A Fuzzy Logic Approach to Experience-Based

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A Fuzzy Logic Approach to Experience-Based Reasoning Zhaohao Sun, Gavin Finnie Faculty of Information Technology, Bond University {zsun, gfinnie}@staff.bond.edu.au

Abstract: Experience-based reasoning (EBR) is a reasoning paradigm used in almost every human activity such as business, military missions, and teaching activities since early human history. However, EBR has not been seriously studied from either a logical or mathematical viewpoint, although casebased reasoning (CBR) researchers have, to some extent, confused CBR with EBR. This paper will attempt to fill this gap by providing a unified fuzzy logic-based treatment of EBR. More specifically, this paper first reviews the logical approach to EBR, in which eight different rules of inference for EBR are discussed. Then the paper proposes fuzzy logic-based models to these eight different rules of inference which constitute the fundamentals for all EBR paradigms from a fuzzy logic viewpoint, and therefore will form a theoretical foundation for EBR. The proposed approach will facilitate research and development of not only EBR but also knowledge management and experience management.

Keywords: : Experience-based reasoning (EBR), experience management (EM), casebased reasoning (CBR), fuzzy logic, fuzzy reasoning, abduction

1. Introduction Experience-based Reasoning (EBR) is a widely used reasoning paradigm based on logical arguments [7]. For example, EBR has been used in help desk systemsa to adapt to new business situations by "learning" from experience, tailoring a help desk to effectively maintain

a. see http://www.phd.com

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critical business systemsa. However there appear to be no theoretical research works for EBR, although there are a lot of empirical works on EBR mainly in the business and commerce fields. As we know, experience is an important asset for a domain expert. However, how to formalize experience is still a big issue. Further, there is no fundamental research to investigate the logical or mathematical foundation of EBR [57]. In the context of CBR, EBR is a model of human decision making and problem solving (Riesbeck & Shank 1989) [44]. However, many CBR researchers have confused EBR with CBR, and they believe that CBR is EBR. For example, Stroulia et al. [44] argued that in EBR, new problems are solved by retrieving and adapting the solutions to similar problems encountered in the past, which should be considered as CBR rather than EBR [49]. This paper will attempt to fill the above gap by providing a unified fuzzy logic-based treatment of EBR, based on our previous work on logical treatment of EBR [57]. More specifically, this paper first reviews the logical approach to EBR, in which eight different rules of inference for EBR are discussed. Then the paper proposes fuzzy logic-based models for these eight different rules of inference which constitute the fundamentals for all EBR paradigms from a fuzzy logic viewpoint. We argue that the proposed methodology of EBR will facilitate the understanding of EBR and its application to knowledge management (KM) and experience management (EM). The rest of this paper is organized as follows: Section 2 examines CBR as a kind of EBR. Section 3 looks at EBR with an interesting example. Section 4 reviews inference rules in EBR

a. see http://www.hallogram.com/service/

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from a logical viewpoint. Section 5 examines fuzzy logic based model for each of the eight inference rules for EBR, and Section 6 ends this paper 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 reasoning paradigms available in EBR. Case-based reasoning (CBR) is a kind of experience-based reasoning (EBR) [57]. In other words, CBR is an EBR that relies on using encapsulated prior experiences as a basis for dealing with similar new situationsa, briefly: CBR: = Experience-based reasoning

(1)

Therefore, the CBR system (CBRS) is an intelligent system based on EBR, which can be modelled as: CBRS = Case Base + CBR Engine

(2)

where the case base (CB) is the set of cases, each of which consists of the previous encountered problem and its solution. The CBR engine is the inference mechanism for performing EBR. As we know, “Two cars with similar quality features have similar prices” is a popular experience principle, which is a summary 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 EBR. Therefore, CBR can be considered as a kind of similarity-based reasoning from a logical viewpoint, that is:

a. See http://www.cs.indiana.edu/~davwils/orals.html

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CBR: = Similarity-based reasoning

(3)

Based on the above discussion, CBR system (CBRS) can be formalized as: CBRS = CB + CBRE

(4)

where CB still denotes the case base. The CBRE is the inference mechanism for performing SBR instead of general EBR, and the former can be formalized as: P' , P → Q----------------------∴ Q'

(5)

where P , P' , Q , and Q' represent compound propositions, P' ∼ P means that P and P' are similara. Q and Q' are also similar. (5) is called generalized modus ponens. This is one essence of any fuzzy reasoning [21]. Strictly speaking, (5) is one of the reasoning rules for performing modus ponens based on fuzzy logic. The goal of the CBRS is to find Q' such that (5) is valid. For example, let p' be the problem description of the customer, p' ∼ p means that p' and p are similar, and p → q is a case retrieved from the case base by the CBRS. The solution in the previous case p → q is only a solution candidate to the problem of the customer, because the q is not the solution to p' but to p, although p' and p are similar. Therefore the CBRS uses case adaptation to find out q' such that q and q' are similar, and q' is the most satisfactory solution to the problem p' [20]. Typical reasoning in CBR, known as the CBR cycle, consists of (case) Repartition, Retrieve, Reuse, Revise and Retain [19][20][62], as shown in Fig. 1. Each of these five components is a complex process. Case base building (repartition), case retrieval and case adaptation are three main stages in CBR, in which SBR plays an important role [19]. Therefore, CBR is a reasoning paradigm, in which SBR dominates each of the main stages. In other words, CBR is a kind of process reasoning, and simulates a kind of EBR.

a. For the discussion of similarity, similarity metrics and similarity measures, please see [8].

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It should be noted that although a generalization of CBRa is experience

retrieved case

problem

ie Retr

Reu se

ve

based reasoning, CBR can not cover all learned case

ise Rev

Retain

Repartition

possibilities of EBR. From a logical

case base

viewpoint, CBR can only considered as one of the reasoning paradigms available in EBR, which will be seen in the

solved case

revised case

Ws

Wp

following sections. Fig. 1. The

5

R model of CBR [19]

3. Experience-based Reasoning In this section we will illustrate experience-based reasoning (EBR) with an example, and then make some comments on EBR at a general level. In order to understand EBR, let us look at an example: Peter Hagen is a famous Professor of Business and Commerce at the University of Tricklandb. He has participated in many international conferences and visited many different countries for academic travel. He teaches his student logistics using modus ponens and modus tollens [17], while he explains some social phenomena using abductive reasoning [58]. When he participates in a business negotiation with his competition, he likes to use modus ponens with trick and modus tollens with trick [57]. He also likes to perform some investment, in which he likes to use inverse modus ponens [57]. When asked for investment advice by people he does not trust, he uses inverse modus ponens with trick and abduction with trick. a. see http://experience.univ-lyon1.fr/liris_contribution/main_issues_of_research.htm b.which is not a real name.

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From this example, we can see that: • Any human professional activities usually involve application of many reasoning paradigms such as abduction, deduction, induction and reasoning with trick • Any person has to perform many different reasoning paradigms in order to cope with different social situations or occasions • A person uses a specific reasoning paradigm depending on his experience in different social occasions. Furthermore, experience is all possible past problems and corresponding solutions that a human has encountered. Therefore only one reasoning paradigm is insufficient to model or simulate experience and this is one reason why expert systems have not reached the goal of researchers. In what follows, we will propose a unified reasoning model which integrates all possible reasoning paradigms available for problem solving. These ideas can be extended to a general experience principle; that is, we first should divide a complex problem into subproblems in order to cope with it using simple reasoning paradigms. This principle leads to the following eight different inference rules for EBR.

4. Inference Rules for Experience Based Reasoning As we know, one of the most important principles of EBR is “divide and conquer”; that is, we first divide a real world problem so simply that we can conquer the divided problem using existing reasoning or methods. Based on this idea, we will classify EBR using inference rules from a logical viewpoint and examine the correspondence between such a classification (inference rules for EBR) and real world problems using an inference rule. In what follows, we review eight inference rules for EBR proposed in [57], which cover all possible EBRs from a

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logical viewpoint. These eight different rules of inference, constitute the fundamentals for all forms of EBR [57]. 4.1 Modus Ponens and Modus Tollens Two popular inference rules in mathematical logic, mathematics and artificial intelligence [57] are modus ponens (MP) and modus tollens (MT) [17][37][65]. The former has a general form: P P →Q ---------------∴Q

(6)

where P , Q represent compound propositions. More specifically, (6) means that if P is true, and P → Q is true, then the conclusion Q is also true. From an EBR viewpoint, (6) is a formalized summary of experience [57]. The general form of modus tollens (MT) is as follows: ¬Q P →Q ---------------∴¬ P

(7)

where P and Q represent compound propositions. More specifically, (7) means that if Q is false, and P → Q is true, then the conclusion ¬P is also true. From an EBR viewpoint, (7) is also a formalized summary of experience. Modus ponens (MP) and modus tollens (MT) belong to rules of deduction [41], which are reasoning paradigms in mathematical logic, mathematics and artificial intelligence (AI). The reasoning (or argument) using them can be considered as valid; that is, no matter what

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particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true [17]. 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 of axiom schemes, while the inference system consists of a set of rules of inference [41]. 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 axiom system consists a formal logical system for deduction [41]. Furthermore, a knowledge base system (KBS), which mainly consists of a knowledge base and an inference engine [40][37], can be considered as a computerized logical system. The computerized counterpart of the axiom system is the knowledge base, while the computerized counterpart of the inference system is the inference engine, as shown in Fig. 2. Knowledge-based system

Knowledge base

Inference engine

An axiom system

An inference system A formal logical system

Fig. 2. Relationship between logical systems and ES

From Fig. 2 we can see that a knowledge base system has a sound theoretical foundation. The user interface is also an important part in a KBS. However, it does not have a counterpart in the corresponding logical system; that is, the user interface has no sound theoretical foundation. This is a reason why the user interface was not emphasized in traditional KBS (see

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[37], p.281). Furthermore, if we consider a logical system as a micro-world, then any change in the inference system of the logical system will change the micro-world into another microworld. 4.2 Abduction The third inference rule for EBR is the rule of abduction which has drawn increasing attention in AI and CBR [58][50]. The general model of abduction as a rule of inference is as follows [39][51][58]: Q P →Q ---------------∴P

(8)

where P and Q represent compound propositions in a general setting.a From an EBR viewpoint, (8) 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 [8][52]. More specifically, abduction is the process of inferring certain facts and/or laws and hypotheses that render some sentences plausible, that explain or discover some (eventually new) phenomenon or observation; it is the process of reasoning in which explanatory hypotheses are formed and evaluated. [31] (p.18). Therefore, abduction is a very useful reasoning paradigm, in particular for reasoning towards explanation in (system) diagnosis [58] and analysis in problem solving, and therefore an important form of EBR.

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

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4.3 Tricky Reasoning The reasoning with trick was introduced by Sun and Weber [24][25], and aims at examining logic with trick and reasoning with tricka. Its general form as an inference rule is called modus ponens with trick and represented as [55][57]: P P →Q ---------------∴¬Q

(9)

The modus ponens with trick is motivated by the following fact: Basically speaking, all knowledge and experience consists of two parts: mathematical knowledge and experience, and non-mathematical knowledge and experience as shown in Section 3. The former constitutes Mathematical Logic

Mathematics knowledge and experience

Experience-based reasoning ?

Non-mathematical knowledge and experience

Fig. 1. 3 Mathematics, logic and EBR the resources for existing mathematics, inference rules in mathematical logic can be considered the summary or abstraction of mathematical methods for solving problems in mathematics. The latter constitutes the resources for existing non-mathematical sciences. Although researchers have been always trying to use approaches provided by existing mathematics and mathematical logic to formalize the concepts in their own domain, there are an enormous number of theories and investigations in non-mathematical sciences that are at an empirical level, and requires new logical and mathematical methodologies. Tricky reasoning a. We 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”).

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belonging to this part. Further, from Fig. 3, we can see that mathematics can be considered one part of human knowledge and experience. Mathematics leads to mathematical logic and then CBR and Artificial Intelligence (from a logical viewpoint), because mathematical logic has heavily affected research and development of AI. The rest after the abstraction are non-mathematical knowledge and experience. We believe that the latter leads to experience-based reasoning. Mathematical logic is a formal meta-mathematics from a current viewpoint, it consists of all possible reasoning paradigms and inference rules occurring in mathematics for problem solving. However, from a fundamental viewpoint, only two inference rules have been included; that is, modus ponens, and modus tollens. However, we believe that at least other six inference rules have not been included in mathematical logic. This is the reason why we believe that EBR is an abstraction of non-mathematical knowledge and experience. Multiagent systems (MAS) are an exciting research field in AI. Cooperation, coordination, communication and negotiation play an important role in MAS. However, from a viewpoint of human society, social behaviors can be divided into two categories: rational behaviors and irrational behaviors. The former includes autonomy and trust, while the latter includes deception and lies. Therefore, investigation into deception and lies among intelligent agents in MAS, trick-based logic [25], and trick-based reasoning [24] in MAS has the same importance as research into autonomy and trust in MAS, and will therefore help not only improving better understanding 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

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modus tollens with trick. Its general form is [57]: ¬Q P →Q ---------------∴P

(10)

Example 1. Modus tollens with trick. We have the knowledge in the knowledge base: 1. If Zhaohao is human, then Zhaohao is mortal 2. Zhaohao is immortal What we wish is to prove “Zhaohao is human”. In order to do so, let • P → Q : If Zhaohao is human, then Zhaohao is mortal, • P: Zhaohao is human • Q: Zhaohao is mortal Therefore, we have P: Zhaohao is human, based on modus tollens with trick (10) and the knowledge in the knowledge base (note that ¬Q : Zhaohao 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 (9), because in the traditional logic, we have ¬Q, P → Q ⇒ ¬P and ¬¬P ⇔ P . However, in the non-traditional logic, for example, in fuzzy logic [65], P → Q ⇔ ¬Q → ¬P and ¬¬P ⇔ P are normally invalid. In particular in EBR, they both can be invalid, therefore modus tollens with trick is still meaningful in order to examine the basic rule of inference in EBR. 4.5 Abduction with Trick As discussed in Section 4.2, abduction is an important reasoning paradigm in EBR. It’s “dual” form is abduction with trick, which is also the summary of a kind of EBRs. The general form of abduction with trick, as a basic rule of inference, is as follows:

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Q P →Q ---------------∴¬ P

(11)

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 to the east and attack in the west”; that is, he gets ¬P rather than P in the abduction as a rule of inference. This also verifies that difference is a necessary condition for performing trick or deception. Furthermore, just as abduction has been used in system diagnosis or medical diagnosis, 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 from specific diseases, while abduction with trick can be used to exclude some possibilities of the diseases of the patient (see Section 5.6). Therefore, abduction with trick is an important complementary part for performing system diagnosis and medical diagnosis based on abduction. 4.6 Inverse Modus Ponens Inverse modus ponens is also a rule of inference in EBR. The general form of inverse modus ponens is as follows: ¬P P→Q ---------------∴¬Q

(12)

The “inverse” in the definition is motivated by the fact that the “inverse” is defined in mathematical logic: “if ¬p then ¬q ”, provided that if p then q is given [17]. Based on this fact, the inverse of P → Q is ¬P → ¬Q , and then from ¬P , ¬P → ¬Q we have ¬Q using modus ponens. Therefore the definition of inverse modus ponens is reasonable. Because P → Q and ¬P → ¬Q are not logically equivalent, the argument based on (12) is not valid in

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mathematical logic and mathematics. To our knowledge, EBR based on inverse modus ponens is a kind of common sense reasoning [57], because there are many cases that follow inverse modus ponens. For example, if Peter has enough money, then Peter will fly to Beijing. Now Peter does not have sufficient money, then we can conclude that Peter will not fly to Beijing. 4.7 Inverse Modus Ponens With Trick The final inference rule for EBR is inverse modus ponens with trick. Its general form is as follows [57]: ¬P P→Q ---------------∴Q

(13)

The difference between inverse modus ponens with trick and inverse modus ponens is again “with trick”, this is because the reasoning performer tries to use the trick of “make a feint to the east and attack in the west;” that is, he gets Q rather than ¬Q in the inverse modus ponens. This is also verifies that difference is the necessary condition for performing trick or deception as mentioned in Section 4.3. 4.8 Summary of Inference Rules for EBR Table 1 summarizes the proposed eight basic rules of inference with respect to EBR. It should be noted that some general forms in the table such as inverse modus ponens (this concept is first introduced in [57]) have received attention from some researchers [17], (p. 36). However, the researchers consider this inference rule as the source of fallacies in the reasoning, while we argue that they are all basic inference rules for EBR. They should also be an important part in experience management.

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So far, we reviewed eight different rules of inference for EBR (see Table 1) from a classic logical viewpoint. All these rules of inference are the abstraction and summary of experience or EBR in the real world problems. Table 1: Experience-based reasoning: Rules of inference

modus ponens

modus ponens with trick

inverse modus ponens with trick

inverse modus ponens

P P →Q ---------------∴Q

P P →Q ---------------∴¬Q

¬P P →Q ---------------∴Q

¬P P →Q ---------------∴¬Q

modus tollens

¬Q P →Q ---------------∴¬ P

modus tollens with trick

abductive inference rule

abduction with trick

¬Q P→Q ---------------∴P

Q P →Q ---------------∴P

Q P →Q ---------------∴¬ P

It should be noted that, fuzzy logic has extended traditional logic and found many significant applications [65]. Therefore, the next section will examine the proposed 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 for EBR from a fuzzy logic viewpoint. Throughout this section we assume that P and Q represent fuzzy propositions. Let F 0 ( x ), F 01 ( x, y )and F 1 ( y ), x ∈ X, y ∈ Y be fuzzy relations in X, X × Y , Y, respectively, which are fuzzy restrictions on x, (x, y), and y, respectively. X and Y are two ordinary empty sets. Let P, Q, P' , and Q' be fuzzy propositions and correspond to F 0 ( x ), F 1 ( y ) , F' 0 ( x ), F' 1 ( y ) respectively, and P → Q corresponds to F 01 ( 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 theory and infinite

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multivalued logic systems [65]. Fuzzy logic has evolved into an important research and development field in many disciplines such as mathematics, logic, artificial intelligence and philosophy since 1965. Fuzzy reasoning in fuzzy logic is basically generalized from deductive reasoning in the traditional logic with the exception of its computational process [20][52]. In other words, fuzzy reasoning is a mixed symbolic/numeric approach to both deductive reasoning and rule-based reasoning [45]. Its reasoning is based on the generalized modus ponens [65]: P→Q P' ---------------∴ Q'

(14)

where P and Q represent fuzzy propositions, P' is approximate to P ; that is, P' ∼ P . In this work we call it fuzzy modus ponens in order to provide EBR with a unified treatment based on fuzzy logic. (14) is also commonly represented in the following form in fuzzy logic [65]: If x is P Then y is Q x is P' ---------------------------------------------------∴ y is Q'

(15)

For instance

IF a tomato is red THEN the tomato is ripe This tomato is very red Conclusion: This tomato is very ripe

(16)

In the fuzzy setting, the above fuzzy reasoning can be performed using the following compositional rule of inference introduced by Zadeh [65]: F' 1 ( y ) = F' 0 ( x ) ° F 01 ( x, y )

(17)

It should be noted that the fuzzy modus ponens will reduce to classical modus ponens when P = P' and Q = Q' , and P and Q degenerate into a classical compound propositions.

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Furthermore, modus ponens and fuzzy modus ponens are closely related to the forward datadriven inference which is particularly useful in fuzzy logic control [22]. 5.2 Fuzzy Modus Tollens Fuzzy modus Tollens has also been investigated in fuzzy logic [34][65]. Its general form is ¬Q' P →Q ---------------∴¬ P'

(18)

(18) is also commonly represented in the following form in fuzzy logic [22]: If x is P Then y is Q x is ¬Q ' ---------------------------------------------------(19) ∴ y is ¬ P'' (18) will degenerate to modus tollens (see Section 4.1) when P = 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 result Peter does not get a very good examination result ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (20) ∴Peter does not study very hard The fuzzy reasoning based on fuzzy modus tollens can be computed using the following formula, based on the above discussion: F' 0 ( x ) = 1 – F 01 ( x, y ) ° ( 1 – F' 1 ( y ) ) Further, let x ∈ X = { x 1, x 2, …, x n } , y ∈ Y = { y 1, y 2, …, y m } , F˜ 01 = ( µ ( x i, y j ) ) ,

(21)

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˜ = ( µ ( x ), µ ( x ), …, µ ( x ) ) , F' 1 2 n 0

˜ = ( µ ( y ), µ ( y ), …, µ ( y ) ) , F' 1 2 m 1

then

using

the

compositional rule of inference for fuzzy conditional inference [65], we have an alternative form of Equation (21) as followsa. µ ( x1 )

µ ( x 1, y 1 ) … µ ( x 1, y m ) 1 – µ ( y 1 )

= 1– … … … … … µ ( xn ) µ ( x n, y 1 ) … µ ( x n, y m ) 1 – µ ( y m )

(22)

It should be noted that both modus tollens and fuzzy modus tollens are closely related to the backward goal-driven inference which is commonly used in expert systems, especially in the realm of medical diagnosis [22]. 5.3 Fuzzy Abductive Reasoning Fuzzy abductive reasoning has drawn attention in medical diagnosis since the 1990’s. For example, Miyata et al. study fuzzy abductive inference on the cause-and-effect relationships [34]. Yamada et al. proposed a fuzzy abduction based on multi-valued logic [34]. In order to keep a consistent style in our work, we still use a fuzzy logic-based approach to fuzzy abductive reasoning in what follows. We believe that the general investigation into fuzzy abduction can cover the work of Miyata et al. The general form of fuzzy abductive reasoning is as follows: Q' P →Q ---------------∴ P'

(23)

(23) can be represented in the following form in a context of fuzzy logic:

a. Such a computational form can be provided, in a similar way, to each of the eight fuzzy inference rules for EBR discussed in this paper.

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If x is P Then y is Q y is Q' ---------------------------------------------------∴ x is P''

(24)

For instance, IF John gets fever THEN John will be dizzy John is a little dizzy Conclusion: John gets a light fever

(25)

The fuzzy reasoning based on fuzzy abduction can be computed using the following formula, based on the above discussion: F' 0 ( x ) = F 01 ( x, y ) ° F' 1 ( y ) More

specifically,

let

D = { d 1, d 2, …, d n }

be

(26) the

set

of

diseases,

and

S = { s 1, s 2, …, s m } the set of symptoms [34]. According to medical experience, disease d i will lead to symptom s j with the certainty membership µ ij ( d i, s j ) ; that is, fuzzy relationship between

diseases

D

and

symptom

S

are

F˜ ( D, S ) = ( µ ( d i, s j ) ) ,

i = 1, 2, …, n ;j = 1, 2, …, m . If a patient is observed to have a fuzzy symptom set, T S˜ p = ( µ ( s j ) ) , where µ ( s j ) a is the certainty membership of the observed symptom belonging

to s j . Therefore, according to Equation (26), the fuzzy disease set of this patient is:

T

( µ ( d 1 ), …, µ ( d n ) ) =

µ ( d 1, s 1 ) … µ ( d 1, s m ) µ ( s 1 ) … … … … µ ( d n, s 1 ) … µ ( d n, s m ) µ ( s m )

(27)

a. µ ( d i, s j ) can be considered as the confirmability of s j for d i , and µ ( s j ) expresses the intensity of symptom s j , for detail see [65], pp 185-188.

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where µ ( d i ) is the certainty membership of the disease of the patient belonging to d i . It should be noted that although fuzzy abductive inference has been applied in medical diagnosis [34][35][51], its application success basically results from the computational methods which are based on equations (26) or (27). Because the relationship between diseases and manifestations are many-to-many, and these relationships constitute the basis for explanation of symptoms, it is of significance to build a complete relationship systems in a concrete domain in order to use fuzzy abductive reasoning effectively. 5.4 Fuzzy Modus Ponens with Trick As mentioned, differences of understanding, knowledge, and experience are the source of tricks or deception. The fuzzy, incomplete features of experience and knowledge are a further important source leading to trick and deception. This is also the reason why experience is closely connected to trick or deception in certain situations: More experienced, more tricks. The general form of fuzzy reasoning with trick (based on modus ponens), called fuzzy modus ponens with trick, is as follows: P' P →Q ---------------∴¬Q'

(28)

(28) can be represented in the following form in the context of fuzzy logic: If x is P Then y is Q x is P' ---------------------------------------------------∴ y is ¬Q' For instance,

(29)

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IF Bill is the smartest THEN Bill will work at Medisoft Bill is very smart Conclusion: Bill does not like to work at Medisoft

(30)

The fuzzy modus ponens with trick can be computed using the following formula, based on the above discussion: F' 1 ( y ) = 1 – F' 0 ( x ) ° F 01 ( x, y )

(31)

Basically speaking, fuzzy reasoning is the fuzzification of “make a feint to the east and attack in the west”. More specifically, the commander likes to perform the deception: “make a feint to the east and perhaps attack in the south-west” according to the changing situation in the battlefield. This fuzzy reasoning with trick often happens in business negotiations and war commands. There are a lot of books and research studies on this reasoning paradigm. However, few attempts have been made to formalize such reasoning from a logical or computational viewpoint so that the understanding of fuzzy reasoning with trick is still at empirical level. If fuzzy logic and fuzzy reasoning have provided a powerful methodology to deal with incomplete, imprecise and fuzzy knowledge and its processing, then a big issue for fuzzy logic and fuzzy reasoning is how to treat fuzzy reasoning with tricka. 5.5 Fuzzy Modus Tollens with Trick A direct development from fuzzy modus ponens with trick is fuzzy modus tollens with trick. Although fuzzy modus tollens has drawn attention in the fuzzy logic community [34][65], nobody has studied this new kind reasoning paradigm. However, the latter is also an important part in EBR. In what follows, we will go into it in some detail. a. It is easy for a reader to provide a concrete example for fuzzy reasoning with trick in a special setting. For detail see [34][65].

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The general form of fuzzy modus tollens with trick is ¬Q' P →Q ---------------∴ P'

(32)

Theoretically speaking, it is a variant of fuzzy modus ponens with trick (28), because using fuzzy modus ponens with trick, we have ¬Q', P → Q ⇒ ¬P' . However, this variant can only be understood in a fuzzy setting. For example, if we assume the membership of P, µ ( P ) = 1 , and µ ( P' ) = 0.4 , then ¬µ ( P' ) = 1 – µ ( P' ) = 1 – 0.4 = 0.6 . In this fuzzy microworld, both P' and ¬P' are the intermediate states between P and ¬P . Therefore, such an intermediate but uncertain state is the space for performing a trick or deception. It is very difficult for anyone 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 for others to recognize such tricks. Therefore, it is significant to examine either fuzzy modus ponens with trick or fuzzy modus tollens with trick in a fuzzy setting, which is a closer approximation to the tricks and deceptions existing in human society. (32) can be represented in the following form in fuzzy logic [22]: If x is P Then y is Q x is ¬Q ' ---------------------------------------------------∴ y is P''

(33)

For instance, IF Bill is the smartest THEN Bill will work at Medisoft Bill will not work at Medisoft Conclusion: Bill is very smart

(34)

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From a logical viewpoint, this reasoning means that we prefer to accept a fuzzy or approximate statement to the premise in the fuzzy conditional proposition (Bill is very smart), if we do not accept the conclusion resulting from performing fuzzy modus tollens. The fuzzy modus tollens with trick can be also computed using the following formula, based on the above discussion: F' 0 ( x ) = F 01 ( x, y ) ° ( 1 – F' 1 ( y ) )

(35)

5.6 Fuzzy Abduction with Trick Fuzzy abduction with trick has not been drawn any attention in either medical diagnosis or system analysis, although fuzzy abduction has been studied and applied in these fields. In fact, it is also an important kind of EBR towards the explanation of any symptoms in clinical practice or system diagnosis, which will be seen later. The general form of fuzzy abduction with trick is as follows: Q' P →Q ---------------∴¬ P'

(36)

Theoretically speaking, fuzzy abduction with trick is a variant of fuzzy abduction. In particular, when an agent A in a multiagent system (MAS) may guess that another agent B in the MAS performs fuzzy abduction based on (23), while agent B actually performs fuzzy abduction with trick based on (36). Herewith agent A and agent B will suffer a trust crisis. How to resolve such a trust crisis is an important issue for MASs and web-based systems. (36) can be represented in the following form in a context of fuzzy logic:

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If x is P Then y is Q x is Q' ---------------------------------------------------∴ y is ¬P'

(37)

For instance, IF John gets fever THEN John will be dizzy John is a little dizzy Conclusion: John does not get any fever

(38)

Every adult has had similar experience in a clinic: The doctor gives a wrong explanation 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 symptoms (see Section 5.3), then for a patient c in a clinical practice, his symptoms (for example, SARS’s symptoms) are a subset of S, S c , and his diseases are a subset of D, D c . Therefore D c ⊆ D , and S c ⊆ S

(39)

The available medical experience can be expressed as a set of (fuzzy) rules, E; that is, E = { f f = IFd Then s, d ∈ D, s ∈ S }

(40)

The possible experience set for this patient is E c = { f f = IFd Then s, d ∈ D c, s ∈ S c } . 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 f 1 ∈ E – E c and he performs a fuzzy abduction with trick. The fuzzy reasoning based on fuzzy abduction with trick can be computed using the following formula, based on the above discussion: F' 0 ( x ) = 1 – F 01 ( x, y ) ° F' 1 ( y )

(41)

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Similar to what was proposed in Section 5.3, Equation (41) can be replaced in a more concrete form as follows: Let D = { d 1, d 2, …, d n } be the set of diseases, and S = { s 1, s 2, …, s m } the set of symptoms [34][65]. According to medical experience, disease d i will lead to symptom s j with certainty membership µ ij ( d i, s j ) ; that is, fuzzy relationship between

diseases

D

and

symptom

S

are

F˜ ( D, S ) = ( µ ( d i, s j ) ) ,

i = 1, 2, …, n ; j = 1, 2, …, m a. If a patient is observed to have a fuzzy symptom set, T S˜ p = ( µ ( s j ) ) , where µ ( s j ) is the certainty membership of the observed symptom belonging

to s j . Therefore, according to Equation (41), the fuzzy disease set of this patient is:

T

( µ ( d 1 ), …, µ ( d n ) ) = 1 –

µ ( d 1, s 1 ) … µ ( d 1, s m ) µ ( s 1 ) … … … … µ ( d n, s 1 ) … µ ( d n, s m ) µ ( s m )

(42)

where µ ( d i ) is the certainty membership of the disease of the patient belonging to d i It should be noted that fuzzy abduction with trick has still not been applied in medical diagnosis. Its research and development will help to understand why many patients suffer misdiagnosis and incorrect treatment. In particular, it can be also to exclude some possibilities of certain diseases of the patient; that is, for a certain k ∈ { 1, 2, …, n } , if µ ( d k ) is approximate to 0, the disease d k can be excluded from the possible diseases from which the

a. µ ( d i, s j ) can be considered as the confirmability of s j for d i , and µ ( s j ) expresses the intensity of symptom s j , for detail see [65], pp 185-188.

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patient suffers. This approach is illustrated by the following example, which is borrowed from an example given in [34] and simplified: Example 2. Fuzzy abduction with trick. Assume the set of diseases D = { d 1, d 2, d 3 } , S = { s 1, s 2, s 3, s 4 } . The fuzzy confirmability of s j for d i , F˜ ( D, S ) = ( µ ( d i, s j ) ) is given as a fuzzy relation, listed in Table 2. The observed symptoms are denoted as a fuzzy set S˜ in S, and the corresponding certainty membership of the observed symptoms belonging to s j , µ ( s j ) , are listed as a vector ( µ ( s 1 ), µ ( s 2 ), µ ( s 3 ), µ ( s 4 ) = ( 0.6, 0.1, 0.9, 0.3 ) ). Using Equation (42), we calculate ( µ ( d 1 ), µ ( d 2 ), µ ( d 3 ) ) (based on min-max operation [65]) and have Table 2: The fuzzy confirmability of s j for d i µ ( d i, s j )

s1

s2

s3

s4

d1

1.0

0.8

0

0.6

d2

0.6

0

1.0

0

d3

0.8

0.6

0.7

0.6

0.6 1.0 0.8 0 0.6 0.6 0.4 0.1 = 1 – 0.9 = 0.1 µ ( d 2 ) = 1 – 0.6 0 1.0 0 0.9 0.8 0.6 0.7 0.6 0.7 0.3 µ ( d3 ) 0.3 µ ( d1 )

(43)

Because µ ( d 2 ) is 0.1, which is approximate to 0, the disease d 2 can be excluded from the possible diseases from which the patient suffers. 5.7 Fuzzy Inverse Modus Ponens Fuzzy inverse modus ponens is another rule of inference for EBR. Its general form is as follows:

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¬P' P →Q ---------------∴¬Q'

(44)

(44) can be represented in the following form in the context of fuzzy logic: If x is P Then y is Q x is ¬P ' ---------------------------------------------------∴ y is ¬Q'

(45)

Example 3. Fuzzy inverse modus ponens: We have the knowledge in the knowledge base: • If the quarter profit is increasing, then Klaus invests in the Project FIMP, • The quarter profit is marginally decreased. What we wish is to prove “Klaus does not intend to invest in the Project FIMP”. To this end, let P → Q : If the quarter profit is increasing, then Klaus invests in the Project FIMP; P : The quarter profit is increasing. Therefore, we have ¬Q' : Klaus does not intend to invest in the Project FIMP based on (44) and the knowledge in the knowledge base (note that ¬P' : The quarter profit is marginally decreased ). In the conclusion of this example, “Klaus does not intend to invest in the Project FIMP” means that Klaus has not yet decided to invest the project FIMP, which is an intermediate state between “Klaus invests in the Project FIMP” and “Klaus does not invest in the Project FIMP”. The fuzzy reasoning based on fuzzy inverse modus ponens can be computed using the following formula, based on the above discussion: F' 1 ( y ) = 1 – ( 1 – F' 0 ( x ) ) ° F 01 ( x, y )

(46)

5.8 Fuzzy Inverse Modus Ponens with Trick Fuzzy inverse modus ponens with trick is the last rule of inference for EBR. Its general form

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is as follows: ¬P' P→Q ---------------∴Q'

(47)

(47) can be represented in the following form in the context of fuzzy logic: If x is P Then y is Q x is ¬P ' ---------------------------------------------------(48) ∴ y is Q'' Example 4. Fuzzy inverse modus ponens with trick: We have the knowledge in the knowledge base: • If the quarter profit is increasing, then Klaus invests in the Project FIMPT, • The quarter profit is not increasing much. What we wish is to prove “Klaus intends to invest in the Project FIMPT”. To this end, let P → Q : If the quarter profit is increasing, then Klaus invests in the Project FIMP; P : The quarter profit is increasing. Therefore, we have Q' : Klaus intends to invest in the Project FIMPT based on (47) and the knowledge in the knowledge base (note that ¬P' : The quarter profit is not increasing much). In the conclusion of this example, “Klaus intends to invest in the Project FIMP” is approximate to “Klaus invests in the Project FIMPT.” The fuzzy reasoning based on fuzzy inverse modus ponens with trick can be computed using the following formula, based on the above discussion: F' 1 ( y ) = ( 1 – F' 0 ( x ) ) ° F 01 ( x, y )

(49)

It should be noted that fuzzy inverse modus ponens and fuzzy inverse modus ponens with trick have not drawn any attention in either fuzzy logic or computer science. We believe that

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the research and development of fuzzy inverse modus ponens can improve our understanding of experience-based reasoning in a fuzzy setting, because it is commons in human society. 5.9 Summary Table 3. summarizes the proposed eight fuzzy inference rules for experience-based reasoning (including the corresponding form). It should be noted that some general forms in the table such as fuzzy modus ponens and fuzzy modus tollens have received some attention from researchers [34][65] (p. 36), while the rest of them have not been studied in fuzzy logic and computer science, although they are all the summarization of EBRs. We argue that they are all the basic rules of inference for EBR. They should also be an important part in experience management. Table 3. Fuzzy rules of inference for experience-based reasoning

modus ponens

modus ponens with trick

inverse modus ponens with trick

inverse modus ponens

Traditional form

P P→Q ---------------∴Q

P P→Q ---------------∴¬Q

¬P P→Q ---------------∴Q

¬P P→Q ---------------∴¬Q

¬Q P →Q ---------------∴¬ P

Fuzzy form

P' P →Q ---------------∴Q'

P' P →Q ---------------∴¬Q'

¬P' P →Q ---------------∴Q'

¬P' P→Q ---------------∴¬Q'

¬Q' P →Q ---------------∴¬ P'

modus tollens

modus tollens with trick

abductive inference rule

abduction with trick

¬Q P →Q ---------------∴P

Q P→Q ---------------∴P

Q P→Q ---------------∴¬ P

¬Q' P → Q' ---------------∴P

Q' P →Q ---------------∴ P'

Q' P →Q ---------------∴¬ P'

6. Concluding Remarks This paper first reviewed the logical approach to EBR, in which eight different rules of inference for EBR are studied. Then the paper proposed a fuzzy logic-based model to each of these eight different rules of inference which constitute the fundamentals for all EBR paradigms, and therefore will be a theoretical foundation for EBR. The proposed approach will

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facilitate research and development of not only EBR systems but also knowledge/experience management. Experience management (EM)a is drawing increasing attention in e-commerce, computer science, and information systems, and knowledge management (KM) [23] which has become one of the latest hot topics in the business world [4]. EM is more useful than KM, because while every one can have a lot of knowledge, only the experience of experts is invaluable. Therefore, EM can facilitate spreading valuable experience, promoting the transition from experience to knowledge, and facilitate KM. In fact, the relationship between experience and knowledge is the basis for the relationship between EM and KM. Their correspondence to intelligent systems are experience based systems (EBS) such as CBR systems (CBRS) and knowledge based systems (KBS) respectively. Furthermore, from a logical viewpoint, experience management is based on experience based reasoning. Therefore, we will apply the proposed approach and inference rules for EBR to EM in future work. Similarity-based reasoning is an important operational form for performing EBR [20]. It is an important “bridge” connecting CBR and EBR [56], because “similar cars have similar prices” is also a very popular experience principle. Therefore we will apply similarity based reasoning to examine EBR and its eight inference rules in future research.

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