Dynamic properties of fuzzy Petri net model and ... - Springer Link

J. Cent. South Univ. (2015) 22: 4717−4723 DOI: 10.1007/s11771-015-3023-7

Dynamic properties of fuzzy Petri net model and related analysis ZHOU Kai-qing(周恺卿)1, Azlan Mohd Zain1, MO Li-ping(莫礼平)2 1. Soft Computing Research Group, Faculty of Computing, Universiti Teknologi Malaysia UTM Skudai, Johor 81310, Malaysia; 2. College of Information Science and Engineering, Jishou University, Jishou 416000, China © Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: Fuzzy Petri net (FPN) has been extensively applied in industrial fields for knowledge-based systems or systems with uncertainty. Although the applications of FPN are known to be successful, the theoretical research of FPN is still at an initial stage. To pave a way for further study, this work explores related dynamic properties of FPN including reachability, boundedness, safeness, liveness and fairness. The whole methodology is divided into two phases. In the first phase, a comparison between elementary net system (EN_system) and FPN is established to prove that the FPN is an extensive formalism of Petri nets using a backwards-compatible extension method. Next, current research results of dynamic properties are utilized to analyze FPN model. The results illustrate that FPN model is bounded, safe, weak live and fair, and can support theoretical evidences for designing related decomposition algorithm. Key words: fuzzy Petri net; elementary net system; backwards-compatible extension method; dynamic properties

1 Introduction Modelling is a simulation technique to replicate, analyze, and capture behaviors of a system. Recently, modelling has grown up to be a useful and indispensable part to research diverse systems in physics, biology, social science, and engineering [1−5]. In general, the correspondence model is designed to simulate the running of complexity system, to estimate the performance, and to prove the validity of the analytic solutions. The relationship between the real system and the relevant model is illustrated in Fig. 1. Petri net (PN) is a strong mathematical tool to model the discrete events or distributed systems [6]. Compared with other similar modelling techniques, PN can demonstrate the system graphically by using reachable marking graph, reachability tree, and coverability graph. Moreover, PN model can also be used to discuss the behaviors of the system by using the incidence matrix and state equation [7−9]. Particularly, in PN theory, the dynamic properties (or behavioral properties), such as reachability, boundedness, safeness, liveness, and fairness, are closely related to the similar characteristics of the running process of real system [10]. For instance, liveness and fairness are proposed to

explore whether deadlock phenomenon exists and to reflect the starvation-free characteristic of actions in the real system, respectively [11−12]. In current literature, the dynamic properties are applied to test the feasibility and validity of decomposition algorithm of PN by analyzing the consistency of the dynamic properties between original PN model and correspondence sub-nets [13−16]. Fuzzy Petri net (FPN), being a kind of high level PNs (HLPNs), is used to model the knowledge-based systems or systems with uncertainty and applied in many industrial areas, such as manufacturing, robotic, power engineering, and traffic engineering, to execute the fault diagnosis, target recognization and process control [17−23]. However, with rapid-growing scale of the real system, the size of corresponding FPN model also increases sharply. Due to lack of theoretical supports of FPN model, the decomposition algorithms of FPN are limited in the existing literature. To pave the way for the theoretic foundation for evaluating the validity of the decomposition algorithm of FPN, in this work, the dynamic properties are discussed based on the following two aspects. A comparison between EN_system and FPN is set up to prove that the FPN is a kind of backwards-compatible extension formalism from the original PN model. The existing results of dynamic

Foundation item: Project(R.J13000.7828.4F721) supported by Soft Computing Research Group (SCRP), Research Management Centre (RMC), UTM and Ministry of Higher Education Malaysia (MOHE) for Financial Support Through the Fundamental Research Grant Scheme (FRGS), Malaysia; Project(61462029) supported by the National Natural Science Foundation of China Received date: 2014−11−03; Accepted date: 2015−01−31 Corresponding author: ZHOU Kai-qing, PhD; Tel: +60−176832829; E-mail: [email protected]

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Fig. 1 Relationship between real system and correspondence model

properties of other PNs are utilized to discuss FPN by considering the unique characteristics of FPN, and some theorems of dynamic properties of FPN are proved.

2 Petri net, EN_ system and fuzzy Petri net PN has been applied widely in various industrial fields to model, analyze, and discuss the behaviors of real distributed systems and discrete events [24−27]. 2.1 Petri net and EN_ system The formation of PN is illustrated as a six-tuple as follows [28]. Definition 1: A PN (PN) is six-tuple PN ()  {S , T ; F , K ,W , M 0 }, where P is a finite set of places (in PN model, the place is represented by a circle); T is a finite set of transitions (in PN model, the transition is represented by a rectangle); F is a finite set of arcs, F  ( P  T )  (T  P ); K is a capacity function of P, K=(1, 2, 3, …); W: F→{1, 2, 3, …} is a weight function; M0: P→{1, 2, … } is the initial marking, which describes the initial state of the real system. Furthermore, a modified PN is called elementary net system (EN_ system) when a PN model fulfills the following three conditions: 1) s  S , K ( s )  1; 2) ( x, y )  F , W ( x, y )  1; 3) s  S , M ( s )  1. EN_system is the most fundamental model in PN family. In EN_system, P is considered as conditions and represented by B, and T is considered as events and represented by E. Formalism of EN_system is demonstrated as follows [29]. Definition 2: An EN_system is a four-tuple EN_system=(B, E, F, c), where B is a finite set of places; E is a finite set of transitions; F  ( P  T )  (T  P ) is a finite set of flow relation; cB is a case of EN_system. In every certain state of EN_system, the status of

conditions (B) can be divided into two classes. Some conditions are true (M(s)=1), and other conditions are false (M(s)=0). Based on this finding, a subset of B (replaced by C) is used to represent these all “true” conditions. 2.2 Fuzzy Petri net To model and analyze the system with uncertainty, LOONY [30] proposed an initial formation of fuzzy Petri net to execute the approximate reasoning. Up to now, there is no unified formal definition of FPN in existing literature. The general formalization of FPN can be demonstrated by an 8-tuple as follows [31]. Definition 3: FPN is an 8-tuple, FPN ()  {P, T , M , I , O,W ,  , CF }, where P  ( p1 , p 2 ,  , pn ) is a finite set of places; T  (t1 , t 2 , , t m ) is a finite set of transitions; M  (m1, m2 ,, mn )T is a vector of fuzzy marking, and mi[0, 1] means the truth degree of pi (i=1, 2, …, n). Moreover, the initial truth degree vector is denoted by M0; I : P T  {0, 1} is an n×m input matrix defining the directed arc from place to transition:  I ( pi , t j )  1, if there is a directed arc from pi to t j   I ( pi , t j )  0, else

where i=1, 2, …, n; j=1, 2, …, m. O : P T  {0, 1} is an n×m output matrix defining the directed arc from transitions to place: O ( pi , t j )  1, if there is a directed arc from t j to pi  O ( pi , t j )  0, else

where i=1, 2, …, n; j=1, 2, …, m. W(i, j) is the weight from pi to tj; μi: μi→(0, 1] represents the threshold value of transition tj; CF(j, i) is the support strength from tj to pi. 2.3 Other concepts Other related concepts of this work are explained as follows.

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Definition 4: Pre-set and post-set For a PN ()  {S , T ; F , K ,W , M 0 }, pre-set and post-set of each element are demonstrated as follows, respectively. The pre-set or input set of x is defined as  x  { y | ( y, x)  F } and the post-set or output set of x is defined as x   { y | ( y, x)  F }, where x, y  P  T . Definition 5: Input place and output place For a PN ()  {S , T ; F , K ,W , M 0 }, input place and output place are demonstrated as: Input place: p  { p  P| p    p   }; Output place: p  { p  P| p    p   }.

3 Relationships of Petri net, EN_system, and fuzzy Petri net In the past few decades, various HLPNs have been presented in different research applications, which include stochastic Petri net (SPN, 1982), colored Petri net (CPN, 1987), time Petri net (TPN, 1987), and fuzzy Petri net (FPN, 1988). 3.1 Two methods of extension of HLPNs HLPN is a type of further extension from the original PN model based on distinct research goal. Methods of extension of HLPNs can be classified into two types, which are completely backwards-compatible (e.g. colored Petri nets) with the original Petri net and other methods (e.g. time Petri net, stochastic Petri net) [32]. In fact, backwards-compatible extension is not a tangible and real extension of the original PN model. In Table 1 A comparison between EN_system and FPN Faction EN_system

contrast, the HLPNs which are extended by this method can be converted back to the original PN model without any meaning strictly. It means that the convenient analytic techniques and research results of PN can also be applied into these HLPNs. Because some properties are not considered to be or cannot be modeled by the basic PN model, some HLPNs such as TPN and SPN are extended by using another method. In this kind of extension, some additional properties are added to the related formalisms. Moreover, these definitions cannot transform back to the original PN model. Naturally, the traditional analysis method and relevant studies are not still fitted to these formalisms. 3.2 Extension method of FPN FPN is a kind of HLPN to model the knowledge-base systems or systems with uncertainty, and to run the inference or computation task. Table 1 illustrates a systemic comparison between EN_system and FPN model from the following aspects, which are capacity function, weight function, marking, threshold function, support strength, enable rule, and fire rule. The details are given in Table 1. From Table 1, it is clear to get a conclusion defined as follows. Theorem 1: FPN is a type of HLPN based on a backwards-compatible extension. Proof: As mentioned in Section 2, EN_system is the most fundamental formalism of PN family. Based on Table 1, the proof is developed from the following two aspects. Aspect 1: The five main parameters (capacity

FPN

Formalism

EN_system=(B, E, F, c)

FPN(Σ)={P, T, M, I, O, W, μ, CF}

Capacity function

K(s)≡1;

K(s)≡1

Weight function

W(x, y)≡1

W(x, y)[0, 1]

Marking

M(s)≡1

M(s)[0, 1]

Threshold function

μi≡1

μi(0, 1]

Support strength

CF(j, i)≡1 For eE, if e  c  e  c   exists, transition e can be enabled

CF(j, i)(0, 1]

Enable rule

Fire rule

Once and only once a transition e is enabled will transition e fire. The result of firing of e is marked as c[e>c', where c  (c c)  c .

For tT, if p t , M ( p)  w(i, j )   exists, transition t can be enabled. i Once and only once a transition t is enabled will transition t fire. The result of firing of t is marked as M[t>M', where 0, p  t \ t   min{M ( pi )  w( pi , t )}, pi  t  ), M ( p)}, p  t  \  t max{F ( w(t , p )   t M   M ( pi )  w( p, t )}  , p  t   t  F ( w(t , p)  t  M ( p ), others 0, a  (0,1) F (a)   1, a  1

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function, weight function, marking, threshold function, and support strength) all exist in these two formalism. The value of each parameter equals one in EN_system. Moreover, being a HLFP to solve the fuzzy, vague, and uncertain information, the value of the corresponding parameter in FPN model is generalized into a fuzzy interval between zero and one. Aspect 2: For EN_system, the trigger condition of enabling rule is e  c  e  c  . If the requirement is reached, the transition is enabled because M(s)· w(s,y)≥μ is also reached due to W(x, y)≡1, M(s)≡1, and μi≡1. Compared with the EN_system, the trigger condition of enable rule of FPN model ignores the requirement of t  due to the fuzziness. Transition of FPN model can also be enabled when p  t and M ( p)  w(i, j )  i exist. By using the similar analysis, it is easy to find enable and fire rules of FPN and to adopt the traditional definition of the original PN model. Based on the two aspects, a conclusion is drawn that the EN_system is an extreme case of FPN. It is also indicated that FPN is not a tangible and real extension of the original PN model. Hence, Theorem 1 is presented. 3.3 Relationship among PN, EN_system, and FPN Considering Theorem 1, the relationship among PN, EN_system, and FPN can be illustrated by Venn diagram, as shown in Fig. 2.

Fig. 2 Relationships among PN, EN_system and FPN

The extension relationship among PN, EN_system, and FPN are also demonstrated in Fig. 3. Figures 2 and 3 indicate that the existing research results of PN and EN_system about the dynamic properties can be applied in exploring FPN safely.

Fig. 3 Extension relationship among PN, EN_system and FPN

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4 Dynamic properties of fuzzy Petri net Dynamic properties of the PN model are used to simulate the performance of the running process of the real system. The dynamic properties are discussed based on the existing achievements of the original PN and the unique highlights of FPN. The definitions of dynamic properties of PN are from Refs. [27, 33]. 4.1 Reachability Reachability is the elementary property to define and explore the related dynamic properties of PN and HLPNs. The reachability of the original is defined as follows. Definition 6: For a PN ()  {S , T ; F , K ,W , M 0 }, the direct reachable condition from M to M' is tT, transition sequence t1, t2, …, tk and marking sequence M1, M2, … , Mk exist, and then M[t1>M1[t2>M2[t3>…>Mk−1[tk>Mk. R(M) is a set of all markings from M. The set of reachability mark of the original PN is defined below. Definition 7: For a PN ()  {S , T ; F , K ,W , M 0 }, M0 is the initial marking. R(M0) is the set of reachability marking, which is the smallest set and meets two conditions: 1) M 0  R( M 0 ); 2) If M  R( M 0 ) and there exists tT, such as M [t  M ' and M ' R ( M 0 ). In FPN theory, the definition of reachability is similar to the original PN. Particularly, each complete transition sequence from the input place to the output place represents an inference path of FPN model. 4.2 Boundedness and safeness The boundedness and safeness of the original PN are illustrated below. Definition 8: For a PN ()  {S , T ; F , K ,W , M 0 }, the sufficient condition of bounded of place p is given as follows. For pP, if there a positive integral B exists such as M  R ( M 0 ) : M ( p )  B, the bound of place p marked as B(p) is the smallest positive integral: B ( p )  min{B | M  R ( M 0 ) : M ( p )  B}

Moreover, p is safe if B(p)=1. Definition 9: For a PN (Σ )  {S , T ; F , K ,W , M 0 }, Σ is bounded when p  P is bounded, and B(Σ)  max{B( p) | p  P}, Σ is safe if B(Σ)=1.

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From Table 1, two theorems of the boundedness and safeness of FPN can be given as follows. Theorem 2: FPN (Σ)  {P, T , M , I , O,W ,  , CF } is bounded and safe. Proof: FPN is a kind of HLPN by using backwards-compatible extension method from the original PN model. The same as EN_system, p  P, k ( s )  1 exists in FPN. It also means that M  R ( M 0 ), M(p)≤1. According to Definitions 8 and 9, the theorem can be proved. Theorem 3: For FPN FPN (Σ )  {P, T , M , I , O ,W ,  ,CF }, R(M0) of Σ is finitely set and | R ( M 0 ) | 2 m (m is the number of places). Proof: Assume S  {s1 , s2 ,, sm }. Depending on Theorem 1, it is found that FPN is bounded. Mark the bound of each place as B ( S1 ), B( S 2 ), , B ( s m ). For M  R( M 0 ), if M ( si )  B( si ) (i  1,2,, m) exists, it means that the maximum number of different values of M(si) is B(si)+1, which are 0, 1, 2, …, B(si). Moreover, every FPN is safe. So, M ( si )  B( si )  1  1  1  2. Furthermore, the maximum number of different value of M  ( M ( s1 ), M ( s 2 ),, M ( s m )) is

m

 ( B(si )  1). i 1

m

To sum up, | R( M 0 ) |  ( B( si )  1)  2 m. i 1

According to the proposed Theorem 3, it is easy to estimate the maximum of number of inference number in a FPN model. For example, Fig. 4 demonstrates a real case study of fault analysis system from Ref. [33]. In Fig. 4, there are 16 places in the entire FPN model. According to Theorem 3, the maximum number

Fig. 4 FPN model of fault analysis system

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of reachability marking is |R(M0)|=216. Then, according to the existing literature about the application by FPN model [17−23, 33], it is found that the completed inference path should include at least one input place and only one output. It is indicated that the minimum number of inference path is two due to the number of output place. So, the range of number of inference path of FPN model in Fig. 4 is between 2 and 216. This conclusion paves initial theoretical evidence to further study from design decomposition algorithm and calculate the exact number of inference path of FPN model. 4.3 Liveness Liveness is an essential property of PN to simulate the deadlock issue. The definition of liveness in the original PN is given below. Definition 10: For a PN ( Σ )  {S , T ; F , K ,W , M 0 }, M0 is an initial marking and tT. The condition that transition t is live can be described as: for any MR(M0), there exists M'R(M) such as M'[t>. Σ is live means that any tT is live. However, the Definition 10 is impractical and too strict for some systems. Thus, a loose definition of different levels of liveness is presented as follows. Definition 11: For a PN ()  {S , T ; F , K ,W , M 0 }, tT. Level-0 live (or dead): if t can never be fired in any firing sequence; Level-1 live: M  R ( M 0 ) : M [t  Level-2 live: For any integral n, if there exists T* such as M0[> and #(t/)≥n. #(t/)≥n is the appearing number of t in sequence. Level-3 live: If there exists an infinite transition

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sequence  such as that of M0[>, and the times that t appears in  is infinite. Level-4 live (or live): If for M  R( M 0 ) : M [t , t is the level-4 live in Σ. In PN theory, liveness means that for any marking m is reachable from m0 (Level-4 live.) Hence, if a PN is live, then there does not exist a deadlock. Furthermore, other levels live is considered as (weak) live. As noted above, FPN is a formalism to model the KBS or systems uncertainty and to implement related reasoning operations for industrial applications. Table 2 reveals the inner correspondence between KBS and FPN. Table 2 Correspondence between KBS and FPN

KBS

FPN

Every fuzzy production rule

Every transition

Pre-condition or conclusion

Place

Weight

Threshold

Weight from place to transition Support strength from transition to place Threshold of transition

Rule can be fired

Transition can be enabled

CF

The main application of FPN in the industrial field is used to implement fault diagnosis, path recognization, and so on. The core application of FPN is to design related reasoning algorithm based on a different background. Based on this factor, the input and output places must exist in FPN model. The input place and output place are demonstrated, as shown in Figs. 5 and 6, respectively. According to the research result in Ref. [34], it is

not difficult to get a conclusion that the transition of Fig. 5 is not live. However, the input places and output places always exist in the FPN model. So, according to Definition 9, the FPN model is dead. As noted above, the liveness of FPN needs to be re-defined as Definition 10. Theorem 4 protects that every FPN is weak live FPN. Theorem 4: FPN (Σ)  {P, T , M , I , O,W ,  , CF } is (weak) live. Proof: According to the correspondence between KBS and FPN in Table 2, it is found that each transition reflects a fuzzy production rule. So, in the process of reasoning, each transition may be fire. In other words, for t  T in FPN, M  R( M 0 ) : M [t  . So, each transition in FPN is Level-1 live or called weak live. Naturally, Σ is not dead. Moreover, Σ is (weak) live. 4.4 Fairness Fairness is a property of PN model to discuss the starvation-free characteristic of actions in real system. The definition of fairness of PN is given below. Definition 12: For a PN (Σ)  {S , T ; F , K ,W , M 0 } and t1·t2T, the condition of t1 and t2 belongs to the fair relation and is given as follows. If there exists a positive integral k, for any MR(M0) and any T*: M[> such as following equation exists: # (qi /  )  0 # (q j /  )  k , i, j  {1, 2}, i  j

where Σ is a fair PN model when any two transitions in Σ belongs to fair relation. Depending on Definition 11, the related theorem of fairness of FPN can be obtained. Theorem 5 PN (Σ)  {S , T ; F , K ,W , M 0 } is fair. Proof: There is no infinite transition sequence in FPN. For t1 , t 2  T in FPN, the integer k exists as the equation which is shown in Definition 12. The FPN is always fair.

5 Conclusions

Fig. 5 Input place of FPN model

Fig. 6 Output place of FPN model

1) FPN is a strong tool to model the KBS or systems with uncertainty and to enforce the reasoning process for related real system. Two essential aspects of FPN are highlighted: the extension method of FPN from the original PN model and the dynamic properties of FPN and the relevant theorems. 2) It is proved that FPN is a bounded, safe, weak live, and fair formalism in the PN family. The result also assists to estimate the feasible and validity of a decomposition algorithm of FPN. Compared with the original PN model, the key point of estimating the correctness of the decomposition algorithm is focused on how to keep consistency of the inference paths between the original FPN model and correspondence subnets.

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