A Unified 2D Representation of Fuzzy Reasoning, CBR, and Experience Based Reasoning

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1 University of Wollongong Research Online Faculty of Commerce - aers (Archive) Faculty of Business 26 A Unified 2D Reresentation of Fuzzy Reasoning, CBR, and Exerience Based Reasoning Zhaohao Sun University of Wollongong, zsun@uow.edu.au G. Finnie Bond University ublication Details This aer was originally ublished as: Sun, Z & Finnie G, A unified 2D reresentation of fuzzy reasoning, CBR, and exerience-based reasoning, in B Gabrys, RJ Howlett & LC Jain (eds), Knowledge-Based Intelligent Information and Engineering Systems (KES 26) art I, Lecture Notes in Comuter Science 4251/26, Sringer Berlin/Heidelberg, 26, The original ublication is available here from Sringerlink. Research Online is the oen access institutional reository for the University of Wollongong. For further information contact the UOW Library: research-ubs@uow.edu.au

2 A Unified 2D Reresentation of Fuzzy Reasoning, CBR, and Exerience Based Reasoning Abstract Fuzzy reasoning, case-based reasoning (CBR) and exerience-based reasoning (EBR) or natural reasoning have been seriously studied for years. However, these studies essentially can be considered as a 1-dimensional aroach, because their reasoning aradigm is 1-dimensional. This aer will roose a 2-dimensional (2D) aroach that reresents fuzzy reasoning, CBR, and EBR in a unified way. This aroach also integrates many reasoning aradigms such as abduction, deduction and similarity-based reasoning into a unified treatment. The roosed aroach will facilitate research and develoment of fuzzy reasoning, knowledge/exerience management, knowledge-based systems, and CBR. Keywords Fuzzy reasoning, case-based reasoning, exerience-based reasoning, abduction, 2-dimensional (2D) reresentation, exerience management Discilines Business Social and Behavioral Sciences ublication Details This aer was originally ublished as: Sun, Z & Finnie G, A unified 2D reresentation of fuzzy reasoning, CBR, and exerience-based reasoning, in B Gabrys, RJ Howlett & LC Jain (eds), Knowledge-Based Intelligent Information and Engineering Systems (KES 26) art I, Lecture Notes in Comuter Science 4251/26, Sringer Berlin/Heidelberg, 26, The original ublication is available here from Sringerlink. This conference aer is available at Research Online: htt://ro.uow.edu.au/commaers/25

3 Sun Z and Finnie G. (26) A unified 2D reresentation of fuzzy reasoning, CBR, and exerience-based reasoning. In B. Gabrys, R.J. Howlett, and L.C. Jain (Eds.) Knowledge-Based Intelligent Information and Engineering Systems (KES 26) art I, LNAI 4251, Sringer Berlin/ Heidelberg, A Unified 2D Reresentation of Fuzzy Reasoning, CBR, and Exerience Based Reasoning Zhaohao Sun, Gavin Finnie* School of Economics and Information Systems, University of Wollongong Wollongong NSW 2522 Australia zsun@uow.edu.au * School of Information Technology, Bond University, Gold Coast ld 4229 Australia gfinnie@staff.bond.edu.au Abstract: Fuzzy reasoning, case-based reasoning (CBR) and exerience-based reasoning (EBR) or natural reasoning have been seriously studied for years. However, these studies essentially can be considered as a 1-dimensional aroach, because their reasoning aradigm is 1- dimensional. This aer will roose a 2-dimensional (2D) aroach that reresents fuzzy reasoning, CBR, and EBR in a unified way. This aroach also integrates many reasoning aradigms such as abduction, deduction and similarity-based reasoning into a unified treatment. The roosed aroach will facilitate research and develoment of fuzzy reasoning, knowledge/ exerience management, knowledge-based systems, and CBR. Keywords: Fuzzy reasoning, case-based reasoning, exerience-based reasoning, abduction, 2-dimensional (2D) reresentation, exerience management. 1 Introduction Fuzzy reasoning [21], case-based reasoning (CBR) [13] and exerience-based reasoning (EBR) [18] or natural reasoning and also traditional reasoning aradigms such as deduction [9], abduction [17] and similarity-based reasoning (SBR) [11] have been seriously studied in comuter science and artificial intelligence (AI) with many alications in engineering [21], commerce [11], business and management [12], to name a few. However, these studies essentially can be considered as a 1-dimensional (1D) aroach, because the fundamental inference rule of each of these reasoning aradigms is reresented in a 1D way. This 1D reresentation limits their own develoment and isolates them from each other, although some interrelationshis between fuzzy reasoning and CBR have been studied [13]. This aer will fill this ga by roosing a 2-dimensional (2D) aroach that reresents fuzzy reasoning, CBR, and EBR in a unified way. This aroach also integrates many reasoning aradigms such as abduction, deduction and SBR into a unified treatment. The roosed aroach will facilitate research and develoment of fuzzy reasoning, knowledge/exerience management, knowledge-based systems, and CBR. 2 Fundamentals Any intelligent system can only give the answers to roblems in a ossible world, which corresonds to a scenario in the real world [11]. Based on this idea, the ossible world of roblems, W, and the ossible world of solutions, W s, are the whole world -1/8-

4 of an agent [9] to use an intelligent systems to do everything that he can. If an agent considers a CBR system as a function h from W to W s, it is meaningless to discuss the image of hx if x W. Therefore, the agent can only know and work in the world W W s. For examle, in a CBR e-sale system, the ossible world of roblems, W, might consist of [5]: roerties of goods Normalized ueries of customers Knowledge of customer behavior General knowledge of business, and so on. And the ossible world of solutions, W s, consists of: rice of goods Customized answers to the ueries of customers General strategies for attracting customers to buy the goods, and so on. It should be noted that if we assume that is the subset of roblem or antecedent descritions and is the subset of solution or action descritions [8], then it is obvious that and are subsets of W and W s resectively. A conditional roosition,, can be denoted as an ordered air, where and. Further, if we assume that W is a dimensional world, then W s is another dimensional world. Therefore, we can use a coordinate system to reresent these two dimensional worlds consisting of W and W s. 3 A 2D Reresentation of Fuzzy Reasoning In roositional logic, reasoning basically belongs to deductive reasoning. Reasoning is erformed by a number of inference rules [11], in which the most commonly used is modus onens (M): (1) where and reresent comound roositions. One of the most imortant features of this reasoning is that it satisfies the transitive law, and then this reasoning can be erformed as many times or stes as reuired with the reservation of validity of the result of the inference. This means that the reasoning in traditional logic is a multiste reasoning. Fuzzy reasoning in fuzzy logic is basically generalized from the deductive reasoning in traditional logic with the excetion of its comutational rocess [17]. Its reasoning is based on the following generalized modus onens [13]: -2/8-

5 ' (2) ' where and reresent fuzzy roositions, ' is aroximate to, that is, '. Model (2) is also commonly reresented in the following form in fuzzy logic [21]: If x is Then y is ---- x is ' (3) y is ' For instance, IF a tomato is red THEN the tomato is rie This tomato is very red Conclusion: This tomato is very rie As we know, inference rules lay a central role in reasoning aradigms of AI. For examle, one of the main inference rules for fuzzy reasoning is the generalized modus onens Model (2). Based on the discussion in Section 2, Model (2) can be reresented in a 2-dimensional (2D) way as shown in Fig. 1, where the node ointed to by an arrow reresents the conclusion of the inference, the small cycle denotes the (fuzzy) roosition '. Further, the oint denoted by ' ', ' ' ' ' and ' ' will be in the 1st, 2nd, 3rd and 4th uadrant resectively (Similar exlanations are valid for other figures in this aer). Therefore, fuzzy reasoning can be considered as a first uadrant reasoning because it only takes the 1st uadrant of the coordinate system. For fuzzy reasoning, means, and ' is a fuzzy roosition aroximate or close to fuzzy roosition, and then the conclusion inferred from the fuzzy reasoning based on Model (2) is '. It should be noted that when ', then ', the generalized modus onens will degenerate to classic modus onens. The 2D reresentation of modus onens will be simler than that in Fig. 1, which we do not discuss any more. 4 A 2D Reresentation of CBR and SBR CBR is a reasoning aradigm based on revious exeriences or cases that are the oerational definition of exeriences; that is, a case-based reasoner solves new roblems by adating solutions that were used to solve old roblems [13]. Therefore, we call CBR a form of exerience-based reasoning [11], briefly, CBR:= Exerience-based reasoning (5) CBR is based on two rinciles about the nature of the world [6]. The first rincile is that the world is regular: similar roblems have similar solutions. Conseuently, solutions to similar rior roblems are a useful starting oint for new roblem solving. The second rincile is that the tyes of roblems that an agent encounter tend to recur. Hence, future roblems are likely to be similar to current roblems. When the two rinciles hold, CBR is an effective reasoning aradigm. (4) -3/8-

6 The first rincile also imlies that EBR is based on the exerience rincile, for examle, in business activities it is usually true that Two cars with similar uality features have similar rices. However, from a logical viewoint, this is a kind of similarity-based reasoning (SBR) [4]. In other words, SBR can be considered as a secial and oerational form of EBR. Therefore, CBR can be considered as a kind of SBR from a logical viewoint [13]. CBR: = Similarity-based reasoning (6) Similar to the inference engine in exert systems (ESs), one can also use a CBR engine (CBRE) to denote the inference engine in CBR system (CBRS); that is, CBRS = CB + CBRE (7) where CB is the case base. CBRE erforms similarity-based reasoning, while the inference engine in ESs erforms traditional deductive reasoning. In CBR, similarity based reasoning is realized by the following model [13]. ' (8) ' where, W and W reresent a roblem descrition and the corresonding solution descrition resectively; that is, ' is the current roblem descrition s which is similar to, ', is a case stored in the CB. The CBR is to find a solution, ', to the current roblem ', and '. Therefore, CBR or SBR based on Model (8) can also be reresented in a 2D way as shown in the Fig. 2. ' (,) ' ' ' (,) ' ' ' Fig. 1. A 2D reresentation of fuzzy reasoning ' Fig. 2. A 2D reresentation of CBR and SBR In this 2D reresentation, ' and, ' and, and are the concrete alication form of ' and, ' and, and in CBR resectively. Comaring Fig. 1 with Fig. 2, we find that the 2D reresentation of fuzzy reasoning and that of CBR and SBR are the same, although they have different semantic interretations and oerational algorithms for erforming their own reasoning aradigms based on different real world scenarios. Further, CBR and SBR can also be considered a first uadrant reasoning. It should be noted that the above 2D reresentation of CBR is an abstract form of the Leake s model of CBR [4][6]. Based on the above examination, the interesting uestion arises: Are there other uadrant reasoning aradigms? -4/8-

7 5 A Unified 2D Reresentation of EBR Exerience-based reasoning (EBR) is a reasoning aradigm based on rior exeriences. EBR as a technology has been used in many alications [12][14]. Taking into account research and develoment of CBR, Sun and Finnie [12][18] roosed eight different inference rules for EBR which cover all ossibilities of EBR, as shown in Table 1. These inference rules are listed in the first row, and their corresonding general forms with resect to classic logic are shown in the second row resectively [12]. Their corresonding fuzzy logic based forms are shown in the third row resectively [16], which can also be considered as similarity based inference rules resectively taking into account SBR. Because four of them, modus onens (M), modus tollens (MT), abduction and modus onens with trick (MT) are well-known in AI and comuter sciences [9][1][11], we do not go into them any more, and focus on reviewing two of the other four inference rules in some detail. Table 1: Exerience-based reasoning: Eight inference rules. M MT abduction MTT AT MT IM IMT ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Inverse modus onens (IM) is an inference rule in EBR [12]. For examle, if John has enough money, then John will fly to China. Now John does not have sufficient money, then we can conclude that John will not fly to China. The last inference rule for EBR is inverse modus onens with trick (IMT) [16]. The difference between IMT and IM is with trick, this is because the reasoning erformer tries to use the trick of make a feint to the east and attack in the west ; that is, he gets rather than in the inverse modus onens. It should be noted that the inference rules with trick such as MTT, AT, MT and IMT are non-traditional inference rules. However, they are really abstractions of some EBR, although few have tried to formalize them. The with trick is only a semantic exlanation for such models. One can give other exlanations for them. For examle, one can use against exectation instead of with trick to exlain results of stock trading. Then s/he can use inference rules of against exectation to identify and discover the against exectation atterns in the stock trading databases [2]. So far, we have examined fuzzy reasoning, CBR, SBR, and EBR. However, their relationshi seems not clear or at least lacks a grahic evidence. Further, the existing studies on them can be essentially considered as one dimensional (1D) aroach, because the fundamental inference rule of each of these reasoning aradigms is reresented in a 1D way. Based on the discussion in Section 3 and Section 4, we reresent -5/8-

8 the other seven inference rules in a coordinate system, in what follows. We will briefly discuss each of the 2D reresentations owing to the sace limitation. ' (,) ' (,) ' ' ' ' ' Fig. 3. A 2D reresentation of fuzzy MT ' Fig. 4. A 2D reresentation of fuzzy abduction Fig. 3 shows the 2D reresentation of fuzzy modus tollens (MT) in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) MT are a non-first uadrant reasoning, because it involves the uantities in the first and third uadrant, we can call it first-third uadrant reasoning. Fig. 4 shows the 2D reresentation of fuzzy abduction in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) abduction are a first uadrant reasoning. (,) ' ' ' ' (,) ' Fig. 5. A 2D reresentation of fuzzy MTT ' ' ' Fig. 6. A 2D reresentation of fuzzy AT Fig. 5 shows the 2D reresentation of (fuzzy or similarity-based) modus tollens with trick (MTT) in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) MTT are a non-first uadrant reasoning, but a first-fourth uadrant reasoning. Fig. 6 shows the 2D reresentation of (fuzzy or similarity-based) abduction with trick (AT) in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) AT are a non-first uadrant reasoning, but a first-second uadrant reasoning. Fig. 7 shows the 2D reresentation of (fuzzy or similarity-based) modus onens with trick (MT) in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) MT are a non-first uadrant reasoning, but a first-fourth uadrant reasoning. -6/8-

9 (,) (,) ' ' ' ' ' ' ' ' Fig. 7. A 2D reresentation of fuzzy MT Fig. 8. A 2D reresentation of fuzzy IM Fig. 8 shows the 2D reresentation of (fuzzy or similarity-based) inverse modus onens (IM) in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) IM are a non-first uadrant reasoning, but a first-third uadrant reasoning. ' ' ' (,) ' Fig. 9 shows the 2D reresentation of (fuzzy or similarity-based) inverse modus onens with trick (IMT) in EBR, which demonstrates that any reasoning aradigms based on (fuzzy or similarity-based) IMT are a non-first uadrant reasoning, but a first-second uadrant reasoning. 6 Concluding Remarks Fig. 9. A 2D reresentation of fuzzy IMT This aer examined fuzzy reasoning, CBR, similarity based reasoning and EBR. It then roosed a 2D aroach that reresents fuzzy reasoning, CBR, and EBR in a unified way. This aroach also integrated many reasoning aradigms such as abduction, deduction and similarity-based reasoning into a unified treatment. The roosed aroach will facilitate research and develoment of fuzzy systems, knowledge/exerience management, knowledge-based systems, and case-based reasoning. In future work, we will examine the 2D reasoning roerties (such as continuity of reasoning aradigms) of fuzzy reasoning, CBR, similarity based reasoning and EBR and look at the alication of the roosed 2D reresentation of reasoning aradigms in AI, e-commerce and e-services. -7/8-

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