illustrated is illustrated as shown as shown in Fig inure Figure 4. 4.
Begin
Redefine all FPRs using the proposed formalism
Optimize the FPRs with Whether exists conclusion-sharing stituation? Y conclusion-sharing situation by using virtual conclusion. N Generate the corresponding reachability tree for each FPR
Obtain the root-nodes set
Get any root-node from the root-nodes set
Expend the reachability tree of the any given root-node by using BFS based on the relationship between conditions(s) and conclusion.
Record all visited nodes. Once the current node has been visited , move on to the visited node for solving the sharing condition. Y The root-nodes set is empty? N Obtain the complete reachability tree
End .
FigureFigure 4. Flowchart of generation algorithm. 4. Experiments and Related Analysis 4. Experiments and Related Analysis In this section, four groups of FPRs with different characteristics are used to test the feasibility of In this section, four groups of FPRs with different characteristics are used to test the feasibility the proposed algorithm., which are: of the proposed algorithm., which are: • FPRs with ‘and’ and ‘or’ relationships; · • FPRs with sharing‘and’ and condition ‘or’ relationships; situation; • FPRs with sharing conclusion situation; • FPR with multi-conclusions. Symmetry 2018, 10, x 6 of 9
· FPRs with sharing condition situation; · FPRs with sharing conclusion situation; Symmetry· FPR2018 with, 10 ,multi 454 -conclusions. 6 of 9
4.1. Experimental Data and Simulation Results 4.1. Experimental Data and Simulation Results Experiment one: in this experiment, four FPRs, includes one simple type, two ‘and’ rules and a Experiment one: in this experiment, four FPRs, includes one simple type, two ‘and’ rules and a ‘or’ rule, are used to generate the corresponding reachability tree. The FPRs are listed in Table 1 and ‘or’ rule, are used to generate the corresponding reachability tree. The FPRs are listed in Table1 and the simulation result is demonstrated in Figure 5. the simulation result is demonstrated in Figure5.
Table 1. Four fuzzy production rules (FPRs) in a knowledge-based system (KBS). Table 1. Four fuzzy production rules (FPRs) in a knowledge-based system (KBS). No FPR Noif R and R and R FPR then Q(,,,,) w w w CF Rule 1 if R1 and R 2 and R 3 then Q 111121311(,,,,) w w w CF Rule 1 i f R1 andif R2 Rand then R3 Qthen( w Q1 1,(w11 , CF, w12 ), w13, µ1, CF1) Rule 2 if R3 then Q 2( w 2 1, 2 , CF 2 ) Rule 2 i f R3 then Q2 (w2 = 1, µ2, CF2) if Q or Q then Q( w 1, w 1, , , CF , CF ) RuleRule 3 3 iif f Q Q11or or Q Q2 2then then Q Q3 3((w w31 31= 1, w 3232 = 1,1,µ 3131 ,, µ 3232 ,, CFCF 3131 ,, CFCF 3232 )) Rule 4 i f Rif7 and R and R6 Rand and R5 Rthen then R1 R ifi f Q Q1 oror Q Q2 thenthen Q Q(,,,,)3 w( ww15 w, w16, w CF17, µ15 , CF16) Rule 4 if R7 and R 6 and R 5 then R 1 if Q 1 or Q 2 then Q 31516171516(,,,,) w w w CF
FigFigureure 5 5.. TheThe obtained obtained reachability reachability tree from Table 1 1..
Experiment two: in this experiment, two FPRs with sharing conditions are used to generate the corresponding reachability reachability tree. tree. The The FPRs FPRs are listed are in listed Table in2 and Table the simulation 2 and the resultsimulation is demonstrated result is indemonstrated Figure6. in Figure 6.
Table 2. TwoTwo FPRs inin aa KBS.KBS.
No No FPRFPR if R and R and R then Q(,,,,) w w w CF Rule 1Rule 1 i fif R1 R1and and R2 R 2and and R3 R 3then then Q1 Q 1(w(,,,,) w11 11, w w12 12, w w13 13, µ 11, CFCF 1111) Rule 2 i fif R1 Ror or R2 Rthen then Q2 Q(w(21 w=1, 1,w w22 = 1,1,µ21 ,, µ22 ,, CFCF21 ,, CFCF22 )) Rule 2 if R1 or R 2 then Q 2( w 21 1, w 22 1, 21 , 22 , CF 21 , CF 22 )
FigFigureure 6. TheThe reachability reachability tree obtained from Table 2 2..
Experiment three:: in this experiment, a FPR with multi-conclusionsmulti-conclusions is used to generate the corresponding reachability reachability tree. tree. The The FPRs FPRs are listed are in listed Table in3 and Table the simulation3 and the result simulation is demonstrated result is indemonstrated Figure7. in Figure 7.
Symmetry 20182018,, 1010,, 454xx 77 of 9 9
Table 3. Two FPRs in a KBS.. Table 3. Two FPRs in a KBS. No FPR Noif R and R and R FPR then Q(,,,,) w w w CF Rule 1 if R11 and R 2 2 and R 3 3 then Q 1 1(,,,,) w 11 11 w 12 12 w 13 13 11 11 CF 11 11 Rule 1 if R ori f R R orand R R thenand Q R ( wthen 1, w Q 1,( w, w 1,,,,, w , µ , CF CF) , CF , CF ) if R11 or R 2 21 or R 3 32 then Q 1 13( w 21 21 1, w 22 221 1, w 23 2311 1,,,,12 21 2113 22 2211 23 23 CF11 21 21 , CF 22 22 , CF 23 23 ) RuleRule 2 2 i f R or R or R then Q (w = 1, w = 1, w = 1, µ , µ , µ , CF , CF , CF ) 1 2 3 1 21 22 23 21 22 23 21 22 23
FigFigureure 7. 7. TheThe reachability reachability tree tree obtained obtained from from Table Table3 33...
Experiment FourFour:: inin thisthisthis experiment, experiment,experiment, two twotwo FPRs FPRsFPRs with withwith a sharinga sharing conclusion conclusion are are used used to generateto generate the thecorrespondingthe corresponding corresponding reachability reachability reachability tree. The tree. tree. FPRs The The are FPRs FPRs listed are are in Tablelisted listed4 inand in Table Table the simulation 4 4 and and the the result simulation is demonstrated result is demonstratedin Figure8. in Figure 8..
Table 4. FPR with multi-conclusions. Table 4. FPR with multi--conclusionsconclusions..
NoNo FPR Rule 1 i fif R Rand and R Rand and R Rthen thenQ Q and Q (,,,,,) w(w w, w w, w, µ CF, CF CF, CF ) Rule 1 if1 R11 and2 R 2 2 and3 R 3 3 then Q1 1 and Q 2 2(,,,,,) w 11 1111 w 12 1212 w 13 1313 11 11 CF11 11 11 CF11 12 12 12
FigFigureure 8. 8. TheThe reachability reachability tree tree obtained obtained from from Table Table4 44...
44.2..2.2.. Analysis of S Simulationimulationimulation R Resultsesultsesults The above above four four experiments experiments test test how how feasible feasiblefeasible is the isis automaticthethe automatic automatic reachability reachability reachability tree generation tree tree generation generation ability abilityin the proposed inin thethe proposedproposed algorithm algorithmalgorithm from four fromfrom aspects, fourfour whichaspects,aspects, are whichwhich generally areare generally FPRs,generally FPRs FPRs,FPRs, with FPRsFPRs sharing withwith conditions, sharingsharing conditions,FPRs with sharing FPRs with conclusion sharing andconclusion FPR with and multi-conclusions. FPR with multi--conclusions. The simulation simulation result result of experiment of experimentiment 1 shows 1 1 shows shows that the that that algorithm the the algorithm algorithm generates generates generatesthe correct thereachability the correct correct reachabilitytreereachability for simple treetree FPRs. forfor simplesimple Then, experimentFPRs.FPRs. Then,Then, two experimentexperiment reveals that twotwo the revealsreveals proposed thatthat thethe algorithm proposedproposed owns algorithmalgorithm the capability ownsowns thetothe find capabilitycapability and merge toto findfind the andand sharing mergemerge thethe conditions sharingsharing conditionsconditions among FPRs. amongamong Based FPRs.FPRs. on Based experiments on experiments one and one two,and two,thetwo, condition-sharing ththe condition--sharing situation situation and conclusion-sharing and conclusion situation--sharing are situation both included are in both experiment includedincluded three inin experimentto verify the threethree correctness to to verify verify of the the the proposed correctness correctness algorithm. of of the the proposed proposed For the conclusion-sharing algorithm. algorithm. For For the the conclusion caseconclusion in experiment--sharing casethree, in a experiment virtual node three, is automatically a virtual node added is automatically to represent the added flow to relationship representt thethe among flowflow FPRs. relationshiprelationship Finally, amongthe FPR FPRs. with multi-conclusions Finally, thethe FPR is with used multi to test--conclusions the algorithm is fromused the to viewpointtest the algorithm of multi-input from and the viewpointmulti-output of cases multi in--inputinput experiment and and multi multi four.--output The four case simulations inin experiment experiment results revealfour. four. The that four the proposed simulation algorithm results revealcanreveal generate thatthat thethe the proposedproposed corresponding algorithmalgorithm reachability cancan generagenera fortete the thethe FPN correspondingcorresponding model correctly. reachabilityreachability forfor thethe FPNFPN modelmodel correctly.
Symmetry 2018, 10, 454 8 of 9
5. Conclusions The current method neglects to record the dependence relationships (‘and’ or ‘or’ relationships) among the places in the neighbouring layers. Focusing on this issue, a modified reachability tree is proposed in this manuscript based on the and/or graph. Furthermore, an automatic generation algorithm is proposed to execute the modelling operation from FPN to the corresponding reachability tree. Four experiments from different cases are employed to test the feasibility of the proposed algorithm. Simulation results show that the proposed algorithm possesses the capability to generate the corresponding reachability tree for the large-scale FPN model automatically.
Author Contributions: K.-Q.Z. conceived and designed the experiments and wrote the paper; L.-P.M. analyzed the data; W.-H.G. revised the manuscript; L.D. performed the experiments. Acknowledgments: This work is supported by the National Natural Science Foundation of China (Nos. 61741205, 61462029) and Post-doctoral Science Foundation of Central South University (No. 175605). Conflicts of Interest: The authors declare no conflict of interest.
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