adaptive fuzzy petri nets for dynamic knowledge representation and inference

Adaptive fuzzy petri nets for dynamic knowledge representationand inference

X. Li, F. Lara-Rosano*

Centre for Instrumentation Research, National University of Mexico (UNAM), Apartado Postal 70-418, Mexico City, D.F. 04510, Mexico

Abstract

Knowledge in some fields like Medicine, Science and Engineering is very dynamic because of the continuous contributions of research anddevelopment. Therefore, it would be very useful to design knowledge-based systems capable to be adjusted like human cognition andthinking, according to knowledge dynamics. Aiming at this objective, a more generalized fuzzy Petri net model for expert systems isproposed, which is called AFPN (Adaptive Fuzzy Petri Nets). This model has both the features of a fuzzy Petri net and the learning ability ofa neural network. Being trained, an AFPN model can be used for dynamic knowledge representation and inference. After the introduction ofthe AFPN model, the reasoning algorithm and the weight learning algorithm are developed. An example is included as an illustration.q 2000 Published by Elsevier Science Ltd.

Keywords: Petri nets; Knowledge-based systems; Fuzzy reasoning; Knowledge representation; Adaptive expert systems; Neural learning

1. Introduction

Petri Nets (PN) models and net theory have become animportant computational paradigm to represent and analyzea broad class of systems. As a computational paradigm forintelligent systems, net theory provides a graphical languageto visualize, communicate and interpret engineeringproblems, as well as a specification and engineeringlanguage which can be used as a development, simulationand implementation tool (Pedrycz & Gomide, 1994). PNhave the ability to represent and analyze in an easy wayconcurrently and synchronization phenomena, like concur-rent evolutions, where various processes that evolve simul-taneously are partially independent. Furthermore, PNapproach can be easily combined with other techniquesand theories such as object-oriented programming, fuzzysets, neural networks, etc. These combined PN are widelyused in computer systems, manufacturing systems, roboticsystems, knowledge-based systems, process control, as wellas other kinds of engineering applications.

Because normal PN cannot deal with vague or fuzzyinformation such as “very high” and “good”, severalFuzzy Petri Nets (FPN) have been introduced. As a modelof knowledge-based systems, FPN are used for fuzzy

knowledge representation and reasoning. In fact, by imple-menting the FPN model, major features offered by the PNmodel, such as correctness, circular rules, consistency, andcompleteness checking, can also be applied. PN have aninherent quality in representing logic in an intuitive andvisual way and also can be implemented to simulate systemsin operation. Therefore, a complex fuzzy expert systemreasoning path can be reduced to a simple sprouting treewhen applying a FPN-based reasoning algorithm as aninference engine. Besides these applications, FPN theoryalso provides means to manipulate imprecise and vagueinformation. So, many FPN models are proposed to supportfuzzy reasoning and decision making (Bugarn & Barro,1994; Cao & Sanderson, 1995; Chen, Ke & Chang, 1990;Looney, 1994; Scarpelli & Gomide, 1993; Scarpelli,Gomide & Yager, 1996; Yeung & Tsang, 1994, 1998).There are some other applications of FPN in knowledge-based systems, such as inconsistency checking (Scarpelli &Gomide, 1994), uncertainty management (Konar & Mandal,1996), and knowledge learning (Looney, 1994). For moredetails, a brief summary and discussion of FPN and itsapplication are given in Section 2.

In this paper, we pay attention to knowledge representa-tion (by weighted fuzzy production rules) and inferencewith FPN. The proposed model is called Adaptive FuzzyPetri Nets (AFPN). This model can not only be implementedto do knowledge inference, but also has a learning abilitylike a neural network, implying that the system knowledgecan be learned.

Expert Systems with Applications 19 (2000) 235–241PERGAMON

Expert Systemswith Applications

0957-4174/00/$ - see front matterq 2000 Published by Elsevier Science Ltd.PII: S0957-4174(00)00036-1

www.elsevier.com/locate/eswa

* Corresponding author. Tel.:152-5622-8601; fax:152-5622-8620.E-mail address:[email protected] (F. Lara-Rosano).

2. Fuzzy petri nets

Generally, a FPN structure is defined as a 8-tuple (Chen etal., 1990)

FPN � { P;T;D; I ;O; f ;a;b} �1�where P� { p1; p2;…; pn} denotes a set of places,T �{ t1; t2;…; tm} denotes a set of transitions,D �{ d1; d2;…;dn} denotes a set of propositions,P > T > D �f and uPu � uDu: I �O� : T ! P∞ is the input (output) func-tion, a mapping from transitions to bags of places.f : T !�0;1� is an association function which assigns a certaintyvalue to each transition.a : P! �0;1� is an associationfunction which assigns a real value between zero to one toeach place, andb : P! D is a bijective mapping betweenthe proposition and place label for each node. (Chen et al.,1990).

Many results prove that FPN are suitable to representmisty logic implication relations. FPN can deal with differ-ent types of compound fuzzy production rules. In Jeffrey,Lobo and Murata (1996) and Chaudhury, Marinescu andWhinston (1993), FPN are also introduced to representHorn clauses or Non-Horn clauses. For this reason, themajority of FPN models are used for fuzzy reasoning anddecision making (Bugarn & Barro, 1994; Cao & Sanderson,1995; Chen et al., 1990; Looney, 1994; Scarpelli & Gomide,1993; Scarpelli et al., 1996; Yeung & Tsang, 1994, 1998;).On the other hand, FPN can also be used for inconsistencychecking and uncertainty management (Garg, Ahson &Gupta, 1991; Konar & Mandal, 1996; Scarpelli & Gomide,1994). As a model for checking inconsistency, FPN has theadvantage of supporting the representation and execution offuzzy rules, and the verification process that allows thedevelopment of automatic verifiers.

By combining with some other techniques, FPNs areproved to be very useful in some actual applications. Forexample, in Hanna (1996)FPN with neural networks (NN)are used to model products’ quality from a CNC-millingmachining centre. In Andreu, Pascal and Valette (1997),FPN-based programmable logic controllers (PLC) aredeveloped, based on a combination of PN with Possibilitytheory. Also, takingG2, a real-time expert system develop-ment package, a continuous FPN (CFPN) has been imple-mented for intelligent process monitoring and control(Tang, Pang & Woo, 1995).

However, as Pedrycz and Gomide (1994) pointed, “theprimary thrust of these attempts is in a proper representationof the semantics of the underlying reasoning mechanisms.They lack of an adjustment (learning) mechanism to copewith potential numerical deficiencies of these models”.Yeung also mentioned this problem (Yeung & Tsang,1998). Henceforth, Hirota and Pedrycz (1994) changed themodeling perspective, and developed a NN with OR/ANDlogic neurons to model fuzzy set concavities. Later, Pedryczand Gomide (1994) proposed a generalized FPN model(GFPN) which can be transformed easily into the NN

proposed by Hirota and Pedrycz. The introduced architec-ture supports an explicit rather than an implicit form ofknowledge representation. Moreover, this approach is aconstructive one as parameters of the corresponding NNcan be learned (trained) rather than assigned via guessprior to the use of the net. Another FPN model with learningability was proposed by Looney (1994). An algorithm foradjusting thresholds is presented, but adjusting the weightsis done by testing, and without assurance of convergency ofthe algorithm. Lara-Rosano (1994b) introduced theWeighted Production Networks (WPN) to model complexknowledge where the antecedents in a production rule weredifferentially weighted to express “the relevance of thesingle factors in producing the result”. In this paper, Lara-Rosano also defines the weight of a Weighted Conjunction(WC) and the weight of a Weighted Disjunction (WD), buthe does not consider thresholds. Lara-Rosano (1994a)models complex soft systems by syncretizing a fuzzy causalimpact NN with a PN modeling approach. However, hismodel is used to model fixed parameters systems. A learningalgorithm is not provided.

To overcome this problem, Yeung proposed a multilevelweighted fuzzy reasoning algorithm, where a FPN isenhanced to include a set of threshold values and weights.He assigns a threshold value to every proposition in theantecedent, so that it fires only when its certainty value isgreater than the threshold. He also assigns a weight to eachconditional proposition in a production rule of the typeIF…THEN, for determining the consequent. Based on thisFPN model, Yeung proposes an algorithm for fuzzy reason-ing. The results and comparisons reveal that this method ismuch better than others (Yeung & Tsang, 1998).

Although Yeung has made his model more flexible, theweights (or the parameters) in the model are still fixed. For afuzzy knowledge (or an expert) system, however, theexperiences and learning are very important. This meansthat the model should be able to be modified according tonew incoming data. Our AFPN model tackles this problemby making weights adjustable like those of a neural network.Therefore, our FPN model has both the dynamic abilities ofan ordinary fuzzy PN and the learning ability of a NN.

3. Weighted fuzzy production rules and adaptive fuzzypetri nets

In this paper, we assume that a fuzzy knowledge-basedsystem is described by weighted fuzzy production rules(WFPR) and the system modeling is realized by mappingthese rules into an Adaptive FPN (AFPN).

3.1. Definition of weighted fuzzy production rules

The definition of a weighted fuzzy production rule(WFPR) that we give here is similar to those given in

X. Li, F. Lara-Rosano / Expert Systems with Applications 19 (2000) 235–241236

(Lara-Rosano, 1994b) or (Yeung & Tsang, 1998) but differsfrom them because it also considers negative weights, thatis, negative propositions are also permitted to appear in arule. WFPRs are categorized into three types defined asfollows: Let ai �or S ai� the ith antecedent proposition ofthe rule, andc the consequent proposition. Each propositionai can have the format “x is fa”, where fa is an element of aset of fuzzy setsF. Letl i be the threshold of the propositionto fire the rule,wi the weight of the antecedentai andm thecertainty factor of the rule:

Type 1. A simple fuzzy production ruleR: IF a �or S a� THEN c, l , w �CF� m�For this type of rule, since there is only one propositiona �or S a� in the antecedent, the weightw is 1 (or 21).Type 2. A composed conjunctive ruleR: IF a1 �or S a1� AND a2 �or S a2� AND …ANDan �or S an� THEN c;l1;l2;…;ln; w1;w2;…; wn

�CF� m�;Type 3. A composed disjunctive ruleR: IF a1 �or S a1� OR a2 �or S a2� OR…ORan �or S an� THEN c;l1;l2;…;ln; w1;w2;…; wn

�CF� m�;

In these types, negative antecedent propositions are consid-ered. When an antecedent proposition is a negative one, anegative real number is assigned to the correspondingweight of its FNPN model.

3.2. Definition of adaptive fuzzy petri nets

Chen’s FPN definition is only a basic structure, it cannotrepresent the above WFPRs completely. Considering thesetwo aspects, we define an Adaptive Fuzzy Petri Net asfollowing:

Definition 1. An AFPN is a 9-tuple

AFPN� { P;T;D; I ;O;a;b;Th;W}

whereP, T, D, I, O, a , b are defined the same as Chen et al.,1990;Th : P! �0;1� is the function which assigns a thresh-old value l i from zero to one to each placei Th�{l1;…li ;…;lm} : For any transitiont, if the certainty factors

associated with the tokens of all its input places are allgreater than their thresholds, then the transition is enabledand fires instantly.

WI : I ! �21;1� andWO : O! �21;1�; are sets of inputweights and output weights which assign weights to all thearcs of a netW �WI < WO: wIj [ WI indicates how much aplace (or an antecedent condition) impacts a following tran-sition (or an event) connected bywIj. A positive value meansa positive impact and a negative value means a negativeimpact. For a transitiont, assumeI �t� � { p1;p2;…;pn} :The corresponding input weights to these places arewI1;wI2;…;wIn: It is reasonable to letwI1 1 wI2 1 …1wIn � 1: Since all these conditions result in one consequent,the sum of their impacts is one. In the same way,wOj [ WO

indicates how much a transition impacts its output places, ifthe transition fires.

Compared to Yeung and Tsang’s (1994) definition, ourstructure combines the advantages of the two definitions ofChen and Yeung, it has a simpler structure (Yeung’s is a 13-tuple) but almost the same description ability (except for thenegative weights in our model). The comparison results inthe following five differences:

1. For any description, just as Chen does, we only give themapping without the target set. For example,a , Th.

2. In our definition, Th is defined as a mapping fromeach place to a threshold value rather than to a set ofthresholds; it equals toTh and g in Yeung’s defini-tion.

3. The set of weightsW in our model is composed oftwo parts: a set of input weights, and a set of outputweights. An input weight is assigned to an arc from aplace to a transition, and an output weight isassigned to an arc from a transition to a place, mean-ing the certainty value of the corresponding proposi-tion (or rule). It equals toW and f in Yeung’sdefinition. On the other hand, weights are added onarcs, meanwhile in Yeung’s definition they are addedon places.

4. In our model, the uncertain reasoning process inte-grates the impact of each weighted branch. However,this reasoning process is taken simply as the

X. Li, F. Lara-Rosano / Expert Systems with Applications 19 (2000) 235–241 237

Fig. 1. Mapping from WFPRs’ to FPNAWs’.

“more-or-less form” in reference (Yeung & Tsang,1994).

5. We consider negative weights, Yeung does not.

3.3. Mapping WFPR into AFPN

Arguments in favor of modeling FPRs by FPNs can befound in some papers, for example, Hanna (1996), Chen etal. (1990) and Yeung and Tsang (1994). A knowledge-basedsystem can be modeled as an AFPN by mapping its WFPRsinto small AFPNs and then connecting them throughcommon places.

To map WFPR into AFPN, we map propositions asplaces; the connections “THEN” between antecedent andconsequent propositions as transitions and arcs (for acomposed disjunctive rule, more transitions are needed);the weights of the antecedent propositions and the certaintyfactor of the rule as the weights of the input and output arcsof the corresponding transition, respectively. The mappingof the three types of weighted fuzzy production rules intothe FPNs is given step by step as follows.

Type 1. A simple fuzzy production ruleR: IF a �or S a� THEN c; l;w �CF� m�Rule R is represented in terms of FPN by Fig. 1aType 2. A composed conjunctive ruleR: IF a1 �or S a1� AND a2 �or S a2� AND …ANDan �or S an� THEN c;l1;l2;…;ln; w1;w2;…;wn

�CF� m�;Rule R is represented in terms of FPN by Fig. 1bType 3. A composed disjunctive ruleR: IF a1 �or S a1� OR a2 �or S a2� OR…ORan �or S an� THEN c;l1; l2;…;ln; w1;w2;…;wn

�CF� m�;Rule R is represented in terms of FPN by Fig. 1c.

4. Implementation of AFPN model

When implemented, an AFPN can be used as an inferenceengine. The implementation of an AFPN is realized by firingtransitions. A transitiont fires instantly as soon as it isenabled.t is enabled if all its input places have tokenswhose certainty factors are greater than their thresholds.

Let I �t� � { pi1; pi2;…;pin} ; with the corresponding inputweightswI1;wI2;…;wIn and thresholdsl1;l2;…;ln and letO�t� � { po1;po2;…;pom} with the corresponding outputweightswO1;wO2;…;wm;

Definition 2. ;t [ T; t is enabled and fired if;pij [ I �t�;the certainty valuea�pij � $ li ; j � 1;2;…;n holds. Afterfiring t, the token inpij is removed, and a token withcertainty factorwOk × �minlij 1

P�a�pij �2 lj�wIj � is putinto each of its output placespok, k � 1; 2;…;m: If a placepok has more than one input transitions (as Fig. 1d) and morethan one of its input transitions fire, then the new certainty

factor of pok is the new token which is produced by thetransition with the maximum output weight.

We assume the readers are familiar with the ordinarysymbols of Petri nets (such as:t; u:tu etc.). In order to describethe fuzzy reasoning algorithm more clearly, we make someother remarks.

_P� PUI 1 Pint 1 PO;

whereP is the set of places, we divideP into three parts,

PUI � { p [ Pu_p� B} ;

Pint � { p [ Pu_p ± B andp_± B} ;

PO � { p_� B} ;

p [ PUI

is called a user input place,

p [ Pint

an interior place,

p [ PO

an output place.Given Tp � { t [ Tu_t > PUI ± B and_t > Pint � B} ;

then t [ Tp is called an initially enabled transition.Given Tf � { t [ Tpu for ;pi [ _t; a�pi� $ li} ; then t [

Tf is called a current enabled transition.

4.1. Fuzzy reasoning algorithm

Step 1. Build the set of user input placesPUI.Step 2. Build the set of initially enabled transitionsTp.Step 3. Find current enabled transitionsTf � { t [Tpu for ;pi [ _t; a�pi� $ li} :Step 4. Fire all current enabled transitions and calculatenew certainty factors which are produced by fired transi-tions according to Definition 2.Step 5. Make token transmission. Assumep is one of theoutput places of a fired transition

5.1 If u_pu � 1; then add a token top with the certaintyfactor which is produced by its input transition;5.2 If u_pu . 1 and more than one of its input transitionsfired, then select the transition with the maximumoutput weight, and add a token top with the certaintyfactor produced by this transition.

Step 6. LetT � T 2 Tf ; P� P 2 _Tf

Step 7. Go to Step2 and repeat, untilTf � À

5. AFPN training

Before training an AFPN, we should be clear as to whatare the system outputs and what its inputs. Assume that asystem is described by WFPRs. We define all the righthands of the rules as system outputs. If we have enough


training data, the parameters can be adjusted well enough.According to different types of system inputs, an AFPN isdivided into four types of sub-structures, which are shown inFig. 1a–d. Therefore, the learning of a whole net can bedecomposed into several simpler learning processes refer-ring to these small subnets. This fact greatly reduces thecomplexity of the learning algorithm.

Given a WFPRR, we suppose that the thresholds of all itsantecedent propositions and its certainty factors are known.But we are not sure about the input weightsWI. Theseweights are to be learned, but only in the situations wherethere are enabled transitions. On the other hand, forType 1 FPRs, the input weights are 1, thus only Type2 and 3 FPRs’ weights need to be learned. Furthermore,according to Definition 2, if we have a set of trainingdata, by comparing the weights of the input arcs, it isnot difficult to know which transition is the dominatingone at this place; thus if the data are available, thentraining can be implemented.

Type 2 FPRs can be translated into a standard neuralnetwork:

y�k� �W��k�TP�k�1 b�

wherek is time, input vectorP�k� � �a�p1��k�2 l1;a�p2� ��k�2 l2�T; the weight vectorW�k� � �wI1�k�;wI2�k��T; biasb� min�l1;l2�; the outputy(k) is the certainty factor of theconclusions as shown in Fig. 2. So, Widrow–Hoff learning

law (Least Mean Square) can be applied as

W�k 1 1� �W�k�1 2de�k�P�k�; e�k� � t�k�2 y�k�;�2�

where t(k) is the goal output (teacher) and the weightvector W(k) is calculated recursively. It is known thatfor a small enough positive constantd , the updating law(2) converges to real values (Hagan, Demuth & Beale,1996).

5.1. Example

{ A;B;C;D;E;F;G} are related propositions of a knowl-edge-based systemG . Between them there exist the follow-ing rules:

R1: If A and B Then E,lA;lB;wA;wB �CF� m1�R2: If C Then F,lC �CF� m2�R3: If F Then G,lF �CF� m3�R4: If D and E Then G,lD;lE;wD;wE �CF� m4�:

Based on above translation principle, we map this systemGinto an AFPN which is shown in Fig. 3.

For this knowledge-based systemG , we assume we knowthe following real data:

lA � 0:5; lB � 0:8; lC � 0:3; lD � 0:8; lE � 0:1;

lF � 0:4; m1 � 0:8; m2 � 0:9; m3 � 0:6; m4 � 0:7


Fig. 2. Adaptable parts.

Fig. 3. FPNAW representation of knowledge-based system.

As the desired weights

wpA � 0:73; wp

B � 0:27; wpD � 0:1; wp

E � 0:9 �3�are unknown, we will use neural networks technique to learnthese weights from actual data. Given any set of inputsY �{a�pA�;a�pB�;a�pC�;a�pD�} ; the human expert can give thecorresponding output C � {a�pE�;a�pF�;a�pG�} : Forexample, suppose that forY1 � {0 :7;0:9; 0:5;0:9} ; thedesired output isC � {0 :5148; 0:45;0:3406}; and forY2 �{0 :4;0:2; 0:9;0:7} ; the desired output isC � {0 ; 0:81;0:648}: The adaptable parts of AFPN (see Fig. 4) may betransformed as Fig. 3b and c.

Let the initial a priori weights be:

wA�0� � 0:5; wB�0� � 0:5; wD�0� � 0:2; wE�0� � 0:8

To apply Widrow–Hoff learning law, we maket(k) thedesired outputsta�pE��k� � �0:5148;0�; ta�pG��k� � �0:3406;0:648�; y�k� � �a�pE�;a�pG��; P1�k� � �a�pA�;a�pB��;P2�k� � �a�pD�;a�pE��; d � 1:7: The learning results areshown in Fig. 4. One can see that after 20 steps, theweights converge to their real values, i.e. AFPN math-ematically models the knowledge-based systemG .

6. Conclusion

The advantages and drawbacks of using FPN in fuzzyknowledge systems are listed in detail (Yeung & Tsang,1994). Our work has the following features which are differ-ent from those of earlier works:

1. We consider the contribution of an antecedent propo-sition to its consequent proposition as a weight takenas an input weight in the AFPN model.

2. Negative impacts are also permitted in our model.This fact is very important to model social oreconomic systems.

3. The fuzzy token transfer rule of AFPN is based on

the accumulation of the contributions of all its inputs(see Definition 2), while most others are based onMin–Max calculation.

4. AFPN can be trained like neural networks, by learn-ing the weights from the data given by experienceand/or experts. This is an innovation over othermodels.

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adaptive fuzzy petri nets for dynamic knowledge representation and inference

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