Aliveness of Fuzzy Dynamic Petri Net Systems

Chengjia Li, Fuling Ding, Ni Li Institute of Operational Research & Cybernetics

Hangzhou Dianzi University Hangzhou, P. R. China, 310018

chengjiali@hdu.edu.cn

Abstract—Fuzzy Petri net system is an important tool to analyze the behavior of concurrent systems. With a formal definition of fuzzy dynamic Petri nets, the concept of fuzzy Petri net language is first proposed in this paper. Based on the analysis of fuzzy Petri net language, the aliveness and the blocking of fuzzy Petri net are studied afterwards. Further, the relationship between the aliveness of language and blocking of the system is proposed. Finally, an example is given to illustrate the importance of aliveness analysis based on fuzzy Petri net Language.

Keywords-Petri nets; fuzzy set; aliveness; fuzzy reasoning;

I. INTRODUCTION The theory of fuzzy sets [1] has been widely used as a

formal framework for the development of methodologies which represent and reason about imprecise and vague information. Petri nets are a graphical and mathematical modeling tool, applicable to many systems such as discrete event systems [2], intelligent systems [3], communication protocols [4] and expert systems etc. They are powerful in modeling information processing systems that are characterized as being concurrent, asynchronous, distributed, parallel and non-deterministic[5]. Since Zisman showed that PNs could be translated into production rule systems [6], they have caused great attention in the field of artificial intelligence. In order to make real-world knowledge suitable for processing by computers, many knowledge representation methods have been developed, such as production rules, fuzzy production rules [7]. Because normal PN cannot deal with vague or fuzzy information such as “very high” and “good”, some Fuzzy Petri Nets (FPN)[8-10] have been introduced. As a model of knowledge-based systems, FPN are used for dealing with imprecise information. In the past few years, there has been an increasing interest at extending the fundamental concepts of Fuzzy Petri nets to incorporate capabilities for handling fuzziness in systems modeling and analysis. In recent years, many researchers focused on fuzzy Petri nets [11-15]. Chen et al. [12] consider a representation of fuzzy production rules with certainty factors. The reasoning algorithm used in that study determines whether there exists an antecedence–consequence relationship between two propositions. If this is the case, the degree of truth of the consequent proposition is evaluated from that of antecedent propositions. In a later

work by Chen [13], the authors extend their previous work with a weighted fuzzy Petri net (WFPN) model, where certainty factors, truth values of propositions and weights of propositions are represented by fuzzy numbers. However, in their work [12，13] only exact matching is allowed. A similar approach taken by Manoj et al. modify Chen’s fuzzy reasoning algorithm[13] after finding out that it does not work with all types of data[14]. Finally, The high-level fuzzy Petri net (HLFPN) model (introduced by the authors in [16] and [17]) as opposite to other approaches found in literature, derives from high-level Petri nets (HLPN) such as pr-t nets [16] and colored Petri nets [17]. Scarpelli et al. [15] propose a high level fuzzy Petri net (HLFPN) for modeling fuzzy reasoning based on compositional rule of inference. Their forward reasoning algorithm consists of the extraction of a subnet from an entire net. However, after extracting the subnet, it is not allowed to have concurrent inference. The relationship between ordinary PN and PN for knowledge representation and reasoning remains unexploited in the previous work so that PN theory could not be used into proposition logic reasoning efficiently. In this paper, the dynamic fuzzy Petri net (FDPN) model was proposed to solve the problem. And we noted that to study the related features of the FDPN and its dynamic behavior with the language of fuzzy dynamic Petri net were rarely mentioned in literature. Therefore, we start to study the language of fuzzy Petri net as the first step.

II. FORMAL DEFINITION OF FUZZY DYNAMIC PETRI NETS Fuzzy Petri nets are extensions of Petri nets, which are

exclusively used in rule-based knowledge representation and reasoning. Different definitions had been given in different situation based on the context of different applications. This section gives a formal definition and execution rules of a fuzzy dynamic Petri nets (FDPN for abbreviation).

A. Definition of FDPN Definition 1 FDPN can be defined as a 6-tuple:

{ }0FDPN= P,T,F,M ,W,τ where we have the following.

1 2{ , , , }mP p p p= is a finite set of places;

2009 International Conference on Artificial Intelligence and Computational Intelligence

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DOI 10.1109/AICI.2009.169

477

1 2{ , , , }nT t t t= is a finite set of transitions; F is a finite set of arc between places and transitions;

[ ]0 : 0,1M P → , is an association function hereafter called fuzzy marking, representing a mapping from places to real values between 0 and 1;

ijw is the set of weights from the thi transition to the

thj place or from the thi place to the thj transition, where i and j are integers;

( ]: 0,1Tτ → , ( )tτ represents a set of threshold values in the interval (0, 1] associated with transitions 1 2{ , , , }nT t t t= , respectively;

Definition 2 { }: 0,1I P T× → is an m n× input matrix defining the directed arcs from places to

transitions, ( ), 1i jI p t = if there is a directed arc

from ip to jt , else ( ), 0i jI p t = ; { }: 0,1O P T× → is

an m n× output matrix defining the directed arcs from

transitions to places, ( ), 1i jO p t = if there is a directed arc

from jt to ip , else ( ), 0i jO p t = for 1, 2, ,i m=

and 1, 2, ,j n= .

Definition 3 ( ]~

: 0,1I P T× → is an m n× input weight

matrix, for ( ) ( )~

, ,i j i jI p t w p t= ;

( ]~

: 0,1O P T× → is an m n× output weight matrix,

for ( ) ( )~

, ,i j j iO p t w t p= .

B. Execution transitions of FDPNs Similar to ordinary PN, the execution transitions of an

FDPN include enabling and firing rules. For t T∀ ∈ is enabled if and only

if p t•∀ ∈ , ( ) ( ) ( ),M p w p t tτ⋅ ≥ , denoted by [M t > ;

Enabled at marking M , t reaches a new one 'M , denoted by [ 'M t M> ,

( )( ) ( ){ }

( ) ( )

( ) ( ) ( )( )

( )

'

0 , if \

min ,max , , ,if \

( ),

, , if

,

i i i

p t t

M p w p t p tF w t p M p p t t

tM p

M p w p tF w t p p t t

t

M p

τ

τ

• •

•• •

• •

∈

⎧ ⎫⎛ ⎞⋅ ∈⎪ ⎪⎜ ⎟⋅ ∈⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭=⎛ ⎞⋅

⋅ ∈ ∩⎜ ⎟⎜ ⎟⎝ ⎠

else

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

where ( ) if 0 1if 1

a aF a

a< <⎧

= ⎨ ≥⎩

，

1，.

( ) ( ){ }( )

min ,( )

i i iM p w p t p tD t

tτ

•⋅ ∈=

t• and t• represent the preset and postset of t respectively; ( )M p represents actual truth value

(credibility) of p ; ( )tτ represents the minimum

requirements to p , for p t•∀ ∈ (threshold). Noted that self-loop execution of the transitions is

included in the network, therefore, as defined in this paper, loops can be the existed in Fuzzy Petri nets, it is more general. FDPN can be widely applied in actual system, such as the global economic system, ecological environment systems, thus, it has practical significance.

III. ANALYSIS OF ALIVENESS BASED ON FDPN LANGUAGE

Now let’s represent [ 'M t M> in a quantified way and

introduce the operator ⊗ : ⊗ :1) when the system is in state M , the transition t is

enabling and when t is fired, the system reaches state 'M , noted 'M t M⊗ = ;

2) when the system is in state M , the transition t is not enabling, noted 0M t⊗ = .

The aliveness is an important feature of Petri net theory. This paper will give aliveness description of dynamic fuzzy Petri net from the perspective of the language. Next, we will give the definition of fuzzy Petri net languages.

A. Basic definitions Definition 4 (Aliveness)

{ }0FPN= P,T,F,M ,W,τ is a fuzzy dynamic Petri net:

(1) the transition t T∈ in FPN is live if and only if

0( )M RS M∀ ∈ , ' ( )M RS M∃ ∈ ， such that ' 0M t⊗ > ; (2) FPN is live if and only if all of the transition t

in FPN are live; (3) FPN is deadlock-free if and only if

0( )M RS M∀ ∈ ， t T∃ ∈ ，such that 0M t⊗ > ;

(4) FPN is deadlock if and only if 0( )M RS M∃ ∈ ,

for t T∀ ∈ ，such that 0M t⊗ = ； Definition 5 Assume that { }0= P,T,F,M ,W,τΣ is a

fuzzy dynamic Petri net, let 01 i

( )( )=Max(M ) imsζ σ σ

≤ ≤Σ ⊗ ×

478

and *Tσ ∈ , [0 1 0]is = represents a vector whose the ith place value is one and others value are zeros, we knowζ is the language of Σ .

Definition 6 ( )( ) ( )( )pref ζ σ ζ σ= . Definition 7

( )( ), , ( )( ) 0( )( )

0 t T t

sprefζ σ ζ σ

ζ σ∃ ∈ >⎧

= ⎨⎩

i f such t hat， el se

Definition 8

*( ), ( ) ( ) 0( )( )

T Tstpref

ζ σ σ σ ζ σ σζ σ′ ′ ′⎧ ∃ ∈ Φ = ∧ >

= ⎨⎩

i f , such t hat0， el se

Note： ( )σΦ represents the letter set whose elements are contained in the stringσ .

B. The Aliveness Description of FDPN Based-on Language

Theorem 1 FPN is deadlock-free if and only if spref ζ ζ= （ ( )ζΦ ≠ ∅）。

Proof: (Necessity) FPN is deadlock-free, so 0( )M RS M∀ ∈ , t T∃ ∈ ，

such that 0M t⊗ > . For *Tσ∀ ∈ , if ( ) 0ζ σ >

then 0M Mσ⊗ = , ( ) max( )Mζ σ = ,

so *t T T∃ ∈ ⊆ ( )Tε ∉ such that 0M t⊗ > then ( ) 0tζ σ > , so ( )( ) ( ) max( )spref Mζ σ ζ σ= = .

(Sufficiency)For σ∀ ， if ( )( ) ( )spref ζ σ ζ σ= : (1) ( )( ) ( ) 0spref ζ σ ζ σ= = , obviously.

(2) ( )( ) ( ) 0spref ζ σ ζ σ= ≠ , then ' *Tσ∃ ∈ such

that '( ) 0ζ σσ > and ' 0σ ≠ ， let ' ''tσ σ= .

So 0( )M RS M∀ ∈ , t T∃ ∈ , such that 0M t⊗ > ，

so FPN is deadlock-free. Theorem 2 FPN is live if and only if stpref ζ ζ= （

( )ζΦ ≠ ∅）。 Proof ： (Necessity) For σ∀ ， ( ) 0ζ σ > :(1)if

*Tσ ′∃ ∈ and '( ) TσΦ = , but '( ) 0ζ σσ = , then,

0( )M RS M∃ ∈ , such that t T∀ ∈ ， 0M t⊗ = ，

so FPN is deadlock; (2)if *Tσ ′∀ ∈ ，'( ) 0ζ σσ > ,

but '( ) TσΦ ≠ . Let '\ ( )t T σ∈ Φ ， 0( )M RS M∃ ∈ ,

for ' ( )M RS M∀ ∈ , ' 0M t⊗ = , so FPN is not live.

(Sufficiency) stpref ζ ζ= , in other word, if *Tσ∀ ∈ ,

( ) 0ζ σ > then *Tσ ′∃ ∈ such

that '( ) 0ζ σσ > and ( ) Tσ ′Φ = , so t T∀ ∈ ,

let tσ σ σ′ ′′ ′′′= , 0( )M RS M∀ ∈ , ' ( )M RS M∃ ∈ such

that 0M Mσ⊗ = and '' 'M Mσ⊗ = and ' 0M t⊗ > .

So FPN is live.

IV. THE BLOCKING ANALYSIS OF FDPN

Definition 9 Assuming that { }0= P,T,F,M ,W,τΣ is a

fuzzy dynamic Petri net, 0( )fG R M∀ ⊆ ,

call , fG< Σ > is nonblocking if and only if

0( , ( ) ) ( , )fR M pref Gζ ζ< Σ > = < Σ > .

, fG< Σ > is nonblocking means that every running track

can extend to a marking state gM ， g fM G∈ .

Definition 10 Assuming that { }0= P,T,F,M ,W,τΣ is a fuzzy

dynamic Petri net, 0( )fG R M∀ ⊆ ,

(1) , fG< Σ > is called strictly nonblocking, if

0( , ( ) ) ( , )fR M spref Gζ ζ< Σ > = < Σ > .

(2) , fG<Σ > is called strongly nonblocking, if

0( , ( ) ) ( , )fR M stpref Gζ ζ< Σ > = < Σ > .

Note:

1) If , fG< Σ > is strictly nonblocking, then every possible

track can approach necessarily a marking state gM ，

g fM G∈ . 2) If , fG< Σ > is strongly nonblocking, then every

possible track can extend to a marking state gM ，

g fM G∈ and each transition in the track has the opportunity to appear in the extension of the track.

Non-blocking is an important feature in the supervisory control theory of discrete event system. In this paper, we extend the notion into fuzzy Petri systems. Next, we will give the relation between aliveness and blockingness.

Theorem 3 Assume that { }0= P,T,F,M ,W,τΣ is a fuzzy

dynamic Petri net, 0( )fG R M∀ ⊆ ,

(1) if , fG< Σ > is strictly nonblocking, then Σ is

deadlock-free;

(2) if , fG< Σ > is strongly nonblocking, then Σ is live.

Proof: (1) , fG< Σ > is strictly nonblocking,

479

So 0( , ( ) ) ( , )fR M spref Gζ ζ< Σ > = < Σ >

because

0 0( , ) ( , ( ) ) ( , ( ) )fspref G spref R M R Mζ ζ ζ< Σ > ⊆ < Σ > ⊆ < Σ >

and 0 0( , ( ) ) ( , ( ) )R M spref R Mζ ζ< Σ > = < Σ > ,

therefore spref ζ ζ= ,i.e., Σ is deadlock-free.

(2)Similar proof can be.

Inference 1

Assume that { }0= P,T,F,M ,W,τΣ is a fuzzy dynamic Petri

net, for 0( )fG R M⊆ ， if , fG< Σ > is strongly

nonblocking, then , fG< Σ > is strictly nonblocking.

Proof: , fG< Σ > is strongly nonblocking,

then 0( , ( ) ) ( , )fR M stpref Gζ ζ< Σ > = < Σ > , and

0( , ) ( , ) ( , ( ) )f fstpref G spref G R Mζ ζ ζ< Σ > ⊆ < Σ > ⊆ < Σ >

Therefore 0( , ) ( , ( ) )fspref G R Mζ ζ< Σ > = < Σ > ,

So , fG< Σ > is strongly nonblocking.

V. ILLUSTRATION EXAMPLE We give aliveness description of FDPN based-on

language above. It can be used to detect the aliveness of FDPN.

Example Fig. 1 and Fig. 2 represent two general fuzzy Petri net respectively.

For FDPNs appear in Fig. 1 and Fig. 2, if the initial marking

is 0 (0.9 0)M = , then FDPN and 0M form a fuzzy Petri

net system, noted 1 1 0( , )FDPN MΣ = and

2 2 0( , )FDPN MΣ = .

(1) { }1 1 1 3 2( ) ( ( ))spref spref t t tζ ζ ∗ ∗Σ = Σ = , so 1Σ is

deadlock-free.

0.8

0.4

0.5

0.3

0.3

0.6 0.6

0.6

0.6

1p 2p

1t 2t

3t Figure 1. FDPN1

0.350.2

0.8

0.4

0.5

0.3

0.3

0.6 0.6

0.6

0.6

0.6

1p 2p

1t 2t

3t

4t Figure 2. FDPN2

(2) 1 3 1( )t tσ ζ∗= ∈ Σ , if 1Tσ ∗′∈ , such

that 1( )σ σ ζ′∈ Σ , then 2tσ ∗′ = , 2 1( ) { }t Tσ′Φ = ⊂ ，

so 1( ( ))stprefσ ζ∉ Σ , in other words,

1 1( ) ( ( ))stprefζ ζΣ = Σ . So 1Σ is not live.

(3) { }2 2 1 3 2 4( ) ( ( )) ( )stpref stpref t t t tζ ζ ∗ ∗ ∗Σ = Σ = ,

then 2Σ is live. It is also deadlock-free. Note: (1)if let 1 1( , ) 0.4w t p = in Fig. 2, as showed in Fig. 3, the rest conditions don’t change, noted

3 3 0( , )FDPN MΣ = , then after firing 1t , there is no transitions can be fired in the fuzzy Petri net system, so is deadlock.

(2) If let 2( ) 0.7tτ = in Fig. 2, as showed in Fig. 4, the

rest conditions don’t change, noted 4 4 0( , )FDPN MΣ = ,

then after firing 3t , the transition 2t cannot be fired any more.

So 3Σ is deadlock-free.

(3) If let 0 (0.7 0)M = in Fig.1 and Fig.2 , then the aliveness of the FDPN showed in Fig.1 and Fig.2 will also be changed, will not repeat them here.

As a result, in the process of FDPN aliveness analysis, we can draw the following conclusions: 1. Aliveness of FDPN is related to the structure of the net. 2. When the structure of the two net are the same, the aliveness of them is related to the threshold of transitions, the weight of the input/output arc and initial marking (the same as ordinary Petri nets).

0.350.2

0.8

0.4

0.5

0.3

0.3

0.6 0.4

0.6

0.6

0.6

1p 2p

1t 2t

3t

4t Figure 3. FDPN3

480

0.350.2

0.8

0.7

0.5

0.3

0.3

0.6 0.6

0.6

0.6

0.6

1p 2p

1t 2t

3t

4t

Figure 4. FDPN4

VI. CONCLUSIONS This paper proposed a model of fuzzy Petri nets with

loop, and the concept of fuzzy Petri net language was first proposed. Combining with the language of fuzzy Petri net, we studied the dynamic behavior of the process and illustrated that the language of fuzzy Petri Nets can effectively delineate the aliveness of the net. Then, we analyzed the blocking of system based on fuzzy dynamic Petri net language. The relationship between the aliveness of language and blocking were proposed and proved. In the future, we will study fuzzy Petri net and the other features of FDPN related with the fuzzy language. Moreover, Study the impact of Synthesis operation on keeping the characteristics of the fuzzy dynamic Petri net’s language. From the language description, we will analyze the behavior features of large-scale of fuzzy systems. It will eventually be effectively combined with applications to solve practical related problems.

ACKNOWLEDGMENT This work was supported in part by Natural Science

Foundation of Zhejiang Province under grant Y7080044, in part by the National Natural Science Foundation of China under grant 60772006.

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