Action Rules Discovery Systems: DEAR1, DEAR2, ARED, ….. by Zbigniew W. Raś

LERS Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 (a, a 1 ) (a, a 2 ) (b, b 1 ) (b,b 2 ) ……….. (d,d 1 ) (d,d 2 ) Decision System S atomic terms r = [[(a, a 2 )*(b, b 1 )] → (d, d 1 )] w  Y → w  Z rule Support: Confidence: Y = {x 2, x 4 } Z = {x 1,x 2,x 3,x 4,x 5,x 7 } sup(r) = 2 conf(r) = 2/2 = 1

abcefd bcefd bcefd Splitting the node using the stable attribute Dom(a) = {1,2,3} & Dom(b) = {1,2,3,4,5} All objects have the same decision value, so this sub-table is not analyzed any further None of the objects contain the desired class “1”, so this sub-table stops splitting any further a = 1 a = 2 a = 3 bcefd ced ced b = 1 b = 5 All the flexible values are the same for both objects, therefore this sub-table is not analyzed any further Partition decision table S Stable:{ a, b} Flexible: {c, e, f} Reclassification direction: 2  1 or 3  1 All objects have the same value 8 for attribute f, so it is crossed out from the sub-table ( this condition is used for stable attributes as well) T1T1 T2T2 T3T3 T4T4 T5T5 Action Rules Discovery (Preprocessing)

Table: Set of rules R with supporting objects Figure of (d, H)-tree T1 Figure of (d, L)-tree T2 Objectsabcd x1, x2, x3, x40L x1, x30L x2, x42L 1L x5, x63L x7, x821H 12H Objectsabc x1, x2, x3, x40 x1, x30 x2, x42 1 x5, x63 Objectsbc x1, x30 x2, x42 1 x5, x63 Objectsb x2, x42 x5, x63 c = 1c = ? c = 0 Objectsbc x1, x2, x3, x4 Objectsb x1, x3 a = 0 Objectsb x2, x4 a = ? Objectsabc x7, x Objectsbc x7, x81 a = 2 Objectsbc x7, x812 a = ? Stable Attribute: {a, c} Flexible Attribute: b Decision Attribute: d T1 T2 T3 T4 (T3, T1) : (a = 2)  (b, 2  1)  ( d, L  H) (a = 2)  (b, 3  1)  ( d, L  H) Objectsb x7, x81 c = ? c = 2 Objectsb x7, x81 c = ? Objectsb x1, x2, x3, x4 T5 T6 System DEAR1

Objectsabcd r1x1, x2, x3, x40L r2x1, x30L r3x2, x42L r4x2, x41L r5x5, x63L r6x7, x821H r7x7, x812H Objectsabcd x1, x2, x3, x40L x1, x30L x2, x42L 1L x5, x63L x7, x821H 12H Stable Attribute: b Flexible Attribute: {a, c} Decision Attribute: d Objectsacd x1, x2, x3, x40L x1, x30L x2, x4L 1L b = 2 Objectsacd x1, x2, x3, x40L x1, x30L x2, x41L x5, x6L b = 3 Objectsacd x1, x2, x3, x40L x1, x30L x2, x41L x7, x82H 2H b = 1 Objectsac x1, x2, x3, x40 x1, x30 x2, x41 Objectsac x7, x82 2 d = L d = H Set of rules R with supporting objects (b = 1)  (a, 0  2)  ( d, L  H) (b = 1)  (c, 0  2)  ( d, L  H) (b = 1)  (c, 1  2)  ( d, L  H) System DEAR2

Cost of Action Rule Action rule r: [(b 1, v 1 → w 1 )  (b 2, v 2 → w 2 )  …  ( b p, v p → w p )](x)  (d, k 1 → k 2 )(x) The cost of r in S: cost S (r) =  {  S (v i, w i ) : 1  i  p} Action rule r is feasible in S, if cost S (r) <  S (k 1, k 2 ). For any feasible action rule r, the cost of the conditional part of r is lower than the cost of its decision part.

Example: r = [(b 1, v 1 → w 1 )  …  (b j, v j → w j )  …  ( b p, v p → w p )](x)  (d, k 1 → k 2 )(x) In R S [(b j, v j → w j )] we find r 1 = [(b j1, v j1 → w j1 )  (b j2, v j2 → w j2 )  …  ( b jq, v jq → w jq )](x)  (b j, v j → w j )(x) Then, we can compose r with r 1 and the same replace term (b j, v j → w j ) by term from the left hand side of r 1 : [(b 1, v 1 → w 1 )  …  [(b j1, v j1 → w j1 )  (b j2, v j2 → w j2 )  …  ( b jq, v jq → w jq )]  …  ( b p, v p → w p )](x)  (d, k 1 → k 2 )(x) Cost of Action Rule

ARED Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 (a, a 1 → a 1 ) (a, a 2 → a 2 ) (b, b 1 → b 1 ) (b, b 2 → b 2 ) ……….. (d, d 1 → d 1 ) (d, d 2 → d 2 ) Decision System S atomic action terms r=[(a, a 2 → a 2 )*(b, b 1 → b 1 )] → (d, d 1 → d 1 ) (w, w)  (Y, Y ) → (w,w)  (Z, Z) action rule Support: Confidence: Y = {x 2, x 4 } Z = {x 1,x 2,x 3,x 4,x 5,x 7 } sup(r) = 2 conf(r) = 2/2 = 1

ARED Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 (a, a 1 → a 1 ) (a, a 2 → a 2 ) (b, b 1 → b 1 ) (b, b 2 → b 2 ) ……….. (d, d 1 → d 1 ) (d, d 2 → d 2 ) Decision System S atomic action terms r=[(a, a 2 → a 1 )*(b, b 1 → b 1 )] → (d, d 1 → d 2 ) (w 1, w 2 )  (Y 1, Y 2 ) → (w 1,w 2 )  (Z 1, Z 2 ) action rule Support: Confidence: Y = {x 2, x 4 } Z = {x 1,x 2,x 3,x 4,x 5,x 7 } sup(r) = ? conf(r) = ? Y=(Y 1,Y 2 ), Z=(Z 1,Z 2 ) w = (w 1,w 2 )

ARED Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 (a, a 1 → a 1 ) (a, a 1 → a 2 ) (b, b 1 → b 2 ) (b, b 2 → b 2 ) ……….. (d, d 1 → d 1 ) (d, d 2 → d 2 ) Decision System S atomic action terms r=[(a, a 2 → a 1 )*(b, b 1 → b 1 )] → (d, d 1 → d 2 ) (Y 1, Y 2 ) (Z 1, Z 2 ) action rule Support: Confidence: Y 1 = {x 2, x 4 } Z 1 = {x 1,x 2,x 3,x 4,x 5,x 7 } Y 2 = {x 1, x 6 } Z 2 = { x 6 } sup(r) = 2 conf(r) = 2/2 = 1 Y 1 → Z 1, Y 2 → Z 2

ARED Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 (a, a 1 → a 1 ) (a, a 1 → a 2 ) (b, b 1 → b 2 ) (b, b 2 → b 2 ) ……….. (d, d 1 → d 1 ) (d, d 2 → d 2 ) Decision System S atomic terms r=[(a, a 2 → a 1 )*(b, b 1 → b 1 )] → (d, d 1 → d 2 ) (Y 1, Y 2 ) (Z 1, Z 2 ) rule Y 1 = {x 2, x 4 } Z 1 = {x 1,x 2,x 3,x 4,x 5,x 7 } Y 2 = {x 1, x 6 } Z 2 = { x 6 } sup(r) = 2 conf(r) = 2/2 = 1

ARED Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 (a, a 1 → a 1 ) (a, a 1 → a 2 ) (b, b 1 → b 2 ) (b, b 2 → b 2 ) ……….. (d, d 1 → d 1 ) (d, d 2 → d 2 ) Decision System S atomic terms r=[(a, a 2 → a 1 )*(b, b 1 → b 1 )] → (d, d 1 → d 2 ) (Y 1, Y 2 ) (Z 1, Z 2 ) rule Y 1 = {x 2, x 4 } Z 1 = {x 1,x 2,x 3,x 4,x 5,x 7 } Y 2 = {x 1, x 6 } Z 2 = { x 6 } sup(r) = 2 conf(r) = 2/2 = 1

ARED Meaning of (d,d 1  d 2 ) in S: N S (d,d 1  d 2 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 stable attribute flexible attributes Object reclassification from class d 1 to d 2 λ1=2, λ2=1/4 Atomic classification terms: (b,b 1  b 1 ), (b,b 2  b 2 ), (b,b 3  b 3 ) (a,a 1  a 2 ), (a,a 1  a 1 ), (a,a 2  a 2 ), (a,a 2  a 1 ) (c,c 1  c 2 ), (c,c 2  c 1 ), (c,c 1  c 1 ), (c,c 2  c 2 ) λ1 - minimum support, λ2 - minimum confidence

ARED Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 stable attribute flexible attributes Object reclassification from class d 1 to d 2 λ1=2, λ2=1/4 Notation: t 1 =(b,b 1  b 1 ), t 2 =(b,b 2  b 2 ), t 3 =(b,b 3  b 3 ), t 4 =(a,a 1  a 2 ), t 5 =(a,a 1  a 1 ), t 6 =(a,a 2  a 2 ), t 7 =(a,a 2  a 1 ), t 8 =(c,c 1  c 2 ), t 9 =(c,c 2  c 1 ), t 10 =(c,c 1  c 1 ), t 11 =(c,c 2  c 2 ), t 12 = (d,d 1  d 2 ). λ1 - minimum support, λ2 - minimum confidence

For decision attribute in S: N S (d,d 1  d 2 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Object reclassification from class d 1 to d 2 λ1=2, λ2=1/4 For classification attribute in S: N S (t 1 ) = N S (b,b 1  b 1 ) = [{x 1,x 2, x 4, x 6 }, {x 1,x 2, x 4, x 6 }] N S (t 2 ) = N S (b,b 2  b 2 ) = [{x 3,x 7, x 8 }, {x 3,x 7, x 8 }] N S (t 3 ) = N S (b,b 3  b 3 ) = [{x 5 }, {x 5 }] N S (t 4 ) = N S (a,a 1  a 2 ) = [{x 1,x 6, x 7, x 8 }, {x 2,x 3, x 4, x 5 }] Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 Not marked λ1=3 Mark “-” λ2=0 Mark “-” λ1=1 Mark “-” λ2=0

For decision attribute in S: N S (d,d 1  d 2 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Object reclassification from class d 1 to d 2 λ1=2, λ2=1/4 For classification attribute in S: Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3 Not marked λ1=2 Mark “-” λ2= 0 Mark “+” λ1=4,λ2=1/4 N S (t 5 ) = N S (a,a 1  a 1 ) = [{x 1,x 6, x 7, x 8 }, {x 1,x 6, x 7, x 8 }] N S (t 6 )= N S (a,a 2  a 2 ) = [{x 2,x 3, x 4, x 5 }, {x 2,x 3, x 4, x 5 }] N S (t 7 )= N S (a,a 2  a 1 ) = [{x 2,x 3, x 4, x 5 }, {x 1,x 6, x 7, x 8 }]

For decision attribute in S: N S (t 12 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Object reclassification from class d1 to d2 λ1=2, λ2=1/4 For classification attribute in S: N S (t 1 )=[{x 1,x 2, x 4, x 6 }, {x 1,x 2, x 4, x 6 }] Not marked λ1=3 N S (t 2 )=[{x 3,x 7, x 8 }, {x 3,x 7, x 8 }] Marked “-” λ2=0 N S (t 3 )=[{x 5 }, {x 5 }] Marked “-” λ1=1 N S (t 4 )=[{x 1,x 6, x 7, x 8 }, {x 2,x 3, x 4, x 5 }] Marked “-” λ2=0 N S (t 5 )=[{x 1,x 6, x 7, x 8 }, {x 1,x 6, x 7, x 8 }] Not marked λ1=2 N S (t 6 )=[{x 2,x 3, x 4, x 5 }, {x 2,x 3, x 4, x 5 }] Marked “-” λ2=0 Mark “+” λ1=4, λ2=1/4 N S (t 7 )=[{x 2,x 3, x 4, x 5 }, {x 1,x 6, x 7, x 8 }] r = [t 7  t 1 ] Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3

For decision attribute in S: N S (t 12 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Object reclassification from class d1 to d2 λ1=2, λ2=1/4 For classification attribute in S: N S (t 8 )= N S (c,c 1  c 2 ) = [{x 1,x 4, x 8 }, {x 2, x 3, x 5, x 6, x 7 }] Not marked Marked “-” N S (t 10 ) = N S (c,c 1  c 1 ) = [{x 1, x 4, x 8 }, {x 1, x 4, x 8 }] Marked “-” N S (t 11 ) = N S (c,c 2  c 2 )= [{x 2, x 3, x 5, x 6, x 7 }, {x 2, x 3, x 5, x 6, x 7 }] Not marked conf = 2/3 *1/5 <λ 2 N S (t 9 ) = N S (c,c 2  c 1 ) = [{x 2, x 3, x 5, x 6, x 7 }, {x 1, x 4, x 8 }] Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3

For decision attribute in S: N S (t 12 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Object reclassification from class d1 to d2λ1=2, λ2=1/4 For classification attribute in S: Marked “+” N S (t 1 *t 11 )=[{x 2, x 6 }, {x 2, x 6 }] Marked “-”, λ1=1 N S (t 5 *t 8 )=[{x 1, x 8 }, {x 6, x 7 }] Marked “-”, λ1=1 Rule r = [t 1 *t 8 →t 12 ], conf = 1/2 ≥ λ 2, sup=2 ≥ λ 1 Now action terms of length 2 from unmarked action terms of length 1 N S (t 1 *t 5 )=[{x 1, x 6 }, {x 1, x 6 }] Marked “-”, λ1=1 N S (t 1 *t 8 )=[{x 1, x 4 }, {x 2, x 6 }] N S (t 5 *t 11 )=[{x 6, x 7 }, {x 6, x 7 }] Marked “-”, λ1=1 N S (t 8 *t 11 )=[ Ø, {x 2, x 3, x 5, x 6, x 7 }] Marked “-” N S (t 1 )=[{x 1,x 2, x 4, x 6 }, {x 1,x 2, x 4, x 6 }], N S (t 5 )=[{x 1,x 6, x 7, x 8 }, {x 1,x 6, x 7, x 8 }], N S (t 8 )=[{x 1,x 4, x 8 }, {x 2, x 3, x 5, x 6, x 7 }], N S (t 11 )= [{x 2, x 3, x 5, x 6, x 7 }, {x 2, x 3, x 5, x 6, x 7 }].

ARED Algorithm For decision attribute in S: N S (t 12 )=[{x 1,x 2, x 3, x 4, x 5, x 7 }, {x 6 }] Object reclassification from class d1 to d2λ1=2, λ2=1/4 For classification attribute in S: Action rules: [[(b,b1 →b1 )*(c,c1 → c2)] → (d, d1→d2) ] [[(a,a2 →a1 ] → (d, d1→d2) ] Xabcd x1x1x1x1 a1a1a1a1 b1b1b1b1 c1c1c1c1 d1d1d1d1 x2x2x2x2 a2a2a2a2 b1b1b1b1 c2c2c2c2 d1d1d1d1 x3x3x3x3 a2a2a2a2 b2b2b2b2 c2c2c2c2 d1d1d1d1 x4x4x4x4 a2a2a2a2 b1b1b1b1 c1c1c1c1 d1d1d1d1 x5x5x5x5 a2a2a2a2 b3b3b3b3 c2c2c2c2 d1d1d1d1 x6x6x6x6 a1a1a1a1 b1b1b1b1 c2c2c2c2 d2d2d2d2 x7x7x7x7 a1a1a1a1 b2b2b2b2 c2c2c2c2 d1d1d1d1 x8x8x8x8 a1a1a1a1 b2b2b2b2 c1c1c1c1 d3d3d3d3

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