Need for Set Intervals Need for Set . . . Elementary Set . . . Towards More Adequate Representation How to Get Exact Set . . . Intermediate Value . . . of Uncertainty: From Intervals to Set Fuzzy Sets Interval-Valued Fuzzy . . . Intervals, with the Possible Addition of Solution Probabilistic Case: In . . . Probabilities and Certainty Degrees Similar Idea for Sets Acknowledgments J. T. Yao 1 , Y. Y. Yao 1 , V. Kreinovich 2 , Title Page P. Pinheiro da Silva 2 , S. A. Starks 2 , G. Xiang 2 , and H. T. Nguyen 3 ◭◭ ◮◮ ◭ ◮ 1 Department of Computer Science, University of Regina, Saskatchewan, Canada Page 1 of 12 2 NASA Pan-American Center for Earth and Environmental Studies Go Back University of Texas, El Paso, TX 79968, USA Full Screen 3 Department of Mathematical Sciences New Mexico State University Close Las Cruces, NM 88003, USA contact email vladik@utep.edu Quit
Need for Set Intervals 1. Need for Set Intervals Need for Set . . . Elementary Set . . . • Ideal case: complete knowledge. How to Get Exact Set . . . • We are interested in: properties P i such as “high fever”, “headache”, Intermediate Value . . . etc. Fuzzy Sets Interval-Valued Fuzzy . . . • Complete: we know the exact set S i of all the objects that satisfy Solution each property P i . Probabilistic Case: In . . . • In practice , we usually only have partial knowledge: Similar Idea for Sets – the set S i of all the objects about which we know that P i Acknowledgments holds, and Title Page – the set S i about which we know that P i may hold (i.e., equiv- ◭◭ ◮◮ alently, that we have not yet excluded the possibility of P i ). ◭ ◮ • Set interval: the only information about the actual (unknown) set S i = { x : P i ( x ) } is that S i ⊆ S i ⊆ S i , i.e., that Page 2 of 12 def S i ∈ S i = [ S i , S i ] = { S i : S i ⊆ S i ⊆ S i } . Go Back Full Screen Close Quit
Need for Set Intervals 2. Need for Set Operations with Set Intervals Need for Set . . . Elementary Set . . . • Main problem: How to Get Exact Set . . . – we have some information about the original properties P i ; Intermediate Value . . . – we would like to describe the set S = { x : P ( x ) } of all the Fuzzy Sets def Interval-Valued Fuzzy . . . values that satisfy some combination P = f ( P 1 , . . . , P n ). Solution • Example (informal): flu ↔ high fever and headache and not rash. Probabilistic Case: In . . . • Example (formal): f ( P 1 , P 2 , P 3 ) = P 1 & P 2 & ¬ P 3 . Similar Idea for Sets Acknowledgments • Ideal case: we know the exact sets S i = { x : P i ( x ) } . Title Page • Solution: ◭◭ ◮◮ – f ( S 1 , . . . , S n ) is composition of union, intersection, and com- ◭ ◮ plement; – apply the corresponding set operation, step-by-step, to the Page 3 of 12 known sets S i . Go Back • General case: describe the class S of all possible sets S corre- sponding to different S i ∈ S i : Full Screen def Close S = { f ( S 1 , . . . , S n ) : S 1 ∈ S 1 , . . . , S n ∈ S n } . Quit
Need for Set Intervals 3. Elementary Set Operations and Their Use Need for Set . . . Elementary Set . . . • Simplest case: n = 2 and f ( P 1 , P 2 ) is an elementary set operation How to Get Exact Set . . . (union, intersection, complement). Intermediate Value . . . • Useful property: elementary set operations are monotonic in ⊆ . Fuzzy Sets • For these operations, formulas for estimating S are known: Interval-Valued Fuzzy . . . Solution [ A, A ] ∪ [ B, B ] = [ A ∪ B, A ∪ B ]; [ A, A ] ∩ [ B, B ] = [ A ∩ B, A ∩ B ]; Probabilistic Case: In . . . − [ A, A ] = [ − A, − A ] . Similar Idea for Sets Acknowledgments • General case: idea (similar to interval computations) Title Page – parse the expression f ( S 1 , . . . , S n ); – replace each elementary set operation by the corresponding ◭◭ ◮◮ operation with interval sets. ◭ ◮ • Result: we get an enclosure for S = [ S, S ]. Page 4 of 12 • Problem: we may get excess width. Go Back • Example: for f ( S 1 ) = S 1 ∪ − S 1 , S 1 = [ ∅ , U ]. Full Screen – actual range: S = { U } ; – enclosure: − S 1 = [ ∅ , U ], so Close S 1 ∪ − S 1 = [ ∅ , U ] ∪ [ ∅ , U ] = [ ∅ , U ] . Quit
Need for Set Intervals 4. How to Get Exact Set Range? How Difficult Is It? Need for Set . . . Elementary Set . . . • Problem: in general, set operations such as S 1 ∪ − S 1 are not How to Get Exact Set . . . ⊆ -monotonic. Intermediate Value . . . • Solution for computing S : Fuzzy Sets Interval-Valued Fuzzy . . . – represent f ( S 1 , . . . , S n ) in a canonical DNF form Solution ( S 1 ∩ − S 2 ∩ . . . ∩ S n ) ∪ ( . . . ) ∪ . . . Probabilistic Case: In . . . Similar Idea for Sets – apply straightforward interval computations: Acknowledgments ( S 1 ∩ − S 2 ∩ . . . ∩ S n ) ∪ ( . . . ) ∪ . . . Title Page • Proof: each element from each conjunction S 1 ∩ − S 2 ∩ . . . ∩ S n is ◭◭ ◮◮ possible. ◭ ◮ • Example: S 1 △ S 2 = ( S 1 ∩ − S 2 ) ∪ ( − S 1 ∩ S 2 )), so S = ( S 1 ∩ − S 2 ) ∪ ( − S 1 ∩ S 2 ) . Page 5 of 12 Go Back • Solution for computing S : use S = − ( − S ), i.e., use CNF. • Problem: turning into DNF or CNF requires exponential time. Full Screen • Comment: the problem of checking whether ∅ ∈ f ( S 1 , . . . , S n ) is Close NP-hard. Quit
Need for Set Intervals 5. Intermediate Value Theorem for Set Intervals Need for Set . . . Elementary Set . . . • Situation: in the range S = f ( S 1 , . . . , S n ), we found the intersec- How to Get Exact Set . . . tion S and the union S of all possible sets. Intermediate Value . . . • Conclusion: S ⊆ [ S, S ]. Fuzzy Sets • Theorem: S = [ S, S ]. Interval-Valued Fuzzy . . . Solution • Equivalent formulation: for every S ∈ [ S, S ], there exist sets Probabilistic Case: In . . . S 1 ∈ [ S 1 , S 1 ] , . . . , S n ∈ [ S n , S n ] Similar Idea for Sets Acknowledgments for which S = f ( S 1 , . . . , S n ). Title Page • Difficulty: values S i ( u ) and S ( u ) are discrete (0 or 1), so the standard intermediate value theorem does not apply. ◭◭ ◮◮ • Solution: we define S i element-by-element. ◭ ◮ • Known: for each u ∈ U , we have S ( u ) ≤ S ( u ) ≤ S ( u ). Page 6 of 12 • Conclusion: S ( u ) = S ( u ) or S ( u ) = S ( u ). Go Back • By definition of S and S , in both cases, there exist sets s ( u ) for i Full Screen which S ( u ) = f ( s ( u ) 1 ( u ) , . . . , s ( u ) n ( u )). Close • We take S i ( u ) = s ( u ) i ( u ). Quit
Need for Set Intervals 6. Fuzzy Sets Need for Set . . . Elementary Set . . . • Previous description: How to Get Exact Set . . . – about some elements u , we know P ( u ); Intermediate Value . . . – about some elements u , we know ¬ P ( u ): Fuzzy Sets Interval-Valued Fuzzy . . . – about other elements u , we know nothing about P ( u ). Solution • Description: sets S and ( − S ) = − S . Probabilistic Case: In . . . Similar Idea for Sets • Additional information: experts may believe that P ( u ) holds with Acknowledgments some certainty α . Title Page • How to describe this information: a nested family of sets S α cor- responding to α : ◭◭ ◮◮ • S 0 = S ; ◭ ◮ • S 1 = S ; Page 7 of 12 • if α < α ′ then S α ⊆ S α ′ . Go Back • Traditional description: µ A ( u ) = max { α : u ∈ S α } . Full Screen • Set operations in terms of µ : µ A ∪ B ( u ) = max( µ A ( u ) , µ B ( u )); µ A ∩ B ( u ) = min( µ A ( u ) , µ B ( u )); µ ¬ A ( u ) = 1 − µ A ( u ). Close Quit
Need for Set Intervals 7. Interval-Valued Fuzzy Sets Need for Set . . . Elementary Set . . . • Situation: for every α , we are not sure which elements belong to How to Get Exact Set . . . S α and which do not. Intermediate Value . . . • Description: S α ⊆ S α . Fuzzy Sets Interval-Valued Fuzzy . . . • Alternative description: interval-valued membership function Solution [ µ A ( u ) , µ A ( u )] . Probabilistic Case: In . . . Similar Idea for Sets • Meaning: for all u , we have µ A ( u ) ∈ [ µ A ( u ) , µ A ( u )], i.e., Acknowledgments Title Page A ⊆ A ⊆ A. ◭◭ ◮◮ • Problem: ◭ ◮ – we know A 1 , . . . , A n , Page 8 of 12 – we know that A = f ( A 1 , . . . , A n ) for some set-expression f ; – find the range of A : Go Back f ( A 1 , . . . , A n ) = { f ( A 1 , . . . , A n ) : A 1 ∈ A 1 , . . . , A n ∈ A n } . Full Screen Close Quit
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