Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion A Study of a Logic for Multiple-source Approximation Systems Mohua Banerjee Md. Aquil Khan Indian Institute of Technology Kanpur ICLA 2009 Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Basic Concepts Related to Rough Set Theory 1 Multiple-source Approximation Systems 2 Different Notions of Lower/Upper Approximations Different Notions of Definability Logic for MSAS(LMSAS) 3 Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics Conclusion 4 Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Out break of some disease. Totally unaffected class − → classes 1 to 8. Totally affected class − → class 9. Partially affected class − → class 10. Q. Is ‘A’, a class 8 student, affected? A. No. Q. Is ‘B’, a class 9 student, affected? A. Yes. Q. Is ‘C’, a class 10 student, affected? A. Possibly, but not certainly. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Pawlak Approximation space [Pawlak’82] ( U , R ), where R is an equivalence relation on U . U
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Pawlak Approximation space [Pawlak’82] ( U , R ), where R is an equivalence relation on U . U X
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Pawlak Approximation space [Pawlak’82] ( U , R ), where R is an equivalence relation on U . U X B ( X ) X R X R Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion U − ve X + ve Boundary element Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion U := Set of students; aRb iff a and b are in the same class; X := Set of affected students. U 1 2 3 4 5 6 7 8 9 10
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion U := Set of students; aRb iff a and b are in the same class; X := Set of affected students. U 1 2 3 4 5 6 7 8 9 10 X
Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion U := Set of students; aRb iff a and b are in the same class; X := Set of affected students. U 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 X Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion Multiple-source Approximation Systems(MSAS) F := ( U , { R i } i ∈ N ), where U is a non-empty set, N an initial segment of the set of positive integers, and each R i , i ∈ N , is an equivalence relation on the domain U . | N | is referred to as the cardinality of F and is denoted by | F | . Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion F := ( U , { R i } i ∈ N ) , X ⊆ U Definition � Strong lower approximation X s := X R i ; i � Weak lower approximation X w := X R i . i � Strong upper approximation X s := X R i ; i � Weak upper approximation X w := X R i . i For MSAS F := ( U , { R } ) X s = X w = X R and X s = X w = X R Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion X s ⊆ X w ⊆ X ⊆ X s ⊆ X w U ( X w ) c X w \ X s X s \ X w X w \ X s X s
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion X s ⊆ X w ⊆ X ⊆ X s ⊆ X w U ( X w ) c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion Proposition 1 X ∩ Y s = X s ∩ Y s ; X ∪ Y w = X w ∪ Y w ; 2 X ∩ Y s ⊆ X s ∩ Y s ; X ∪ Y w ⊇ X w ∪ Y w ; 3 X c s = ( X w ) c ; X c w = ( X s ) c ; 4 X cs = ( X w ) c ; X cw = ( X s ) c ; 5 X w = ( X w ) w ; X s = ( X s ) s ; 6 X w = ( X w ) w = ( X s ) w ; 7 ( X s ) w ⊆ X w ; Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion X ⊆ U is lower definable if X s = X w . U ( X w ) c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve X is lower definable iff the sets of +ve elements in all approximations spaces are identical. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion X ⊆ U is upper definable if X s = X w . U ( X w ) c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve X is upper definable iff the sets of -ve elements in all approximations spaces are identical. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion X ⊆ U is weak definable if X s = X w . U ( X w ) c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve X is weak definable iff X does not have certain boundary element. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion X ⊆ U is strong definable if X s = X w . U ( X w ) c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve X is strong definable iff every element of U is either certain +ve or certain -ve. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion Proposition X is upper definable iff X c is lower definable. Arbitrary union (intersection) of upper (lower) definable sets is also upper (lower) definable. (Collection of upper (lower) definable sets is not closed under intersection (union)). Collection of all strong definable sets forms a complete field of sets. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
Outline Basic Concepts Related to Rough Set Theory Different Notions of Lower/Upper Approximations Multiple-source Approximation Systems Different Notions of Definability Logic for MSAS(LMSAS) Conclusion Proposition The following are equivalent: X is strong definable. X is both lower and upper definable and X is definable in some approximation space. X is definable in each approximation space. X s = X w = X = X s = X w . Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems
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