fuzzy-xcs: an accuracy-based fuzzy classiﬂer...

Fuzzy-XCS: An Accuracy-Based Fuzzy Classifier System ∗

Jorge CasillasDept. Computer Science & Artif. Intell.

University of GranadaGranada E-18071, Spain

[email protected]

Brian Carse, Larry BullFact. Comput., Eng. & Mathematical Sci.

University of the West of EnglandBristol BS16 1QY, UK

Brian.Carse, [email protected]

Abstract

The issue of rule generalization has received agreat deal of attention in the discrete-valuedlearning classifier system field. In particular,the accuracy based XCS is the subject of ex-tensive ongoing research.

However, the same issue does not appear tohave received a similar level of attention inthe case of the fuzzy classifier system. Thismay be due to the difficulty in extending thediscrete-valued system operation to the con-tinuous case.

The intention of this contribution is to pro-pose an approach to properly develop a fuzzyXCS system.

Keywords: learning classifier systems,XCS, evolutionary algorithm, Michigan-stylelearning fuzzy systems, fuzzy implications.

1 Introduction

The fuzzy classifier system is a machine learning sys-tem which employs linguistic rules and fuzzy sets inits representation and an evolutionary algorithm (EA)for rule discovery. It therefore combines an easily un-derstood representation (as opposed to, for example,neural networks approaches) with a general purposesearch method. In order to exploit the fuzzy repre-senting to the full, the ability to learn generalizationis of great importance.

Generalized rules allow more compact rule bases, scal-ability to higher dimensional spaces, faster inference,

∗ Part of this research has been developed during aresearch stay of J. Casillas at the Faculty of Computing,Engineering and Mathematical Sciences of the Universityof the West of England (Bristol, UK) supported by theSecretarıa General de Universidades e Investigacion de laJunta de Andalucıa (Spain)

and better linguistic interpretability. The issue of rulegeneralization, and the interplay between general andspecific rules in the same evolving population, has re-ceived a great deal attention in the discrete-valuedclassifier system research community (e.g. [8]). Thesame issue does not appear to have received a similarlevel of attention in the case of fuzzy classifier sys-tems [1, 2, 3, 4, 5, 6, 7].

Traditional Michigan-style classifier systems have been“strength-based” in the sense that a classifier ac-crues strength during interaction with the environment(through rewards and/or penalties). This strength canthen be used for two purposes: resolving conflicts be-tween simultaneously matched classifiers during learn-ing episodes; and as the basis of fitness for the EA. Acompletely different approach can be taken in which aclassifier’s fitness, from the point of view of the EA, isbased on its “accuracy,” i.e. how well a classifier pre-dicts payoff whenever it fires. Such an accuracy-basedapproach offers a number of advantages such as avoid-ing over-general classifiers, obtaining optimally generalclassifiers, and learning of a complete “covering map.”This accuracy-based classifier system, called XCS, wasproposed in [8] and it is currently of major interest tothe research community in this field.

This work aims at proposing a new approach to achieveaccuracy-based Michigan-style fuzzy classifier systems.The proposal, Fuzzy-XCS, is based on XCS but prop-erly adapted to fuzzy systems. The fuzzy inferenceand the reinforcement component are changed to con-sider a different, competitive interaction among rulesthat allow an accuracy-based operation.

The paper is organized as follows. Section 2 intro-duces some difficulties to develop an accuracy-basedfuzzy classifier system. Section 3 describes a compet-itive inference mechanism. Section 4 introduces theproposed Fuzzy-XCS system. Section 5 shows someexperimental results. Section 6 concludes.

XII CONGRESO ESPAÑOL SOBRE TECNOLOGÍAS Y LÓGICA FUZZY 369

2 Difficulties in Accuracy-BasedFuzzy Classifier Systems

Most of the difficulty in accuracy-based fuzzy clas-sifier system is the fuzzy inference process, there-fore, it is briefly explained in the next subsection.Then, the problems when using strength-based Michi-gan fuzzy classifier systems, the advantages of con-sidering accuracy-based fitness, and the difficulties indoing that are introduced.

2.1 Cooperation-Based Fuzzy Inference

This is the most commonly used fuzzy inferencemethod. Linguistic (or Mamdani-type) fuzzy rule-based systems are formed by linguistic fuzzy rules withthe following structure:

...alsoRr−1: IF X1 is Ar−1

1 and . . . and Xn is Ar−1n

THEN Y1 is Br−11 and . . . and Ym is Br−1

m ,alsoRr: IF X1 is Ar

1 and . . . and Xn is Arn

THEN Y1 is Br1 and . . . and Ym is Br

m ,also...

with Xi and Yj being input and output linguistic vari-ables respectively, and with Ai and Bj being linguisticlabels with associated fuzzy sets defining their mean-ing. These linguistic labels use a global semantic defin-ing the set of possible fuzzy sets used for each variable.

The inference process, which obtains an output as re-sponse to a specific input, consists basically of the fol-lowing steps: for each fired (or matched) fuzzy rule,firstly the conjunction (and) operator is used to ob-tain the matching degree of the rule; secondly, the im-plication (then) operator is applied to scale the out-put fuzzy set to a degree according to the matching;finally, the last step is to apply the aggregation (also)operator to reduce the combined output of all the rulesacting together to a single fuzzy set.

If we need a real-valued output, a last stage (defuzzi-fication), converts the output fuzzy set to a number.Center of gravity, center of sums, or mean of maximaare some possibilities to do that.

The most widely inference scheme used in linguisticfuzzy systems is that proposed by Mamdani and Assil-ian in the first fuzzy controller of 1975. It involves thecombination of operators (min,min,max ), i.e., mini-mum for conjunction, minimum for implication, andmaximum for aggregation. This inference is usuallynamed Max-Min. Alternatively, sometimes it uses an-

other t-norm (e.g. product) to play the role of con-junction and/or implication, or another t-conorm (e.g.bounded sum) to the also operator.

This is the most commonly used approach (we couldsay even the only one) followed in engineering prob-lems such as fuzzy control and fuzzy modeling. It isbecause of, with these operators, the fuzzy rules “co-operate” to generate the output, in the sense that anInterpolative Reasoning is performed to define the out-put in the zones where several rules work to a mediummatching degree. Indeed, the fact of using a t-conormto aggregate the information involves that we are us-ing a union and, therefore, the effect of each rule isadded to the final consensual output.

2.2 Problems with “Strength-Based”Michigan Fuzzy Classifier Systems

As mentioned in the introduction, the strength-basedfuzzy classifier system is characterized by using thesame parameter (i.e. strength) to resolve conflicts be-tween matched classifiers and to compute their fitness.A number of problems arise from this dual use of clas-sifier strength. These include:

1. The cooperation/competition problem. High-strength, potentially cooperative classifiers go on tocompete under the action of the EA.

2. Over-general rules with relatively high (but incon-sistent) payoff can come to dominate the population.

3. In some environmental states, the maximum payoffachievable (by performing the best possible action forthat state) may be relatively low. Although a classifiermight be the best that can exist for that state, it can beeradicated from the population by other classifiers thatachieve higher rewards in other states. This results ingaps in the system’s “covering map.”

2.3 Advantages of using “Accuracy-Based”Fitness

A completely different approach can be taken in whicha classifier’s fitness, from the point of view of the EA,is based on its “accuracy,” i.e. how well a classifier pre-dicts payoff whenever it fires. Such an accuracy-basedapproach offers a number of advantages. Firstly, it candistinguish between accurate and over-general classi-fiers: an over-general classifier will have relatively lowaccuracy since payoff will vary according to the inputstates covered by the classifier. Indeed, it has beenshown (in the discrete valued case) that the accuracy-based approach can lead to evolution of optimally gen-eral classifiers. Additionally it can maintain both con-sistently correct and consistently incorrect classifierswhich allows learning of a complete “covering map.”

370 XII CONGRESO ESPAÑOL SOBRE TECNOLOGÍAS Y LÓGICA FUZZY

A potential drawback of the accuracy-based approachis that it is likely to require larger populations of clas-sifiers.

2.4 Difficulties in Moving to“Accuracy-Based” Fuzzy ClassifierSystems

Firstly, in a traditional fuzzy classifier system severalrules fire in parallel (this is how the system achievesinterpolation); credit assignment is much more diffi-cult in the fuzzy case and it may well be that ap-portioning credit in proportion to a fuzzy classifier’sactivation level is not appropriate. A further difficultyis measuring the accuracy of a rule’s predicted payoffsince (particularly early in the search) a fuzzy rule willfire with many different other fuzzy rules at differenttime-steps, giving very different payoffs. Yet anotherdifficulty is that the payoff a fuzzy rule receives de-pends on the input vector — an active fuzzy rule willreceive different payoffs for different inputs. This fur-ther complicates payoff predictions used as the basisfor accuracy-based fitness.

3 Competitive Fuzzy Inference

The problems explained in the previous section aremainly due to the fact that, usually, all the matchedrules cooperate to define the final solution in a inter-polative behavior as explained in Section 2.1. If it isthe problem, why not to change our approach and lookfor competitive interaction? This section introducessuch an approach.

3.1 Why Use Competitive Inference?

Michigan-style learning classifier systems in general,and XCS in particular, do not consider the interactionof the different rules as a cooperative action, insteadthey consider that each rule competes with the restto be the best one for a particular input vector. Thisinvolves that the action is only due to a set of rulesthat were the winners among the rules composing thematch set. Extending this approach to fuzzy modeling,it makes sense to deal with competitive fuzzy rules, inwhich case, we need to know how to work with suchrules.

In off-line fuzzy system learning, it seems that coop-erative inference has some clear advantages that makeit be more suitable for this problem. In this case, weknown the whole data set a priori and our objective isto build a landscape of the output data. To do that,it seems a good approach to fill the gaps between datawith interpolation, and this is the reason why cooper-ative inference is successful.

However, in on-line fuzzy system learning, why is itnecessary to interpolate the actions of the rules? Theobjective in this problem should not be to define alandscape of the output, but a landscape of the reward.This means taking the best action in each state. Forthis purpose, a competitive action makes more sensesince our objective is to optimize the rules individuallyto have the best reward.

With competitive fuzzy inference we propose a deci-sion making process where the matched rules competeamong themselves, and only one is the winner. Theoutput finally obtained is mainly due to the winnerrule. That is, loser rules do not have an important in-fluence on the output. Nevertheless, competition doesnot involve a loss of interaction. There is an interac-tion, but with a selfish objective. To perform compet-itive inference we only need to change the roles of theinference operators.

3.2 Fuzzy Operators for CompetitiveInference

First of all, since each rule competes with the rest, weshould use an intersection (t-norm) as also operator,instead of the union approach followed by the cooper-ative inference. We use the minimum in this paper.However, this change does not allow us to use the samekind of implication operator.

Analyzing the classical Boolean implication p ⇒ q,equivalent to ¬p ∨ q, it is true when p is false. Thatis, when we have not information about the certaintyof the fact (antecedent), we assume the “optimistic”view of thinking that the implication is true.

However, in cooperative inference, a t-norm (intersec-tion) is used as implication. It is a “pessimistic” viewthat assumes the falseness of the implication when thefact is false. It works if we use a union (t-conorm)to aggregate the effect of the different rules. This be-havior could be interpreted that the belief of the rulesresides in the set of them, instead of in each of them.

To develop a competitive inference where a inter-section is considered as aggregation, the implicationshould follow the classical Boolean if-then view assum-ing that each rule has all the information about therelationship between fact and consequence. Thus, thebelief develops separately for each rule. This family ofimplications are known as logical fuzzy implications,instead of the engineering fuzzy implications based onthe t-norm. Table 1 includes some of the best knownlogical fuzzy implications.

To understand better the different behavior betweencooperative and competitive inference, Figure 1 showsthe output generated by two fuzzy systems that only


Table 1: Some logical fuzzy implicationsOperator µR(x, y)

Kleene-Dienes max{1− µA(x), µB(y)}Zadeh max{min{µA(x), µB(y)}, 1− µA(x)}ÃLukasiewicz min{1, 1− µA(x) + µB(y)}

Dubois-Prade

{1− µA(x) if µB(x) = 0µB(x) if µA(x) = 11 otherwise

Godel

{1 if µA(x) ≤ µB(x)µB(x) otherwise

Goguen min{µB(y)/max{µA(y), µB(y)}}

differ in the inference mechanism. In the cooper-ative inference, minimum, minimum, and maximumare used as conjunction, implication, and aggregationoperators, respectively. In the competitive inference,minimum, ÃLukasiewicz, and minimum are used as con-junction, implication, and aggregation operators, re-spectively. The center of gravity is used as defuzzifi-cation process in both cases. Both systems consist oftwo input variables and one output variable, with 5triangular fuzzy sets uniformly distributed in the uni-verse of discourse [0,1] for each variable. Among the25 fuzzy rules considered, only 7 have a consequentdifferent from the medium linguistic term (with ver-tex at 0.5). As we can see, both approaches generatea smooth output but, while cooperative inference fillsthe gaps by interpolating, competitive inference iso-lates the effect of each rule.

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Figure 1: Output generated by cooperative (top) andcompetitive (bottom) inferences

On the other hand, Figure 2 illustrates some examplesof competitive inference depending on the used impli-cation operator. The minimum aggregation operatoris used in all the cases. We can see how the actionstrength of the winner rule (the one with the highestmatching degree) not only depends on its matchingdegree but also on the difference in the matching de-gree with respect to fuzzy rules with different actions(consequents), i.e., rival fuzzy rules. The more dif-ferent these matching degrees, the higher the actionstrength of the selected rule. This involves that ac-tually, the competition is performed among differentlinguistic actions. Therefore, we should say that com-petitive inference leads to competitive actions, but notnecessarily competitive rules.

The competition among rules with the same action isonly performed when using the minimum operator asaggregation, since in this case only the best rule ofthat action is considered in the inference. The use ofother aggregation operators, such as the product, willlead the inference to a cooperation among rules withthe same actions. According to XCS, this approach isnot desired since it aims at obtaining the X ×A ⇒ Pmap with the best classifier for each situation-actioncombination [8].

We can characterize slightly different interactions ap-proaches among rules with different actions (i.e. rivalrules) depending on the logical fuzzy implication op-erator used; these are shown in Table 2.

Table 2: Type of competition among rules with differ-ent actions depending on the used implication operator(a t-norm is used as also operator)Implication Type of competition

Kleene-Dienes Rival rules reduce the importance of thewinner rule in non-overlapped areas

Zadeh Like Kleene-Dienes. No interactionamong rules if the winner rule has amatching degree lower than 0.5

ÃLukasiewicz Like Kleene-Dienes, but with higherpreservation of overlapped areas than it

Dubois-Prade Like Kleene-Dienes, but with the high-est preservation of overlapped areas

Godel Output always in the overlapped area.No output with more than two differ-ent actions. If matchings greater thancross-point, output in the center of theoverlapped area

Goguen Like Godel. Better with mean of maxi-mum defuzzification

Finally, we should say that the consideration of com-petitive actions seems to fit properly with the XCSaim. Other fuzzy learning classifier systems, likeBonarini’s work [1, 2], focus on the interaction among


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rules in the antecedent (state) instead of the conse-quent (action). He proposes a competition amongrules with the same antecedent but a cooperationamong rules with different ones. Besides, the implica-tion and aggregation considered in his proposal leadsimplicitly to a cooperation between the consequents.

4 Fuzzy-XCS

The use of competitive fuzzy rules seems to solve theproblems shown by the cooperative approach (Sec-tion 2.4). Let us now see how Fuzzy-XCS works.

4.1 Generalization Representation

First of all, a representation of fuzzy classifiers to allowproper generalization must be done. We propose theuse of disjunctive normal form (DNF) fuzzy rules withthe following structure:

IF X1 is A1 and . . . and Xn is An THEN Y is B

where each input variable Xi takes as a value a set oflinguistic terms Ai = {Ai1∨. . .∨Aili}, whose membersare joined by a disjunctive (t-conorm) operator, whilstthe output variable remains a usual linguistic variablewith a single label associated. We use the bounded sum(min{1, a + b}) as t-conorm.

This structure uses a more compact description thatallows rules with different generalization degrees.Moreover, the structure naturally supports the ab-sence of some input variables in each rule (simply mak-ing Ai be the whole set of linguistic terms).

In order to use this representation in Fuzzy-XCS, wepropose a binary coding scheme for the antecedent ofthe fuzzy rule with size equal to the sum of the num-ber of linguistic terms used in each input variable. Theallele ‘1’ means that the corresponding linguistic termis used in the corresponding variable. For the con-sequent of the rule, an integer coding scheme is usedwhere each gene contains the index of the linguisticterms used for the corresponding output variable. Forexample, assuming we have three linguistic terms (S,M, and L) for each input/output variable, the fuzzyrule [IF X1 is S and X2 is {M or L} THEN Y1 is Mand Y2 is L] is encoded as [100|011||23].

4.2 Performance Component

In XCS [8], the performance component consists ofthree stages: match set construction, prediction arraycomputation, and action set selection. This processhas the final objective of inferring an specific actionfrom the set of classifiers that matches the currentstate. In Fuzzy-XCS the process is different, as de-scribed as follows:


•Match set : To avoid an excessive number of matchedrules, only those rules with a matching degree greateror equal to a specific value (θM ) are included in thematch set.

• Computation of candidate subsets: This stage couldbe equivalent to the prediction array computation. Itis necessary in discrete output systems like XCS togroups the matched classifiers according to the differ-ent actions, since it is not possible to select severalactions. However, in real-valued output systems likeFuzzy-XCS, several “linguistic actions” (consequents)could be considered together. Thus, Fuzzy-XCS re-defines the concept of prediction array computation.We can assume that what should not be accepted inour case is to have an action set with inconsistentrules, i.e. rules with the same antecedent and differ-ent consequent. Since DNF-type rules are considered,to have the same antecedent also involves rules wherethe antecedent of some of them are contained in oth-ers. Therefore, different groups of consistent fuzzy ruleset (with the maximum number of rules in each group)are formed.

• Action set selection: The action set selection choosesthe consistent classifier set with the highest mean pre-diction. In learning classifier systems the criterionused to select the action set alternately varies betweenexploration and exploitation approaches. According tothis, our deterministic selection could be considered asa purely exploitation approach. Although we have ex-perimented with several possibilities, the best resultswere obtained with this procedure.

4.3 Reinforcement Component

The pj (prediction), εj (prediction error), and Fj (fit-ness) values are adjusted by the reinforcement learn-ing standard techniques used in XCS (Q-learning,Widrow-Hoff, and MAM) for each fuzzy classifier Cj .

However, an important difference is considered inFuzzy-XCS: the distribution among the classifiersmust be made proportionally to the degree of contri-bution of each classifier to the obtained output. Thisis a crucial issue because the interaction among theclassifiers (with competition actions in our case) is de-veloped here.

The reinforcement performed in Fuzzy-XCS acts onthe action set AS. The following subsections detail thereinforcement distribution process and the adjustmentof the parameters.

4.3.1 Distribution of the Reinforcement

The reinforcement distribution among the classifiersof the action set AS is made by analyzing the con-

tribution of each classifier to generate the aggregatedoutput fuzzy set. Let B′

j be the scaled output fuzzyset generated by the fuzzy rule Rj :

B′j = I(µARj

(x), Bj), (1)

where µARj(x) is the matching degree of the rule Rj ,

I is the used logical fuzzy implication operator (seeTable 1), and Bj the fuzzy set of the consequent ofthe rule Rj . Let R1 ∈ AS be the winner rule, i.e.,µAR1

(x) ≥ µARj(x), ∀Rj ∈ AS − {R1}.

The process involves analyzing the area that the rivalfuzzy rules (rules with lower matching degrees thanR1) “bite” into the area generated by the winner rule.

Thus, the weight of the winner rule is:

w1 =

∫ ∧|AS|j=1 µB′

j(y) dy∫

µB′1(y) dy, (2)

with |AS| being the action set size and ∧ the t-normused as aggregation operator (the minimum in ourcase), while the weights of the rival rules are computedas follows:

wj =(1− w1) ·

(∫µB′1(y) dy − ∫

µB′1(y) ∧ µB′j(y) dy

)

∑|AS|i=2

(∫µB′1(y) dy − ∫

µB′1(y) ∧ µB′i(y) dy

)

(3)

This distribution is designed to take into account thatcompetitive inference is being considered. To illustratethis behavior, we can see that, from the example shownin Figure 2(a), the competitive inference generates theweights w1 = 0.711, w2 = 0.289, and w3 = 0, whilea cooperative-based distribution proportional to thematching degrees generates the weights w1 = 0.409,w2 = 0.318, and w3 = 0.273. Indeed, the selectionpressure in the competitive inference is higher, thusallowing to discriminate between good and bad rules.

4.3.2 Adjustment of the Parameters

To adjust the parameters of each classifier, firstly theP (payoff) value is computed with the Q-learning tech-nique as follows: P = r + (γ · µAR1

(x)), with r beingthe external reward from the previous time-step, andγ ∈ [0, 1] a constant decreasing factor.

Then, the following adjustment process is performedfor each fuzzy classifier belonging to the action set:

1. Firstly, adjust the error values εj using the standardWidrow-Hoff delta rule with learning rate parameterβ (0 < β ≤ 1) toward |P − pj | considering the weightswj computed in eqs. (2) and (3) to distribute the ad-justment, i.e.

εj ← εj + β · wj · (|P − pj | − εj). (4)


The MAM (moyenne adaptive modifiee) technique isused by adjusting εj to the average of the |P − pj |values, instead of the above equation, during the first1/β times that the corresponding classifier is adjusted.

2. Then, adjust prediction values

pj ← pj + β · wj · (P − pj). (5)

Again, weights are considered to distribute the rein-forcement. MAM technique is also used here duringthe 1/β first adjustments.

3. Finally, recalculate the fitness values Fj from theupdated values of εj , i.e.

Fj ← Fj + β · (k′j − Fj), (6)

with

k′j =kj∑

Ri∈AS ki, kj =

{(εj/ε0)−ν εj > ε01 otherwise.

(7)

No weights wj are considered to update the fitnesssince it depends on the prediction error instead of thereceived payoff. Again, MAM technique is used.

4.4 Discovery Component

The EA for Fuzzy-XCS acts only on the action set. Itselects two classifiers with probabilities proportional totheir fitness, applies crossover and mutation operatorswith probabilities χ and µchom (per chromosome), re-spectively, and inserts the offspring in the population.If the population contains the maximum number offuzzy classifiers allowed, two individuals are deleted tomake room. They are randomly selected proportion-ally to the prediction error weighted by the mean ac-tion set sizes where each fuzzy classifier was involved.

A simple two-point crossover operator that only actson the antecedent part of the chromosomes (binarycoding scheme) is considered. Prediction, predictionerror, and fitness values of offspring are initialized tothe mean values of the parents.

The mutation randomly selects an input/output vari-able of the rule. If an input variable is selected, one ofthe three following possibilities is applied: expansion,which flips to 1 a gene of the selected variable; contrac-tion, which flips to 0 a gene of the selected variable; orshift, which flips to 0 a gene of the variable and flipsto 1 the gene immediately before or after it. The se-lection of one of these mechanisms is made randomlyamong the available choices (e.g., contraction can notbe applied if only a gene of the selected variable has theallele 1). If an output variable is selected, the muta-tion operator simply increases or decreases the integer

value. Prediction, prediction error, and fitness valuesof mutated classifiers are not changed.

An EA subsumption is performed. Thus, if the off-spring is logically contained by either of its parents andthis parent is sufficiently experienced (it has been up-dated a threshold θGA number of times), the offspringis not added to the pool but the parent’s numerosityis incremented.

When no fuzzy rules cover the state with the highestmatching degree, a covering mechanism is used to in-clude a fuzzy classifier with the input linguistic termset that best matches the state and a random action.

5 Experimental Results

Experiments have been performed to test the behaviorof Fuzzy-XCS. We have developed a laboratory prob-lem to play a similar role as the multiplexer problemfor discrete-valued classifier systems. Thus, we havegenerated a example data set from a previously definedrule base with different degrees of generalization. Twoinput variables and one output variable are considered.A total of 576 examples uniformly distributed in theinput space (24 × 24) were generated. Five linguisticterms are considered for each variable. Uniformly dis-tributed triangular-shaped membership functions areused. The rule base considered to generate the dataset is shown in Table 3. The reward depends inverselyon the difference between the inferred and the desiredoutput in a non-linear way. The objective is to obtainthe set of rules that best approximate the data withthe highest degree of generalization, i.e., a rule base asaccurate and compact as possible.

Table 3: Rule base used to generate the data setX1 X2 Y

VS S M L VL VS S M L VL VS S M L VLR1 x x x x xR2 x x x x x xR3 x x x x xR4 x x x x xR5 x x x x x x x

The used parameter values are the following: maxi-mum number of classifiers = 200, β = 0.2, ν = 3,ε0 = 0.05, α = 0.1, θDel = 30, θGA = 30, χ = 0.8,µchrom = 0.1, θtimestep = 20, θM = 0.3, also operator= min, and then operator = ÃLukasiewicz. γ parameteris not considered since it is a simple-step problem.

Figure 3 shows the average behavior of 10 runs ofFuzzy-XCS with competitive inference and reinforce-ment. The upper figure depicts the relative numeros-ity of the five optimum fuzzy classifiers. It shows thecapability of the algorithm to find and keep the op-


timum solution. The bottom figure depicts the meanapproximation error of the last 50 iterations. It showsthe capability of the algorithm to provide the appropri-ate action (output) to the corresponding state (input).Note that the optimum solution is properly found.

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Figure 3: Results of Fuzzy-XCS with competitive in-ference and reinforcement approach

On the other hand, Figure 4 shows the behavior ofan algorithm following a cooperative inference and re-inforcement. To do that, the only changes made tothe algorithm proposed in this paper are the use ofminimum as implication and maximum as aggrega-tion in the inference engine, and a distribution of thereward proportional to the matching degrees in the re-inforcement component. This algorithm is tested witha data set generated with the same cooperative in-ference (Max-Min) to avoid biasing the experiment.Notice that the cooperative approach is unable to findthe solution.

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Figure 4: Results of Fuzzy-XCS with cooperative in-ference and reinforcement approach

6 Concluding Remarks

The paper has presented a proposal to properly de-velop an accuracy-based fuzzy classifier system. It ismainly based on a different inference approach thatconsiders the interaction among fuzzy classifiers (rules)from a competitive point of view. The reinforcementcomponent, based on XCS, is adapted to allow this be-havior. The approach has the advantage of performinga higher selection pressure that results in a proper dis-crimination between good and bad fuzzy classifiers.

Promising results of the proposal have been obtainedin a simple laboratory problem. Current and futurework involves investigating the behavior of the pro-posal in multi-step and real-world problems.

References

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