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equivalence relation. The example clause can be viewed as indicating the value of each feature the extent to which the feature discriminates between the two objects i and j . The core of the dataset is de ned as Core(C) = {a C| Cij , Cij (a) > 0, f {C a} Cij (f ) = 0} (9.25)

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9.2.5.2 Fuzzy Discernibility Function As with the crisp approach, the entries in the matrix can be used to construct the fuzzy discernibility function:

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fD (a1 , . . . , am ) = { Cij |1 j < i |U|}

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(9.26)

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where Cij = {ax |ax Cij }. The function returns values in [0, 1], which can be seen to be a measure of the extent to which the function is satis ed for a given assignment of truth values to variables. To discover reducts from the fuzzy discernibility function, the task is to nd the minimal assignment of the value 1 to the variables such that the formula is maximally satis ed. By setting all variables to 1, the maximal value for the function can be obtained as this provides the most discernibility between objects. Crisp discernibility matrices can be simpli ed by removing duplicate entries and clauses that are supersets of others. A similar degree of simpli cation can be achieved for fuzzy discernibility matrices. Duplicate clauses can be removed as a subset that satis es one clause to a certain degree will always satisfy the other to the same degree.

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9.2.5.3 Decision-Relative Fuzzy Discernibility Matrix As with the crisp discernibility matrix, for a decision system the decision feature must be taken into account for achieving reductions; only those clauses with different decision values are included in the crisp discernibility matrix. For the fuzzy version, this is encoded as

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fD (a1 , . . . , am ) = { {{ Cij } qN ( Rq (i,j )) }|1 j < i |U|}

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(9.27)

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for decision feature q, where denotes fuzzy implication. This construction allows the extent to which decision values differ to affect the overall satis ability of the clause. If Cij (q) = 1 then this clause provides maximum discernibility (i.e., the two objects are maximally different according to the fuzzy similarity measure). When the decision is crisp and crisp equivalence is used, Cij (q) becomes 0 or 1.

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9.2.5.4 Reduction For the purposes of nding reducts, use of the fuzzy intersection of all clauses in the fuzzy discernibility function may not provide enough information for evaluating subsets. Here it may be more informative to

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consider the individual satisfaction of each clause for a given set of features. The degree of satisfaction of a clause Cij for a subset of features P is de ned as SATP (Cij ) =

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{ Cij (a)}

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(9.28)

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Returning to the example, if the subset P = {a, c} is chosen, the resulting degree of satisfaction of the clause is SATP (Cij ) = {0.4 0.2} = 0.6 using the ukasiewicz t-conorm, min(1, x + y). For the decision-relative fuzzy indiscernibility matrix, the decision feature q must be taken into account also: SATP ,q (Cij ) = SATP (Cij ) Cij (q) (9.29)

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For the example clause, if the corresponding decision values are crisp and are different, the degree of satisfaction of the clause is SATP ,q (Cij ) = SATP (Cij ) 1 = 0.6 1 = 0.6 For a subset P , the total satis ability of all clauses can be calculated as SAT(P ) =

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i,j U,i=j i,j U,i=j

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SATP ,q (Cij ) SATC,q (Cij )

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(9.30)

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where C is the full set of conditional attributes, and hence the denominator is a normalizing factor. If this value reaches 1 for a subset P , then the subset is a fuzzy-rough reduct. A proof of the monotonicity of the function SAT(P ) can be found in Section 9.4. Many methods available from the literature for the purpose of nding reducts for crisp discernibility matrices are applicable here also. The Johnson reducer [253] is extended and used herein to illustrate the concepts involved. This is a simple greedy heuristic algorithm that is often applied to discernibility functions to nd a single reduct. Subsets of features found by this process have no guarantee of minimality, but are generally of a size close to the minimal. The algorithm begins by setting the current reduct candidate, P , to the empty set. Then each conditional feature appearing in the discernibility function is evaluated according to the heuristic measure used. For the standard Johnson algorithm this is typically a count of the number of appearances a feature makes