Rough Set Overview of Rough Sets Theoretical Aspects

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Rough Set Overview of Rough Sets (Theoretical Aspects of Reasoning about Data) by Zdzislaw

Rough Set Overview of Rough Sets (Theoretical Aspects of Reasoning about Data) by Zdzislaw Pawlak

Contents 1. Introduction 2. Basic concepts of Rough Sets § § § § information

Contents 1. Introduction 2. Basic concepts of Rough Sets § § § § information system equivalence relation / equivalence class / indiscernibility relation set approximation (Lower & Upper Approximations) - accuracy of Approximation - extension of the definition of approximation of sets dispensable & indispensable - reducts and core dispensable & indispensable attributes - independent - relative reduct & relative core dependency in knowledge - partial dependency of attribute (knowledge) - significance of attributes discernibility Matrix 3. Example § § decision Table dissimilarity Analysis 2

Introduction § Rough Set theory § by Zdzislaw Pawlak in the early 1980’s §

Introduction § Rough Set theory § by Zdzislaw Pawlak in the early 1980’s § use : AI, information processing, data mining, KDD etc. ex) feature selection, feature extraction, data reduction, decision rule generation and pattern extraction (association rules) etc. § theory for dealing with information with uncertainties. reasoning from imprecision data. more specifically, discovering relationships in data. § The idea of the rough set consists of the approximation of a set(X) by a pair of sets, called the low and the upper approximation of this set(X) 3

Information System § Knowledge Representation System ( KR-system , KRS ) § Information systems

Information System § Knowledge Representation System ( KR-system , KRS ) § Information systems § I = < U, Ω> § a finite set U of objects , U={x 1, x 2, . . , xn} ( universe ) § a finite set Ω of attributes , Ω = {q 1, q 2, . . , qm} ={ C, d } § C : set of condition attribute § d : decision attribute § ex) I = < U, {a, c, d}> U a c d x 1 x 2 x 3 x 4 x 5 x 6 x 7 1 4 yes 1 1 no 2 2 no 2 3 1 3 2 Yes 3 no 3 yes 3 no 4

Equivalence Relation § An equivalence relation R on a set U is defined as

Equivalence Relation § An equivalence relation R on a set U is defined as i. e. a collection R of ordered pairs of elements of U, satisfying certain properties. 1. Reflexive: x. Rx for all x in U , 2. Symmetric: x. Ry implies y. Rx for all x, y in U 3. Transitive: x. Ry and y. Rz imply x. Rz for all x, y, z in U 5

Equivalence Class § R : any subset of attributes( ) § If R is

Equivalence Class § R : any subset of attributes( ) § If R is an equivalence relation over U, then U/R is the family of all equivalence classes of R § : equivalence class in R containing an element ex) an subset of attribute ‘R 1={a}’ is equivalence relation the family of all equivalence classes of {a} : U/ R 1 ={{ 1, 2, 6}{3, 4}{5, 7}} equivalence class U 1 2 3 4 5 6 7 a c d 1 1 2 2 3 1 3 4 1 2 2 3 3 3 yes no no Yes no yes no : ※ A family of equivalence relation over U will be call a knowledge base over U 6

Indiscernibility relation § If and then , is also an equivalence relation, ( IND(R)

Indiscernibility relation § If and then , is also an equivalence relation, ( IND(R) ) and will be called an indiscernibility relation over R ※ : intersection of all equivalence relations belonging to R § equivalence class of the equivalence relation IND(R) : § U/IND(R) : the family of all equivalence classes of IND(R) ex) an subset of attribute ‘R={a, c}’ the family of all equivalence classes of {a} : U/{a}={{ 1, 2, 6}{3, 4}{5, 7}} the family of all equivalence classes of {b} : U/{c}={{ 2}{3, 4}{5, 6, 7}{1}} the family of all equivalence classes of IND(R) : U/IND(R) ={{2}{6}{1} {3, 4}{5, 7}}=U/{a, c} U 1 2 3 4 5 6 7 a c d 1 1 2 2 3 1 3 4 yes 1 no 2 Yes 3 no 3 yes 3 no 7

§ I = <U, Ω> = <U, {a, c}> U is a set (called

§ I = <U, Ω> = <U, {a, c}> U is a set (called the universe) Ω is an equivalence relation on U (called an indiscernibility relation). § U is partitioned by Ω into equivalence classes, U a c elements within an equivalence class are indistinguishable in I. 1 1 4 2 § An equivalence relation induces a partitioning of the universe. 3 4 § The partitions can be used to build new subsets of the universe. 6 5 7 1 2 2 1 3 3 ※ equivalence classes of IND(R) are called basic categories (concepts) of knowledge R ※ even union of R-basic categories will be called R-category 8 1 2 2 3 3 3

Set Approximation § Given I = <U, Ω> § Let and R : equivalence

Set Approximation § Given I = <U, Ω> § Let and R : equivalence relation § We can approximate X using only the information contained in R by constructing the R-lower( ) and R-upper( ) approximations of X, where or § X is R-definable (or crisp) if and only if ( i. e X is the union of some R-basic categories, called R-definable set, R-exact set) § X is R-undefinable (rough) with respect to R if and only if ( called R-inexact, R-rough) 9

U U U/R R : subset of attributes set X ∴ X is R-definable

U U U/R R : subset of attributes set X ∴ X is R-definable set X ∴ X is R-rough (undefinable) § R-positive region of X : § R-borderline region of X : § R-negative region of X : 10

EX) I = <U, Ω>, let R={a, c} , X={x | d(x) = yes}={1,

EX) I = <U, Ω>, let R={a, c} , X={x | d(x) = yes}={1, 4, 6} ► approximate set X using only the information contained in R the family of all equivalence classes of IND(R) : U/IND(R) = U/R = {{1}{ 2}{6} {3, 4}{5, 7}} R-lower approximations of X : U 1 a c d 1 4 yes 2 1 1 no 3 4 5 6 7 2 2 3 1 3 2 no 2 Yes 3 no 3 yes 3 no ※ The set X is R-rough since the boundary region is not empty 11

Lower & Upper Approximations {x 2, x 5, x 7} {x 3, x 4}

Lower & Upper Approximations {x 2, x 5, x 7} {x 3, x 4} yes {x 1, x 6} yes/no no 12

Accuracy of Approximation § accuracy measure αR(X) : the degree of completeness of our

Accuracy of Approximation § accuracy measure αR(X) : the degree of completeness of our knowledge R about the set X § If , the R-borderline region of X is empty and the set X is R-definable (i. e X is crisp with respect to R). § If , the set X has some non-empty R-borderline region and X is R-undefinable (i. e X is rough with respect to R). ex) let R={a, c} , X={x | d(x) = yes}={1, 4, 6} 13

§ R-roughness of X : the degree of incompleteness of knowledge R about the

§ R-roughness of X : the degree of incompleteness of knowledge R about the set X ex) let R={a, c} , X={x | d(x) = yes}={1, 4, 6} Y={x | d(x) = no}={2, 3, 6, 7} U/IND(R) = U/R = {{1}{ 2}{6} {3, 4}{5, 7}} 14

Extension of the definition of approximation of sets § F={X 1, X 2, .

Extension of the definition of approximation of sets § F={X 1, X 2, . . . , Xn} : a family of non-empty sets and => R-lower approximation of the family F : R-upper approximation of the family F : ex) R={a, c} F={X, Y}={{1, 4, 6}{2, 3, 5, 7}} , X={x | d(x) = yes}, Y={x | d(x) = no} U/IND(R) = U/R = {{1}{ 2}{6} {3, 4}{5, 7}} U 1 a c d 1 4 yes 2 1 1 no 3 4 5 6 7 2 2 2 no 2 Yes 3 3 no 1 3 3 3 yes no 15

§ the accuracy of approximation of F : the percentage of possible correct decisions

§ the accuracy of approximation of F : the percentage of possible correct decisions when classifying objects employing the knowledge R § the quality of approximation of F : the percentage of objects which can be correctly classified to classes of F employing the knowledge R ex) R={a, c} F={X, Y}={{1, 4, 6}{2, 3, 6, 7}} , X={x | d(x) = yes}, Y={x | d(x) = no} 16

Dispensable & Indispensable § § Let R be a family of equivalence relations let

Dispensable & Indispensable § § Let R be a family of equivalence relations let if IND(R) = IND(R-{a}), then a is dispensable in R if IND(R) ≠ IND(R-{a}), then a is indispensable in R § the family R is independent if each is dependent is indispensable in R ; otherwise R ex) R={a, c} U/{a}={{ 1, 2, 6}{3, 4}{5, 7}} U/{b}={{ 2}{3, 4}{5, 6, 7}{1}} U/IND(R) ={{1}{ 2}{6} {3, 4}{5, 7}}=U/{a, b} ∴ a, b : indispensable in R ∴ R is independent (∵ U/IR ≠ U/{b}, U/IR ≠ U/{a}) U 1 2 3 4 5 6 7 a c 1 4 1 2 2 3 3 1 3 3 3 17

Core & Reduct § the set of all indispensable relation in R => the

Core & Reduct § the set of all indispensable relation in R => the core of R , ( CORE(R) ) § is a reduct of R if Q is independent and IND(Q) = IND(R) , ( RED(R) ) ex) a family of equivalence relations R={P, Q, R} U/P ={{1, 4, 5}{2, 8}{3}{6, 7}} U/{P, Q}= U/R =>R is dispensable in R U/Q ={{1, 3, 5}{6}{2, 4, 7, 8}} U/{P, R} }= U/R => Q is dispensable in R U/R ={{1, 5}{6}{2, 7, 8}{3, 4}} U/{Q, R} }≠ U/R => P is indispensable in R U/{P, Q}={{1, 5}{4}{{2, 8}{3}{6}{7}} ∴DORE(R) ={P} U/{P, R}={{1, 5}{4}{2, 8}{3}{6}{7}} ∴RED(R) = {P, Q} and {P, R} U/{Q, R}={{1, 5}{3}{6}{2, 7, 8}{4}} U/R={{1, 5}{6}{2, 8}{3}{4}{7}} (∵ U/{P, Q}≠U/{P} , U/{P, Q}≠U/{Q} U/{P, R}≠U/{P} , U/{P, R}≠U/{R} ) ※ a reduct of knowledge is its essential part. ※ a core is in a certain sense its most important part. 18

Dispensable & Indispensable Attributes Let R and D be families of equivalence relation over

Dispensable & Indispensable Attributes Let R and D be families of equivalence relation over U, if , then the attribute a is dispensable in I , if , then the attribute a is indispensable in I , The R-positive region of D : 19

ex) R={a, c} D={d} U a c d U/{a}={{ 1, 2, 6}{3, 4}{5, 7}}

ex) R={a, c} D={d} U a c d U/{a}={{ 1, 2, 6}{3, 4}{5, 7}} 1 1 4 yes U/{c}={{ 2}{3, 4}{5, 6, 7}{1}} 2 1 1 no 3 2 2 no 4 2 2 Yes 5 3 3 no 6 1 3 yes 7 3 3 no U/D={{1, 4, 6}{2, 3, 5, 7}} U/IND(R) ={{1}{ 2}{6} {3, 4}{5, 7}}=U/{a, c} => the relation ‘a’ is indispensable in R (‘a’ is indispensable attribute) => the relation ‘c’ is indispensable in R (‘c’ is indispensable attribute) 20

Independent § If every c in R is D-indispensable, then we say that R

Independent § If every c in R is D-indispensable, then we say that R is D-independent (or R is independent with respect to D) ex) R={a, c} D={d} ∴ R is D-independent ( , ) U a c d 1 1 4 yes 2 1 1 no 3 2 2 no 4 2 2 Yes 5 3 3 no 6 1 3 yes 7 3 3 no 21

Relative Reduct & Relative Core § The set of all D-indispensable elementary relation in

Relative Reduct & Relative Core § The set of all D-indispensable elementary relation in R will be called the D-core of R, and will be denoted as CORED(R) ※ a core is in a certain sense its most important part. § The set of attributes is called a reduct of R, if C is the D-independent subfamily of R and => C is a reduct of R ( REDD(R) ) ※ a reduct of knowledge is its essential part. ※ REDD(R) is the family of all D-reducts of R ex) R={a, c} D={d} CORED(R) ={a, c} REDD(R) ={a, c} 22

An Example of Reducts & Core POSR(D)={{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4,

An Example of Reducts & Core POSR(D)={{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4, 5, 6} POSR-{a}(D)={{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4, 5, 6} POSR-{b}(D)={{{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4, 5, 6} POSR-{c}(D)={{5}}={5} • relation ‘a’, ‘b’ is dispensable • relation ‘c’ is indispensable => D-core of R =CORED(R)={c} U={U 1, U 2, U 3, U 4, U 5, U 6} =let {1, 2, 3, 4, 5, 6} Ω={headache, Muscle pan, Temp, Flu}={a, b, c, d} to find reducts of R={a, b, c} • {a, c} is D-independent and POS{a, c}(D)=POSR(D) condition R={a, b, c}, decision D={d} (∵POS{a}(D)={} ≠POS{a, c}(D) U/{a}={{1, 2, 3}{4, 5, 6}} POS{c}(D)={1, 4, 3, 6} ≠POS{a, c}(D) ) U/{b}={{1, 2, 3, 4, 6}{5}} U/{c}={{1, 4}{2, 5}{3, 6}} U/{a, b}={1, 2, 3}{4, 6}{5}} U/{a, c}={{1}{2}{3}{4}{5}{6}} U/{b, c}={{1, 4}{2}{3, 6}{5}} U/R={{1}{4}{2}{5}{3}{6}} U/D={{1, 4, 5}{2, 3, 6}} • {b, c} is D-independent and POS{b, c}(D)=POSR(D) => {a, c} {b, c} is the D-reduct of R POSR-{ab}(D)={{1, 4}{3, 6}}={1, 4, 3, 6} POSR-{ac}(D)={{5}}={5} POSR-{bc}(D)={} 23

Reduct 1 = {Muscle-pain, Temp. } CORE = {Headache, Temp} ∩ {Muscle Pain, Temp}

Reduct 1 = {Muscle-pain, Temp. } CORE = {Headache, Temp} ∩ {Muscle Pain, Temp} Reduct 2 = {Headache, Temp. }    = {Temp} 24

Dependency in knowledge § Given knowledge P, Q § U/P={{1, 5}{2, 8}{3}{4}{6}{7}} § U/Q={{1,

Dependency in knowledge § Given knowledge P, Q § U/P={{1, 5}{2, 8}{3}{4}{6}{7}} § U/Q={{1, 5}{2, 7, 8}{3, 4, 6}} § If , then Q depends on P (P⇒Q) 25

Partial Dependency of knowledge § I=<U, Ω> and § Knowledge Q depends in a

Partial Dependency of knowledge § I=<U, Ω> and § Knowledge Q depends in a degree k (0≤k≤ 1 ) from knowledge P (P⇒k Q) ex) U/Q={{1}{2, 7}{3, 6}{4}{5, 8}} U/P={{1, 5}{2, 8}{3}{4}{6}{7}} POSP(Q) = {3, 4, 6, 7} the degree of dependency between Q and P : § § (P⇒ 0. 5 Q ) If k = 1 we say that Q depends totally on P. If k < 1 we say that Q depends partially (in a degree k) on P. 26

Significance of attributes ex) R={a, b, c}, decision D={d} U/{a}={{1, 2, 3}{4, 5, 6}}

Significance of attributes ex) R={a, b, c}, decision D={d} U/{a}={{1, 2, 3}{4, 5, 6}} U/{a, b}={1, 2, 3}{4, 6}{5}} U/{b}={{1, 2, 3, 4, 6}{5}} U/{a, c}={{1}{2}{3}{4}{5}{6}} U/{c}={{1, 4}{2, 5}{3, 6}} U/{b, c}={{1, 4}{2}{3, 6}{5}} U/R={{1}{4}{2}{5}{3}{6}} U/D={{1, 4, 5}{2, 3, 6}} POSR(D)={{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4, 5, 6} POSR-{a}(D)={{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4, 5, 6} POSR-{b}(D)={{{1, 4, 5}{2, 3, 6}}={1, 2, 3, 4, 5, 6} POSR-{c}(D)={{5}}={5} significance of attribute ‘a’ : significance of attribute ‘b’ : significance of attribute ‘c’ : ∴ the attribute c is most significant, since it most changes the positive region of U/IND(D) 27

Discernibility Matrix § Let I = (U, Ω) be a decision table, with U={x

Discernibility Matrix § Let I = (U, Ω) be a decision table, with U={x 1, x 2, . . , xn} C={a, b, c} : condition attribute set , D={d} : decision attribute set § By a discernibility matrix of I, denoted M(I)={mij}n×n § mij is the set of all the condition attributes that classify objects xi and xj into different classes. < Decision Table > < Discernibility Matrix > U 1 U 2 U 3 U 4 c c - a, c U 5 - a, b, c - - - c U 6 a, c U 4 U 5 U 6 (a) (b) (c) (d) b, c - : same equivalence classes of the relation IND(d) 28

Compute value cores and value reducts from the M(I) § the core can be

Compute value cores and value reducts from the M(I) § the core can be defined now as the set of all single element entries of the discernibility matrix, § is the reduct of R, if B is the minimal subset of R such that for any nonempty entry c ( U 1 U 2 U 3 ) in M(I) U 4 U 2 c U 3 c - U 4 - a, c U 5 - a, b, c - U 6 a, c - - c d-reducts : {a, c} {b, c} U 5 b, c d-CORE(R) 29

CHAPTER 6. Decision Tables 30

CHAPTER 6. Decision Tables 30

 • Proposition 6. 2 Each decision table can be uniquely decomposed into two

• Proposition 6. 2 Each decision table can be uniquely decomposed into two decision tables such that in and where and in , – compute the dependency between condition and decision attributes – decompose the table into two subtables 31

 • Example 1. condition attribute decision attribute Table 2 U a b c

• Example 1. condition attribute decision attribute Table 2 U a b c d e 3 2 0 0 1 1 0 2 2 0 4 1 1 0 2 2 2 0 1 1 1 2 6 2 2 0 1 1 3 2 0 0 1 1 7 2 1 1 1 2 4 1 1 0 2 2 5 1 0 2 0 1 U a b c d e 6 2 2 0 1 1 0 2 2 0 7 2 1 1 1 2 2 0 1 1 1 2 8 0 1 1 0 1 5 1 0 2 0 1 8 0 1 1 0 1 Table 3 • Table 2 is consistent, Table 3 is totally inconsistent → All decision rules in Table 2 are consistent All decision rules in Table 3 are inconsistent 32

 • simplification of decision tables : reduction of condition attributes • steps 1)

• simplification of decision tables : reduction of condition attributes • steps 1) Computation of reducts of condition attributes which is equivalent to elimination of some column from the decision tables 2) Elimination of duplicate rows 3) Elimination of superfluous values of attributes 33

decision attribute condition attribute • Example 2 U a b c d e U

decision attribute condition attribute • Example 2 U a b c d e U a b d e 1 1 0 0 1 1 2 1 0 0 0 1 2 1 0 0 1 3 0 0 0 3 0 0 4 1 1 1 0 5 1 1 0 2 2 5 1 1 2 2 6 2 1 0 2 2 6 2 1 2 2 7 2 2 2 2 remove column c üe-dispensable condition attribute is c. let R={a, b, c, d}, D={e} üCORED(R) ={a, b, d} üREDD(R) ={a, b, d} 34

 • we have to reduce superfluous values of condition attributes in every decision

• we have to reduce superfluous values of condition attributes in every decision rules U a b d → compute the core values 1 1 0 1 1. In the 1 st decision rules • the core of the family of sets e 1 2 1 0 0 1 3 0 0 4 1 1 1 0 5 1 1 2 2 6 2 1 2 2 7 2 2 • the core value is 35

2. In the 2 nd decision rules • the core of the family of

2. In the 2 nd decision rules • the core of the family of sets U a b d e 1 1 0 1 1 2 1 0 0 1 3 0 0 4 1 1 1 0 5 1 1 2 2 • the core value is 6 2 1 2 2 7 2 2 3. In the 3 rd decision rules • the core of the family of sets • the core value is 36

4. In the 4 th decision rules • the core of the family of

4. In the 4 th decision rules • the core of the family of sets U a b d e 1 1 0 1 1 2 1 0 0 1 3 0 0 4 1 1 1 0 5 1 1 2 2 • the core value : 5. In the 5 th decision rules • the core of the family of sets • the core value is 6 2 1 2 2 7 2 2 37

U a b d e 6. In the 6 th decision rules • the

U a b d e 6. In the 6 th decision rules • the core of the family of sets 1 1 0 1 1 2 1 0 0 1 3 0 0 4 1 1 1 0 5 1 1 2 2 6 2 1 2 2 • the core value : not exist 7. In the 7 th decision rules • the core of the family of sets 7 2 2 U a b d e 1 - 0 - 1 2 1 - - 1 3 0 - - 0 4 - 1 1 0 5 - - 2 2 • the core value : not exist 6 - - - 2 7 - - - 2 38

 • to compute value reducts – let’s compute value reducts for the ~

• to compute value reducts – let’s compute value reducts for the ~ 1. 1 st decision rules of the decision table – 2 value reducts 1. 2. – Intersection of reducts : → core value 39

2. 2 nd decision rules of the decision table – 2 value reducts :

2. 2 nd decision rules of the decision table – 2 value reducts : – Intersection of reducts : → core value 3. 3 rd decision rules of the decision table – 1 value reduct : – Intersection of reducts : → core value 40

4. 4 th decision rules of the decision table – 1 value reduct :

4. 4 th decision rules of the decision table – 1 value reduct : – Intersection of reducts : → core value 5. 5 th decision rules of the decision table – 1 value reduct : – Intersection of reducts : → core value 41

6. 6 th decision rules of the decision table – 2 value reducts :

6. 6 th decision rules of the decision table – 2 value reducts : – Intersection of reducts : → core value : not exist 42

7. 7 th decision rules of the decision table – 3 value reducts –

7. 7 th decision rules of the decision table – 3 value reducts – Intersection of reducts : → core value : not exist U a b d 1 1 0 Ⅹ 1 1′ Ⅹ 0 1 2 1 Ⅹ 0 3 0 Ⅹ Ⅹ 0 4 Ⅹ 1 1 0 5 Ⅹ Ⅹ 2 2 6′ 2 Ⅹ Ⅹ 2 7″ 2 1 Ⅹ Ⅹ 2 7′ Ⅹ 2 = 24 solutions to our problem 1 2′ 1 7 – reducts : 0 e 2 Ⅹ Ⅹ 2 43

One solution Another solution U a b d e U a b d 1

One solution Another solution U a b d e U a b d 1 1 0 Ⅹ 1 2 1 Ⅹ 0 1 2 1 0 Ⅹ 1 3 0 Ⅹ Ⅹ 0 4 Ⅹ 1 1 0 5 Ⅹ Ⅹ 2 2 6 Ⅹ Ⅹ 2 2 7 2 Ⅹ Ⅹ 2 7 Ⅹ Ⅹ 2 2 e identical minimal solution U a b d 1, 2 1 0 Ⅹ 1 3 0 Ⅹ Ⅹ 0 4 Ⅹ 1 5, 6, 7 e 1 0 Ⅹ Ⅹ 2 2 enumeration is not essential U a b d 1 1 0 Ⅹ 1 2 0 Ⅹ Ⅹ 0 3 Ⅹ 1 1 0 4 Ⅹ Ⅹ 2 2 44 e

10. 4 Pattern Recognition [ Table 10 ] : Digits display unit in a

10. 4 Pattern Recognition [ Table 10 ] : Digits display unit in a calculator assumed to represent a characterization of “hand written” digits U 0 1 2 3 4 5 6 7 8 9 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 c 1 1 0 1 1 1 1 d 1 0 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 G 0 0 1 1 1 0 1 1 a f b g c e d ▶ Out task is to find a minimal description of each digit and corresponding decision algorithm. 45

compute the core attributes U 0 1 2 3 4 5 6 7 8

compute the core attributes U 0 1 2 3 4 5 6 7 8 9 b c d e 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 [ drop attribute a f 1 0 0 0 1 1 1 0 1 1 g 0 0 1 1 1 0 1 1 U 0 1 2 3 4 5 6 7 8 9 ] a 1 0 1 1 1 c 1 1 0 1 1 1 1 d 1 0 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 g 0 0 1 1 1 0 1 1 [ drop attribute b ] decision rules are inconsistent Rule 1 : b 1 c 1 d 0 e 0 f 0 g 0 → a 0 b 1 c 1 d 0 e 0 f 0 g 0 Rule 7 : b 1 c 1 d 0 e 0 f 0 g 0 → a 1 b 1 c 1 d 0 e 0 f 0 g 0 46

U 0 1 a 1 0 b 1 1 d 1 0 e 1

U 0 1 a 1 0 b 1 1 d 1 0 e 1 0 f 1 0 g 0 0 2 3 4 5 6 7 8 9 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 U a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 c 1 1 0 1 1 1 1 d 1 0 1 1 f 1 0 0 0 1 1 1 0 1 1 g 0 0 1 1 1 0 1 2 3 4 5 6 7 8 9 U ←[drop attribute c] [drop attribute d]→ decision rules are consistent 0 1 2 3 4 5 6 7 8 9 U 0 1 ←[drop attribute e] 2 3 [drop attribute f]→ 4 5 6 decision rules are 7 inconsistent 8 9 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 c 1 1 0 1 1 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 g 0 0 1 1 1 0 1 1 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 c 1 1 0 1 1 1 1 d 1 0 1 1 e 1 0 0 0 1 047 g 0 0 1 1 1 0 1 1

U 0 1 2 3 4 5 6 7 8 9 a 1 0

U 0 1 2 3 4 5 6 7 8 9 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 c 1 1 0 1 1 1 1 d 1 0 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 ∴ attribute c, d : dispensable attribute a, b, e, f, g : indispensable the set {a, b, e, f, g} : core sole reducts : {a, b, e, f, g} [drop attribute g] decision rules are inconsistent 48

compute reduct • all attribute set : {a, b, c, d, e, f, g}

compute reduct • all attribute set : {a, b, c, d, e, f, g} • core : {a, b, e, f, g} • reduct : {a, b, e, f, g} U 0 1 2 3 4 5 6 7 8 9 c 1 1 0 1 1 1 1 d 1 0 1 1 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 G 0 0 1 1 1 0 1 1 49

compute the core values of attributes for table 11 [Table 12 : Removing the

compute the core values of attributes for table 11 [Table 12 : Removing the attribute a ] U 0 1 2 3 4 5 6 7 8 9 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 G 0 0 1 1 1 0 1 1 The core value in rule 1 and 4 : a 0 The core value in rule 7 and 9 : a 1 [Table 13 : Removing the attribute b ] U 0 1 2 3 4 5 6 7 8 9 b 1 1 1 0 0 1 1 1 a 1 0 1 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 G 0 0 1 1 1 0 1 1 The core value in rule 5 and 6 : b 0 The core value in rule 8 and 9 : b 1 50

[ Table 14 : Removing the attribute e ] [ Table 15 : Removing

[ Table 14 : Removing the attribute e ] [ Table 15 : Removing the attribute f ] U 0 1 2 3 4 e 1 0 0 a 1 0 1 1 0 b 1 1 1 f 1 0 0 0 1 G 0 0 1 1 1 5 6 7 8 9 0 1 0 1 1 1 1 1 0 1 1 U 0 1 2 3 4 5 6 7 8 9 f 1 0 0 0 1 1 1 0 1 1 a 1 0 1 1 1 b 1 1 1 0 0 1 1 1 e 1 0 0 0 1 0 G 0 0 1 1 1 0 1 1 U The core value in rule 2 , 6 and 8 : e 1 The core value in rule 3 , 5 and 9 : e 0 The core value in rule 2 and 3 : f 0 The core value in rule 8 and 9 : f 1 0 1 2 3 4 5 6 7 8 9 g 0 0 1 1 1 0 1 1 a 1 0 1 1 1 [ Table 16 : Removing the attribute g ] b 1 1 1 0 0 1 1 1 e 1 0 0 0 1 0 f 1 0 0 0 1 1 1 0 1 1 51

core value for all decision rules [Table 17]: all core value for table 11

core value for all decision rules [Table 17]: all core value for table 11 U a b e 0 1 2 3 4 5 6 7 8 9 _ 0 _ _ 1 _ 0 0 _ 1 1 _ _ 1 0 _ 0 1 _ 1 0 f G _ 0 _ 0 1 _ _ _ _ 0 1 1 1 _ [Table 11] U a b e 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 0 0 1 1 1 • rule 2, 3, 5, 6, 8, 9 are consistent. • rule 0, 1, 4, 7 are inconsistent. 1 0 0 0 1 0 f G 1 0 0 1 0 1 1 1 1 0 0 1 1 [Table 18] All possible value reduct for table 11 U 0 0’ 1 1’ 2 3 4 4’ 5 6 7 7’ 8 9 a x x 0 0 x x 1 1 x 1 b x x x x 0 0 x x 1 1 e 1 x x x 1 0 x x 0 1 0 x 1 0 f x 1 0 x 0 0 1 x x 0 1 1 g 0 0 x 1 X 1 x x 0 0 1 x • to make the rules consistent ⇒ adding proper additional attributes 52

Table 18 U 0 0’ 1 1’ 2 3 4 4’ 5 6 7

Table 18 U 0 0’ 1 1’ 2 3 4 4’ 5 6 7 7’ 8 9 a x x 0 0 x x 1 1 x 1 b x x x x 0 0 x x 1 1 e 1 x x x 1 0 x x 0 1 0 x 1 0 f x 1 0 x 0 0 1 x x 0 1 1 g 0 0 x 1 X 1 x x 0 0 1 x • We have 16=24 minimal decision algorithms. • One of the possible reduced algorithms : e 1 g 0 (f 1 g 0) → 0 a 0 f 0 (a 0 g 0) → 1 a e 1 f 0 → 2 e 0 f 0 g 1 → 3 a 0 f 1 (a 0 g 1) → 4 f b g b 0 e 0 → 5 b 0 e 1 → 6 a 1 e 0 g 0 (a 1 f 0 g 0) → 7 b 1 e 1 f 1 g 1 → 8 c e d a 1 b 1 e 0 f 1 → 9 53