a fuzzy logic approach to experience-based reasoning

A Fuzzy Logic Approach toExperience-Based ReasoningZhaohao Sun,1,* Gavin Finnie2,†

1Department of Computer Science, College of Mathematics and InformationScience, Hebei Normal University, Shijiazhuang 050016, China2School of Information Technology, Bond University, Gold Coast Qld 4229,Australia

Experience-based reasoning ~EBR! is a reasoning paradigm that has been used in almost everyhuman activity such as business, military missions, and teaching activities since early humanhistory. However, EBR has not been seriously studied from either a logical or mathematicalviewpoint, although case-based reasoning ~CBR! researchers have paid attention to EBR to someextent. This article will attempt to fill this gap by providing a unified fuzzy logic-based treat-ment of EBR. More specifically, this article first reviews the logical approach to EBR, in whicheight different rules of inference for EBR are discussed. Then the article proposes fuzzy logic-based models to these eight different rules of inference that constitute the fundamentals for allEBR paradigms from a fuzzy logic viewpoint, and therefore will form a theoretical foundationfor EBR. The proposed approach will facilitate research and development of EBR, fuzzy sys-tems, intelligent systems, knowledge management, and experience management. © 2007 WileyPeriodicals, Inc.

1. INTRODUCTION

Experience-based reasoning ~EBR! is a widely used reasoning paradigm basedon logical arguments.1 For example, EBR has been used in help desk systems2 toadapt to new business situations by “learning” from experience, tailoring a helpdesk to effectively maintain critical business systems. However, there appears tobe no theoretical research works for EBR, although there are a lot of empiricalworks on EBR mainly in the business and commerce fields. As we know, experi-ence is an important asset for a domain expert. However, how to formalize expe-rience is still a big issue. Further, there is no fundamental research to investigatethe logical or mathematical foundation of EBR.3,4

In the context of case-based reasoning ~CBR!, EBR is a model of humandecision making and problem solving.5,6 CBR researchers have studied EBR to

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 22, 867–889 ~2007!© 2007 Wiley Periodicals, Inc. Published online in Wiley InterScience~www.interscience.wiley.com!. • DOI 10.1002/int.20220

some extent. For example, Stroulia and Goel6 argued that in EBR, new problemsare solved by retrieving and adapting the solutions to similar problems encoun-tered in the past.7 However, CBR can only be considered as one part of EBR,because EBR includes many different reasoning paradigms.3,4

We attempt to fill the above gap by providing a unified fuzzy logic-basedtreatment of EBR, based on our previous work on logical treatment of EBR.3,4,8

More specifically, we first review the logical approach to EBR, in which eightdifferent rules of inference for EBR are discussed. Then we propose fuzzy logic-based models for these eight different rules of inference that constitute the funda-mentals for all EBR paradigms from a fuzzy logic viewpoint. We argue that theproposed methodology of EBR will facilitate the understanding of EBR and itsapplications to intelligent systems, knowledge management ~KM!, experience man-agement ~EM!, and e-commerce.

In what follows, our attention will focus primarily on fuzzy logic-based treat-ment of EBR. Although our approach might be of significance in areas such ase-commerce, decision processes, business negotiation, agent societies, multi-agent systems ~MAS!, hybrid intelligent systems, and recognition of fraud anddeception, we shall make no attempt in this article to discuss its applications inthese areas. Because this work was motivated when we attempted to develop log-ical EBR and its applications to detecting and recognizing fraud and deception ine-commerce and MAS,9 we will use e-commerce, MAS, and recognition of fraudand deception as scenarios, if required.

The rest of this article is organized as follows: Section 2 examines CBR as akind of EBR. Section 3 looks at EBR with an interesting example. Section 4 reviewsinference rules in EBR from a logical viewpoint. Section 5 examines a fuzzy-logic-based model for each of the eight inference rules for EBR, and Section 6 ends thisarticle with some concluding remarks.

2. CASE-BASED REASONING AS A KIND OF EBR

This section argues that CBR is a kind of EBR, but it is only one of the rea-soning paradigms available in EBR.

CBR is a kind of EBR.3 In other words, CBR is an EBR that relies on usingencapsulated prior experiences as a basis for dealing with similar new situations.a

The CBR system ~CBRS! is an intelligent system based on EBR, which canbe modeled as

CBRS � Case Base � CBR Engine ~1!

where the case base ~CB! is the set of cases, each of which consists of the previousencountered problem and its solution. The CBR engine is the inference mecha-nism for performing CBR.

As we know, “Two cars with similar quality features have similar prices” is apopular experience principle used in the CBR community, which is a summary

aSee http://www.cs.indiana.edu/;davwils/orals.html.

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International Journal of Intelligent Systems DOI 10.1002/int

based on many individual experiences of buying cars. This is a kind of similarity-based reasoning ~SBR!. In other words, SBR is a concrete realization of CBR.Therefore, CBR can be considered as a kind of similarity-based reasoning from alogical viewpoint, that is,10

CBR :� Similarity-based reasoning ~2!

Based on the above discussion, CBRS can be formalized as

CBRS � CB � CBRE ~3!

where CB still denotes the case base. The CBRE is the inference mechanism forperforming SBR instead of general CBR and EBR, and the former can be formal-ized as

P ', Pr Q

�Q '~4!

where P, P ', Q, and Q ' represent compound propositions and P '; P means that Pand P ' are similar.b Q and Q ' are also similar. Expression ~4! is called generalizedmodus ponens. This is one essence of any fuzzy reasoning.12 Strictly speaking, ~4!is one of the most fundamental reasoning rules for performing modus ponens basedon fuzzy logic. The goal of the CBRS is to find Q ' such that ~4! is valid.8 Forexample, let p ' be the problem description of the customer, p ' ; p means that p '

and p are similar, and pr q is a case retrieved from the case base by the CBRS.The solution in the previous case pr q is only a solution candidate to the problemof the customer, because the q is not the solution to p ' but to p, although p ' and pare similar. Therefore the CBRS uses case adaptation to find out q ' such that q andq ' are similar, and q ' is the most satisfactory solution to the problem p ' .10

Typical reasoning in CBR, known as the CBR cycle, consists of ~case! Repar-tition, Retrieve, Reuse, Revise, and Retain,10,13,14 as shown in Figure 1. Each ofthese five components is a complex process. Case base building and repartition,case retrieval, and case adaptation are three main stages in CBR, in which SBRplays an important role.13 Therefore, CBR is a reasoning paradigm, in which SBRdominates each of the main stages. In other words, CBR is a kind of process rea-soning,10 and simulates a kind of EBR.

It should be noted that although a generalization of CBRc is experience-based reasoning, CBR cannot cover all possibilities of EBR. From a logical view-point, CBR can only be considered as one of the reasoning paradigms available inEBR, which will be seen in the following section.

bFor a discussion of similarity, similarity metrics, and similarity measures, please seeRef. 11.

cSee http://experience.univ-lyon1.fr/liris_contribution/main_issues_of_research.htm.

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3. EXPERIENCE-BASED REASONING

In this section we will illustrate EBR with an example and then make somecomments on EBR at a general level.

To understand EBR, let us look at an example.Dr. Peter Hagen is a famous Professor of Business and Commerce at the Uni-

versity of Trickland.d He has participated in many international conferences andvisited many different countries for academic travel. He teaches his students logis-tics using modus ponens and modus tollens,11 and he explains some social phenom-ena using abductive reasoning.15 When he participates in a business negotiationwith his competitor, he likes to use modus ponens with trick and modus tollenswith trick.4 He also likes to perform some investment, in which he likes to useinverse modus ponens.3,4 When asked for investment advice by people he does nottrust, he uses inverse modus ponens with trick and abduction with trick.4

From this example, we can see that:

• Any human professional activities usually involve application of many reasoning para-digms such as abduction, deduction, induction, and reasoning with trick.

• Any person has to perform many different reasoning paradigms in order to cope withdifferent social situations or occasions.

• A person uses a specific reasoning paradigm depending on his experience in differentsocial occasions.

dThis is an invented name.

Figure 1. The R5 model of CBR.13

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Furthermore, experience is all possible past problems and corresponding solu-tions that a human has encountered. Therefore, only one reasoning paradigm likeCBR, which only simulates an experience principle “similar problems have simi-lar solutions,”16,17 is insufficient to model or simulate all experiences or all kindsof EBR, as shown in Figure 2.

Generally speaking, one of the significant contributions of CBR research anddevelopment is that it points out the importance of experience and EBR, and pro-vides some methodologies such as case reuse and case retention, which can beused in experience reuse and experience retention in EBR and EM.

It should be noted that any EBR is based on certain inference rules, just as thebasis for any reasoning paradigm discussed in artificial intelligence ~AI! and math-ematical logic is inference rules. Therefore, it is necessary to discuss inferencerules for EBR in order to improve the understanding of EBR.

4. INFERENCE RULES FOR EXPERIENCE-BASED REASONING

As we know, one of the most important principles of EBR is “divide andconquer”; that is, we first divide a real-world problem so simply that we can con-quer the divided problem using existing reasoning or methods. Based on this idea,we will classify EBR using inference rules from a logical viewpoint and examinethe correspondence between such a classification ~inference rules for EBR! andreal-world problems using an inference rule. In what follows, we review eightinference rules for EBR proposed in Ref. 3, which cover all possible EBRs from alogical viewpoint. These eight different rules of inference constitute the fundamen-tals for all forms of EBR.3,4

From a logical viewpoint, the current AI models and other computational mod-els for problem solving are basically based on the first four inference rules: modusponens ~MP!,18 modus tollens ~MT!,19 abduction,15,20 and modus ponens with trick~MPT!.8,21 Because these four inference rules are well known in AI, computerscience, and fuzzy logic,8,18,19 we will first review them in this article and thenturn to examining the other four inference rules, modus tollens with trick ~MTT!,4

abduction with trick ~AT!,3 inverse modus ponens ~IMP!,3 and inverse modusponens with trick ~IMPT!,4 in some detail. These inference rules will be consid-ered as the four new inference rules of EBR, because they are nontraditional andhave not been studied in mathematics, logic, fuzzy logic, and AI, although they arereally abstractions of some EBR. The following formalization for them is the firstattempt in this direction, to our knowledge.3,4

Figure 2. CBR, EBR, and deception.4



4.1. Modus Ponens and Modus Tollens

Two popular inference rules in mathematical logic, mathematics and AI3 aremodus ponens ~MP! and modus tollens ~MT!.11,18,19 The former has a general form

PPr Q

�Q~5!

where P, Q represent compound propositions. More specifically, ~5! means that ifP is true and P r Q is true, then the conclusion Q is also true. From an EBRviewpoint, ~5! is a formalized summary of experience.3

The general form of MT is as follows:

¬QPr Q

�¬P~6!

where P and Q represent compound propositions. More specifically, ~6! meansthat if Q is false and Pr Q is true, then the conclusion ¬P is also true. From anEBR viewpoint, ~6! is also a formalized summary of experience.

Modus ponens and modus tollens belong to rules of deduction,22 which arereasoning paradigms in mathematical logic, mathematics, and AI. The reasoning~or argument! using them can be considered as valid; that is, no matter what par-ticular statements are substituted for the statement variables in its premises, if theresulting premises are all true, then the conclusion is also true.11

It should be noted that a formal logical system largely consists of two parts:an axiom system and an inference system. The axiom system consists of a set ofaxiom schemes, and the inference system consists of a set of rules of inference.22

The simplest inference system is a singleton, which consists only of modus ponens~or modus tollens!. In other words, modus ponens ~or modus tollens! and an axiomsystem constitutes a formal logical system for deduction.22

Furthermore, a knowledge-based system ~KBS!, which mainly consists of aknowledge base and an inference engine,18,23 can be considered as a computerizedlogical system. The computerized counterpart of the axiom system is the knowl-edge base, and the computerized counterpart of the inference system is the infer-ence engine, as shown in Figure 3.

From Figure 3 we can see that a knowledge base system has a sound theoret-ical foundation. The user interface is also an important part in a KBS. However, itdoes not have a counterpart in the corresponding logical system; that is, the userinterface has no sound theoretical foundation. This is a reason why the user inter-face was not emphasized in traditional KBS ~see Ref. 18, p. 281!. Furthermore, ifwe consider a logical system as a microworld, then any change in the inferencesystem of the logical system will change the microworld into another microworld.Correspondingly, any change of either axioms or inference rules will lead to acritical change of the formal logical system. Because an expert system is an attempt

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to automate a human expert, we can assert that any change of either knowledge orreasoning methods of a human will lead to changes in his/her behavior or action.

4.2. Abduction

The third inference rule for EBR is the rule of abduction, which has drawnincreasing attention in AI and CBR.15,24 The general model of abduction as aninference rule is as follows25–27:

QPr Q

�P~7!

where P and Q represent compound propositions in a general setting.e

From an EBR viewpoint, ~7! is also a formalized summary of experience.Abduction is the term currently used in the AI community for generation of

explanations for a set of events from a given domain theory.27–30 More specifi-cally, abduction is the process of inferring certain facts and/or laws and hypoth-eses that render some sentences plausible, that explain or discover some ~eventuallynew! phenomenon or observation; it is the process of reasoning in which explan-atory hypotheses are formed and evaluated ~Ref. 25, p. 18!. Therefore, abductionis a very useful reasoning paradigm, in particular for reasoning toward explana-tion in ~system! diagnosis15 and analysis in problem solving, and therefore an impor-tant form of EBR.

eFrom now on, we do not mention this fact about P and Q any more when we introduce anew rule of inference.

Figure 3. Relationship between logical systems and KBS.



4.3. Reasoning with Trick

The reasoning with trick was introduced by Sun and Weber21,31 and aims atexamining logic with trick and reasoning with trick.f Its general form as an infer-ence rule is called modus ponens with trick and is represented as7,31

PPr Q

�¬Q~8!

The modus ponens with trick is motivated by the following fact: Basicallyspeaking, all knowledge and experience consists of two parts—mathematicalknowledge and experience and nonmathematical knowledge and experience, asshown in Figure 4. The former constitutes the resource for existing mathematics;inference rules in mathematical logic can be considered the summary or abstrac-tion of mathematical methods for solving problems in mathematics. The latter con-stitutes the resource for existing nonmathematical sciences. Although researchershave always been trying to use approaches provided by existing mathematics andmathematical logic to formalize the concepts in their own domain, there are anenormous number of theories and investigations in nonmathematical sciences thatare at an empirical level and require new logical and mathematical methodologies.Tricky reasoning belongs to this part.

Furthermore, from Figure 4, we can see that mathematics can be consideredone part of human knowledge and experience. Mathematics has heavily affectedmathematical logic, CBR, and AI. The rest after the abstraction are nonmathemat-ical knowledge and experience. We believe that the latter leads to experience-based reasoning. Mathematical logic is a formal meta-mathematics from atheoretical viewpoint; it consists of all possible reasoning paradigms and infer-ence rules occurring in mathematics for problem solving. However, from a funda-mental viewpoint, only two inference rules, like the atoms of Boolean algebra,11

fWe use the term trick to cover several reasoning approaches including deception ~such as“play a trick on”! or heuristic reasoning ~“the tricks of the trade”!.

Figure 4. Mathematics, logic, and EBR.

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have been included, that is, modus ponens and modus tollens. At least six othermentioned inference rules have not been included in mathematical logic.22 This isthe reason why we believe that EBR is an abstraction of nonmathematical knowl-edge and experience.

Multiagent systems ~MAS! are an exciting research field in AI.9 Coopera-tion, coordination, communication, and negotiation play an important role in MAS.8

However, from the viewpoint of human society, social behaviors can be dividedinto two categories: rational behaviors and irrational behaviors. The former includesautonomy and trust and the latter includes deception and lies. Therefore, investi-gation into deception and lies among intelligent agents in MAS, trick-based logic,31

and trick-based reasoning21 in MAS has the same importance as research into auton-omy and trust in MAS,4 and will therefore help not only improve better understand-ing of human intelligence but also intelligent EBR systems and MAS.

4.4. Modus Tollens with Trick

We have discussed modus tollens in Section 4.1. Now we examine its “dual”form, named modus tollens with trick ~MTT!. Its general form is3,4

¬QPr Q

�P~9!

Example 1: Modus tollens with trick. We have the following knowledge in theknowledge base:

~1! If Peter is human, then Peter is mortal.~2! Peter is immortal.

What we wish is to prove is “Peter is human.” To do so, let

• Pr Q: If Peter is human, then Peter is mortal.• P: Peter is human.• Q: Peter is mortal.

Therefore, we have P: Peter is human, based on modus tollens with trick ~9!and the knowledge in the knowledge base ~note that ¬Q: Peter is not mortal!.

From this example we can see that modus tollens with trick is a kind of EBR.Theoretically speaking, it is also a variant of modus ponens with trick ~8!,

because in the traditional logic, we have ¬Q, Pr Q ] ¬P and ¬¬P? P. How-ever, in the nontraditional logic, for example, in fuzzy logic,19 Pr Q? ¬Qr¬P and ¬¬P ? P are normally invalid. In particular in EBR, they both can beinvalid; therefore modus tollens with trick is still meaningful in order to examinethe basic rule of inference in EBR.



4.5. Abduction with Trick

As discussed in Section 4.2, abduction is an important reasoning paradigm inEBR. Its “dual” form is abduction with trick ~AT!, which is also the summary of akind of EBRs. The general form of abduction with trick, as a basic rule of infer-ence, is as follows:

QPr Q

�¬P~10!

The difference between abduction with trick and abduction is “with trick.”This is because the reasoning performer tries to use the trick of “make a feint tothe east and attack in the west”21 ; that is, he gets ¬P rather than P in the abductionas a rule of inference. This also verifies that difference is a necessary condition forperforming trick or deception.4

Furthermore, just as abduction has been used in system diagnosis or medicaldiagnosis,15 abduction with trick can be also used in these fields. For example,abduction can be used to explain that the symptoms of the patients result fromspecific diseases, whereas abduction with trick can be used to exclude some pos-sibilities of the diseases of the patient ~see Section 5.6!. Therefore, abduction withtrick is an important complementary part for performing system diagnosis andmedical diagnosis based on abduction.

4.6. Inverse Modus Ponens

Inverse modus ponens is also a rule of inference in EBR. The general form ofinverse modus ponens ~IMP! is as follows4:

¬PPr Q

�¬Q~11!

The inverse in the definition is motivated by the fact that the inverse is definedin mathematical logic: “if ¬p, then ¬q,” provided that if p then q is given.11 Basedon this fact, the inverse of Pr Q is ¬Pr ¬Q, and then from ¬P, ¬Pr ¬Q wehave ¬Q using modus ponens. Therefore the definition of inverse modus ponens isreasonable. Because Pr Q and ¬Pr ¬Q are not logically equivalent, the argu-ment based on ~11! is not valid in mathematical logic and mathematics.

To our knowledge, EBR based on inverse modus ponens is a kind of commonsense reasoning,3 because there are many cases that follow inverse modus ponens.For example, if Robert has enough money, then Robert will fly to Beijing. NowRobert does not have sufficient money; then we can conclude that Robert will notfly to Beijing.

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4.7. Inverse Modus Ponens with Trick

The final inference rule for EBR is inverse modus ponens with trick ~IMPT!.Its general form is as follows3,4:

¬PPr Q

�Q~12!

The difference between inverse modus ponens with trick and inverse modusponens is again “with trick”; this is because the reasoning performer tries to usethe trick of “make a feint to the east and attack in the west”; that is, he gets Qrather than ¬Q in the inverse modus ponens. This once again verifies that differ-ence is a necessary condition for performing trick or deception as mentioned inSection 4.3.

4.8. Summary of Inference Rules for EBR

Table I summarizes the proposed eight basic rules of inference with respectto EBR. It should be noted that some general forms in the table such as inversemodus ponens ~this concept is first introduced in Ref. 3! have received attentionfrom some researchers ~Ref. 11, p. 36!. However, the researchers consider thisinference rule as the source of fallacies in the reasoning, whereas we argue thatthey are all basic inference rules for EBR. They should also be an important partin experience management.

So far, we reviewed eight different rules of inference for EBR ~see Table I!from a classic logical viewpoint. All these inference rules are the abstraction andsummary of experience or EBR in real-world problems.

It should be noted that fuzzy logic has extended traditional logic and foundmany significant applications.19 Therefore, the next section will examine the pro-posed approach based on fuzzy logic and fuzzy set theory.

5. EXPERIENCE-BASED REASONING WITH FUZZY REASONING

In this section we will examine these eight different rules of inference forEBR from a fuzzy logic viewpoint. Throughout this section we assume that P and

Table I. Experience-based reasoning: Rules of inference.

Modusponens

Modusponens

with trick

Inversemodusponens

with trick

Inversemodusponens

Modustollens

Modustollens

with trick

Abductiveinference

rule

Abductionwithtrick

PPr Q

�Q

PPr Q

�¬Q

¬PPr Q

�Q

¬PPr Q

�¬Q

¬QPr Q

�¬P

¬QPr Q

�P

QPr Q

�P

QPr Q

�¬P



Q represent fuzzy propositions. Let F0~x!, F01~x, y!, and F1~ y!, x � X, y � Y befuzzy relations in X, X � Y, and Y, respectively, which are fuzzy restrictions on x,~x, y!, and y, respectively. X and Y are two ordinary nonempty sets. Let P, Q,P ' , and Q ' be fuzzy propositions and correspond to F0~x!, F1~ y!, F0

' ~x!, F1' ~ y!,

respectively, and PrQ corresponds to F01~x, y!. � is a fuzzy composition operation.

5.1. Fuzzy Reasoning: Fuzzy Modus Ponens

Fuzzy logic is an extension of mathematical logic based on fuzzy set theoryand infinite multivalued logic systems.19 Fuzzy logic has evolved into an importantresearch and development field in many disciplines such as mathematics, logic,artificial intelligence, and philosophy since 1965.32–34

Fuzzy reasoning in fuzzy logic is basically generalized from deductive rea-soning in traditional logic with the exception of its computational process.10,30 Inother words, fuzzy reasoning is a mixed symbolic/numeric approach to both deduc-tive reasoning and rule-based reasoning.8 Its reasoning is based on the generalizedmodus ponens19:

Pr QP '

�Q '~13!

where P and Q represent fuzzy propositions and P ' is approximate to P; that is,P ';P. In this work we call it fuzzy modus ponens ~FMP! in order to provide EBRwith a unified treatment based on fuzzy logic. Expression ~13! is also commonlyrepresented in the following form in fuzzy logic19:

If x is P Then y is Qx is P '

�y is Q '~14!

For instance,

IF a tomato is red THEN the tomato is ripeThis tomato is very red

Conclusion: This tomato is very ripe~15!

In the fuzzy setting, the above fuzzy reasoning can be performed using thefollowing compositional rule of inference introduced by Zadeh19:

F1' ~ y! � F0

' ~x! � F01~x, y! ~16!

It should be noted that the fuzzy modus ponens will reduce to classical modusponens when P � P ' and Q � Q ' , and P and Q degenerate into a classical com-pound propositions. Furthermore, modus ponens and fuzzy modus ponens are closelyrelated to the forward data-driven inference that is particularly useful in fuzzylogic control.35

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5.2. Fuzzy Modus Tollens

Fuzzy modus tollens ~FMT! has also been investigated in fuzzy logic.19,36 Itsgeneral form is

¬Q '

Pr Q

�¬P '~17!

Expression ~17! is also commonly represented in the following form in fuzzylogic35:

If x is P Then y is Qx is ¬Q '

�y is ¬P '~18!

Expression ~17! will degenerate to modus tollens ~see Section 4.1! whenP � P ' and Q � Q ' , and P and Q degenerate into a classical compound propositions.

An example of fuzzy modus tollens is

IF Peter studies hard THEN Peter will get a good examination resultPeter does not get a very good examination result

� Peter does not study very hard~19!

The fuzzy reasoning based on fuzzy modus tollens can be computed using thefollowing formula, based on the above discussion:

F0' ~x! � 1 � F01~x, y! � ~1 � F1

' ~ y!! ~20!

Further, let x � X � $x1, x2, . . . , xn %, y � Y � $ y1, y2, . . . , ym %, EF01 � ~m~xi , yj !!,EF0'� ~m~x1!,m~x2 !, . . . ,m~xn !!, and EF1

'� ~m~ y1!,m~ y2 !, . . . ,m~ ym !!; then usingthe compositional rule of inference for fuzzy conditional inference,19 we have analternative form of Equation ~20! as followsg:

�m~x1!

J

m~xn !� � 1 � �

m~x1, y1! J m~x1, ym !

J J J

m~xn , y1! J m~xn , ym !� �

1 � m~ y1!

J

1 � m~ ym !� ~21!

It should be noted that both modus tollens and fuzzy modus tollens are closelyrelated to the backward goal-driven inference that is commonly used in expertsystems, especially in the realm of medical diagnosis.35

gSuch a computational form can be provided, in a similar way, to each of the eight fuzzyinference rules for EBR discussed in this article.



5.3. Fuzzy Abductive Reasoning

Fuzzy abductive reasoning has drawn attention in medical diagnosis since the1990s. For example, Miyata et al.36 study fuzzy abductive inference on cause-and-effect relationships. Yamada and Mukaidono37 proposed a fuzzy abductionbased on multi-valued logic. To keep a consistent style in our work, we still use afuzzy logic-based approach to fuzzy abductive reasoning in what follows. Webelieve that the general investigation into fuzzy abduction can cover the work ofMiyata et al.38

The general form of fuzzy abductive reasoning is as follows:

Q '

Pr Q

�P '~22!

Expression ~22! can be represented in the following form in a context of fuzzylogic:

If x is P Then y is Qy is Q '

�x is P '~23!

For instance,

IF John gets fever THEN John will be dizzyJohn is a little dizzy

Conclusion: John gets a light fever~24!

The fuzzy reasoning based on fuzzy abduction can be computed using thefollowing formula, based on the above discussion:

F0' ~x! � F01~x, y! � F1

' ~ y! ~25!

More specifically, let D � $d1, d2, . . . , dn % be the set of diseases, and S �$s1, s2, . . . , sm % the set of symptoms.36 According to medical experience, disease di

will lead to symptom sj with the certainty membership m ij~di , sj !; that is, fuzzyrelationships between diseases D and symptom S are EF~D, S! � ~m~di , sj !!, i �1, 2, . . . , n; j � 1, 2, . . . , m. If a patient is observed to have a fuzzy symptom set,DSp � ~m~sj !!

T , where m~sj !h is the certainty membership of the observed symp-

tom belonging to sj . Therefore, according to Equation ~25!, the fuzzy disease setof this patient is

~m~d1!, . . . ,m~dn !!T � �

m~d1, s1! J m~d1, sm !

J J J

m~dn , s1! J m~dn , sm !� �m~s1!

J

m~sm !� ~26!

hm ij~di , sj ! can be considered as the confirmability of sj for di , and m~sj ! expresses theintensity of symptom sj ; for details see Ref. 19, pp. 185–188.

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where m~di ! is the certainty membership of the disease of the patient belongingto di .

It should be noted that although fuzzy abductive inference has been appliedin medical diagnosis,27,36,38 its application success basically results from thecomputational methods that are based on Equation ~25! or ~26!. Because the rela-tionship between diseases and manifestations are many-to-many, and these rela-tionships constitute the basis for explanation of symptoms, it is of significance tobuild a complete relationship system in a concrete domain in order to use fuzzyabductive reasoning effectively.

5.4. Fuzzy Modus Ponens with Trick

As mentioned, differences of understanding, knowledge, and experience arethe source of tricks or deception.4 The fuzzy, incomplete features of experienceand knowledge are a further important source leading to trick and deception. Thisis also the reason why experience is closely connected to trick or deception incertain situations: more experienced, more tricks.

The general form of fuzzy reasoning with trick ~based on modus ponens!,called fuzzy modus ponens with trick ~FMPT!, is as follows:

P '

Pr Q

�¬Q '~27!

Expression ~27! can be represented in the following form in the context offuzzy logic:

If x is P Then y is Qx is P '

�y is ¬Q '~28!

For instance,

IF Bill is the smartest THEN Bill will work at MedisoftBill is very smart

Conclusion: Bill does not like to work at Medisoft~29!

The fuzzy modus ponens with trick can be computed using the following for-mula, based on the above discussion:

F1' ~ y! � 1 � F0

' ~x! � F01~x, y! ~30!

Basically speaking, fuzzy reasoning with trick is the fuzzification of “make afeint to the east and attack in the west.” More specifically, the commander likes toperform the deception: “make a feint to the east and perhaps attack in the south-west” according to the changing situation in the battlefield.31 This fuzzy reasoningwith trick often happens in business negotiations and war commands. There aremany books and research studies on this reasoning paradigm. However, few



attempts have been made to formalize such reasoning from a logical or computa-tional viewpoint so that understanding of fuzzy reasoning with trick is still at anempirical level. If fuzzy logic and fuzzy reasoning have provided a powerful meth-odology to deal with incomplete, imprecise, and fuzzy knowledge and its process-ing, then a big issue for fuzzy logic and fuzzy reasoning is how to treat fuzzyreasoning with trick.i

5.5. Fuzzy Modus Tollens with Trick

A direct development from fuzzy modus ponens with trick is fuzzy modus tol-lens with trick ~FMTT!. Although fuzzy modus tollens has drawn attention in thefuzzy logic community,19,36 nobody has studied this new kind reasoning para-digm. However, the latter is also an important part in EBR. In what follows, wewill go into it in some detail.

The general form of fuzzy modus tollens with trick is

¬Q '

Pr Q

�P '~31!

Theoretically speaking, it is a variant of fuzzy modus ponens with trick ~27!,because, using fuzzy modus ponens with trick, we have ¬Q ', P r Q ] ¬P ' .However, this variant can only be understood in a fuzzy setting. For example,if we assume the membership of P, m~P !� 1, and m~P ' !� 0.4, then ¬m~P ' !�1 � m~P ' ! � 1 � 0.4 � 0.6. In this fuzzy microworld, both P ' and ¬P ' are theintermediate states between P and ¬P. Therefore, such an intermediate but uncer-tain state is the space for performing a trick or deception. It is very difficult foranyone to perform a trick or deception in a pure two-valued world ~true or false!.Even though one could perform tricks or deceptions in this world, it is easy forothers to recognize such tricks. Therefore, it is significant to examine either fuzzymodus ponens with trick or fuzzy modus tollens with trick in a fuzzy setting, whichis a closer approximation to the tricks and deceptions existing in human society.

Expression ~31! can be represented in the following form in fuzzy logic35:

If x is P Then y is Qx is ¬Q '

�y is P '~32!

For instance,

IF Bill is the smartest THEN Bill will work at MedisoftBill will not work at Medisoft

Conclusion: Bill is very smart~33!

iIt is easy for a reader to provide a concrete example for fuzzy reasoning with trick in aspecial setting. For details, see Refs. 19 and 36.

882 SUN AND FINNIE


From a logical viewpoint, this reasoning means that we prefer to accept a fuzzy orapproximate statement to the premise in the fuzzy conditional proposition ~Bill isvery smart!, if we do not accept the conclusion resulting from performing fuzzymodus tollens.

The fuzzy modus tollens with trick can be also computed using the followingformula, based on the above discussion:

F0' ~x! � F01~x, y! � ~1 � F1

' ~ y!! ~34!

5.6. Fuzzy Abduction with Trick

Fuzzy abduction with trick ~FAT! has not drawn any attention in either med-ical diagnosis or system analysis, although fuzzy abduction has been studied andapplied in these fields. In fact, it is also an important kind of EBR toward theexplanation of any symptoms in clinical practice or system diagnosis, which willbe seen later.

The general form of fuzzy abduction with trick is as follows:

Q '

Pr Q

�¬P '~35!

Theoretically speaking, fuzzy abduction with trick is a variant of fuzzy abduc-tion, in particular, when an agent A in a MAS9 may guess that another agent B inthe MAS performs fuzzy abduction based on ~22!, whereas agent B actually per-forms fuzzy abduction with trick based on ~35!. Herewith agent A and agent B willsuffer a trust crisis. How to resolve such a trust crisis is an important issue forMAS and web-based systems.

Expression ~35! can be represented in the following form in a context of fuzzylogic:

If x is P Then y is Qx is Q '

� y is ¬P '~36!

For instance,

IF John gets fever THEN John will be dizzyJohn is a little dizzy

Conclusion: John does not get a light fever~37!

Every adult has had a similar experience in a clinic: the doctor gives a wrongexplanation for the symptoms. The wrong explanation leads to wrong treatment,because they sometimes do not really use fuzzy abduction.

More formally, if we assume D is the set of diseases and S is the set of symp-toms ~see Section 5.3!, then for a patient c in a clinical practice, his symptoms



~e.g., SARS’s symptoms! are a subset of S, Sc , and his diseases are a subset of D,Dc . Therefore,

Dc � D and Sc � S ~38!

The available medical experience can be expressed as a set of ~fuzzy! rules,E; that is,

E � $ f 6 f � IF d Then s, d � D, s � S % ~39!

The possible experience set for this patient is Ec � $ f 6 f � IF d Then s, d �Dc , s � Sc % . However, a doctor normally can not use such a medical experience-based system and performs experience-based reasoning by himself. In this case,he uses any experience f1 � E � Ec and he performs a fuzzy abduction with trick.

The fuzzy reasoning based on fuzzy abduction with trick can be computedusing the following formula, based on the above discussion:

F0' ~x! � 1 � F01~x, y! �F1

' ~ y! ~40!

Similar to what was proposed in Section 5.3, Equation ~40! can be replaced ina more concrete form as follows: let D � $d1, d2, . . . , dn % be the set of diseases andS � $s1, s2, . . . , sm % the set of symptoms.19,36 According to medical experience,disease di will lead to symptom sj with certainty membership m ij~di , sj !; that is,fuzzy relationship between diseases D and symptom S are EF~D, S!� ~m~di , sj !!,i �1, 2, . . . , n; j �1, 2, . . . , m. If a patient is observed to have a fuzzy symptom set,DSp � ~m~sj !!

T , where m~sj ! is the certainty membership of the observed symptombelonging to sj . Therefore, according to Equation ~40!, the fuzzy disease set of thispatient is

~m~d1!, . . . ,m~dn !!T � 1 � �

m~d1, s1! J m~d1, sm !

J J J

m~dn , s1! J m~dn , sm !� �m~s1!

J

m~sm !� ~41!

where m~di ! is the certainty membership of the disease of the patient belongingto di .

It should be noted that fuzzy abduction with trick has still not been applied inmedical diagnosis. Its research and development will help to understand why manypatients suffer misdiagnosis and incorrect treatment. In particular, it can also beused to exclude some possibilities of certain diseases of the patient; that is, for acertain k � $1, 2, . . . , n% , ifm~dk! is approximate to 0, the disease dk can be excludedfrom the possible diseases from which the patient suffers. This approach is illus-trated by the following example, which is borrowed from an example given inRef. 36 and simplified.

Example 2: Fuzzy abduction with trick. Assume the set of diseases D � $d1, d2, d3%,S � $s1, s2, s3, s4 % . The fuzzy confirmability of sj for di , EF~D, S! � ~m~di , sj !! isgiven as a fuzzy relation, listed in Table II. The observed symptoms are denoted as

884 SUN AND FINNIE


a fuzzy set DS in S, and the corresponding certainty membership of the observedsymptoms belonging to sj ,m~sj !, are listed as a vector ~m~s1!,m~s2!,m~s3!,m~s4!�~0.6,0.1,0.9,0.3!!. Using Equation ~41!, we calculate ~m~d1!,m~d2 !,m~d3 !! ~basedon min-max operation19 ! and have

�m~d1!

m~d2 !

m~d3 !� � 1 � �

1.0 0.8 0 0.6

0.6 0 1.0 0

0.8 0.6 0.7 0.6� �

0.6

0.1

0.9

0.3� � 1 � �

0.6

0.9

0.7� � �

0.4

0.1

0.3� ~42!

Because m~d2 ! is 0.1, which is approximate to 0, the disease d2 can be excludedfrom the possible diseases from which the patient suffers.

5.7. Fuzzy Inverse Modus Ponens

Fuzzy inverse modus ponens ~FIMP! is another rule of inference for EBR. Itsgeneral form is as follows:

¬P '

Pr Q

�¬Q '~43!

Expression ~43! can be represented in the following form in the context of fuzzylogic:

If x is P Then y is Qx is ¬P '

� y is ¬Q '~44!

Example 3: Fuzzy inverse modus ponens. We have the following knowledge inthe knowledge base:

• If the quarter profit is increasing, then Klaus invests in the Project ANF.• The quarter profit is marginally decreased.

What we wish is to prove “Klaus does not intend to invest in the Project ANF.” Tothis end, let P r Q: if the quarter profit is increasing, then Klaus invests in theProject ANF; P: the quarter profit is increasing. Therefore, we have ¬Q ' : Klaus

Table II. The fuzzy confirmability of sj for di .

m~di , sj ! s1 s2 s3 s4

d1 1.0 0.8 0 0.6d2 0.6 0 1.0 0d3 0.8 0.6 0.7 0.6



does not intend to invest in the Project ANF based on ~43! and the knowledge inthe knowledge base ~note that ¬P ' : the quarter profit is marginally decreased!.

In the conclusion of this example, “Klaus does not intend to invest in theProject ANF ” means that Klaus has not yet decided to invest in the project ANF,which is an intermediate state between “Klaus invests in the Project ANF ” and“Klaus does not invest in the Project ANF.”

The fuzzy reasoning based on fuzzy inverse modus ponens can be computedusing the following formula, based on the above discussion:

F1' ~ y! � 1 � ~1 � F0

' ~x!! � F01~x, y! ~45!

5.8. Fuzzy Inverse Modus Ponens with Trick

Fuzzy inverse modus ponens with trick ~FIMPT! is the last rule of inferencefor EBR. Its general form is as follows:

¬P '

Pr Q

�Q '~46!

Expression ~46! can be represented in the following form in the context of fuzzylogic:

If x is P Then y is Qx is ¬P '

� y is Q '~47!

Example 4: Fuzzy inverse modus ponens with trick. We have the following knowl-edge in the knowledge base:

• If the quarter profit is increasing, then Klaus invests in the Project ANA.• The quarter profit is not increasing much.

What we wish is to prove “Klaus intends to invest in the Project ANA.” To thisend, let Pr Q: if the quarter profit is increasing, then Klaus invests in the ProjectANA; P: the quarter profit is increasing. Therefore, we have Q ' : Klaus intends toinvest in the Project ANA based on ~46! and the knowledge in the knowledge base~note that ¬P ' : the quarter profit is not increasing much!. In the conclusion of thisexample, “Klaus intends to invest in the Project ANA” is approximate to “Klausinvests in the Project ANA.”

The fuzzy reasoning based on fuzzy inverse modus ponens with trick can becomputed using the following formula, based on the above discussion:

F1' ~ y! � ~1 � F0

' ~x!! � F01~x, y! ~48!

It should be noted that fuzzy inverse modus ponens and fuzzy inverse modus ponenswith trick have not drawn any attention in either fuzzy logic or computer science.We believe that the research and development of fuzzy inverse modus ponens can

886 SUN AND FINNIE


improve our understanding of experience-based reasoning in a fuzzy setting,because it is common in human society.

5.9. Summary

Table III summarizes the proposed eight fuzzy inference rules for experience-based reasoning ~including the corresponding form!. It should be noted that somegeneral forms in the table such as fuzzy modus ponens and fuzzy modus tollenshave received some attention from researchers ~see Refs. 36 and 19, p. 36!, whereasthe rest of them have not been studied in fuzzy logic and computer science, althoughthey are all the summarization of EBRs. We argue that they are all the basic rulesof inference for EBR. They should also be an important part in experience man-agement and intelligent systems.

6. CONCLUDING REMARKS

This article first reviewed the logical approach to EBR, in which eight differ-ent rules of inference for EBR were studied. Then we proposed a fuzzy logic-based model to each of these eight different rules of inference that constitute thefundamentals for all EBR paradigms and therefore will be a theoretical foundationfor EBR. The proposed approach will facilitate research and development of EBRsystems, intelligent systems, and knowledge/experience management.

Experience management ~EM!16 is drawing increasing attention ine-commerce, computer science, information systems, and knowledge manage-ment ~KM!,39 which has become one of the latest hot topics in the business world.16

EM is more useful than KM, because although every one can have much knowl-edge, only the experience of experts is invaluable. Therefore, EM can facilitatespreading valuable experience, promoting the transition from experience to knowl-edge, and facilitate KM. In fact, the relationship between experience and knowl-edge is the basis for the relationship between EM and KM. Their correspondenceto intelligent systems is experience-based systems ~EBS! such as CBR systems~CBRS! and knowledge-based systems ~KBS!, respectively. Furthermore, from a

Table III. Fuzzy rules of inference for experience-based reasoning.

Modusponens

Modusponens

with trick

Inversemodusponens

with trick

Inversemodusponens

Modustollens

Modustollens

with trick

Abductiveinference

rule

Abductionwithtrick

Traditionalform

PPr Q

�Q

PPr Q

�¬Q

¬PPr Q

�Q

¬PPr Q

�¬Q

¬QPr Q

�¬P

¬QPr Q

�P

QPr Q

�P

QPr Q

�¬P

Fuzzyform

P '

Pr Q

�Q '

P '

Pr Q

�¬Q '

¬P '

Pr Q

�Q '

¬P '

Pr Q

�¬Q '

¬Q '

Pr Q

�¬P '

¬Q '

Pr Q '

�P '

Q '

Pr Q

�P '

Q '

Pr Q

�¬P '



logical viewpoint, experience management is based on experience-based reason-ing. Therefore, we will apply the proposed approach and inference rules for EBRto EM in future work.

Similarity-based reasoning is an important operational form for performingEBR.10 It is an important “bridge” connecting CBR and EBR,8 because “similarcars have similar prices” is also a very popular experience principle. Therefore wewill apply similarity-based reasoning to examine EBR and its eight inference rulesin future work.

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a fuzzy logic approach to experience-based reasoning

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