Outline Representing a . . . Analysis of the Situation Relation to Possibility . . . Membership Functions Why Not Use . . . Representing a Number vs. A Natural Question This Question Is Non- . . . Representing a Set: Proof of Main Result Auxiliary Result: . . . Unique Reconstruction Home Page Title Page Hung T. Nguyen 1 , 2 , Vladik Kreinovich 3 , and Olga Kosheleva 3 ◭◭ ◮◮ ◭ ◮ 1 Department of Mathematical Sciences New Mexico State University, Las Cruces, New Mexico 88008, USA Page 1 of 29 2 Faculty of Economics, Chiang Mai University, Thailand Email: hunguyen@nmsu.edu Go Back 3 University of Texas at El Paso, El Paso, Texas 79968, USA Full Screen vladik@utep.edu, olgak@utep.edu Close Quit
Outline Representing a . . . 1. Outline Analysis of the Situation • In some cases, a membership function µ ( x ) represents Relation to Possibility . . . an unknown number. Why Not Use . . . A Natural Question • In many other cases, it represents an unknown crisp This Question Is Non- . . . set. Main Result • In this case, for each crisp set S , we can estimate the Auxiliary Result: . . . degree µ ( S ) to which this set S is the desired one. Home Page • A natural question is: Title Page – once we know the values µ ( S ) corresponding to all ◭◭ ◮◮ possible crisp sets S , ◭ ◮ – can we reconstruct the original membership func- Page 2 of 29 tion? Go Back • We show that the membership function µ ( x ) can in- Full Screen deed be uniquely reconstructed from the values µ ( S ). Close Quit
Outline Representing a . . . 2. Representing a Number vs. Representing a Set Analysis of the Situation • In some cases, a fuzzy set is used to represent a number . Relation to Possibility . . . Why Not Use . . . • Example: we ask a person how old is Mary, and this A Natural Question person replies that Mary is young. This Question Is Non- . . . • There is an actual number representing age, but we do Main Result not know this number. Auxiliary Result: . . . Home Page • Instead, we have a membership function µ ( x ) that de- scribes our uncertain knowledge about this value. Title Page • µ ( x ) is our degree of confidence that the x has the ◭◭ ◮◮ desired property – e.g., that a person of age x is young. ◭ ◮ • In other cases, a fuzzy set is used to represent not a Page 3 of 29 single crisp value , but rather a whole crisp set . Go Back Full Screen Close Quit
Outline Representing a . . . 3. Representing a Number vs. Representing a Set Analysis of the Situation (cont-d) Relation to Possibility . . . • Example of a fuzzy number representing a set: Why Not Use . . . A Natural Question – when designing a control system for an autonomous This Question Is Non- . . . car, Main Result – we can ask a driver which velocities are safe on a Auxiliary Result: . . . certain road segment. Home Page • In reality, there is a (crisp) set of such values. However, Title Page we do not know this set. ◭◭ ◮◮ • Instead, we have a fuzzy set that describes our uncer- ◭ ◮ tain knowledge about this unknown set. Page 4 of 29 • Here, µ ( x ) is our degree of confidence that this element x belongs to the (unknown) set U . Go Back • Thus, our degree of confidence that the element y does Full Screen not belong to the actual set U is equal to 1 − µ ( y ). Close Quit
Outline Representing a . . . 4. Analysis of the Situation Analysis of the Situation • Fuzziness means that we do not know the actual set U Relation to Possibility . . . exactly. Why Not Use . . . A Natural Question • In other words, several different crisp sets S are possi- This Question Is Non- . . . ble candidates for the unknown actual set U . Main Result • For each crisp set S , let us estimate our degree of con- Auxiliary Result: . . . fidence µ ( S ) that this set S is the set U . Home Page • The equality S = U means that: Title Page – for every x ∈ S , we have x ∈ U , and ◭◭ ◮◮ – for every y �∈ S , we have y �∈ U . ◭ ◮ • In other words, if we consider all x i ∈ S and all y j �∈ S , Page 5 of 29 then S = U means that Go Back x 1 ∈ U and x 1 ∈ U and . . . and y 1 �∈ U and y 2 �∈ U and . . . Full Screen Close Quit
Outline Representing a . . . 5. Analysis of the Situation (cont-d) Analysis of the Situation • Our degree of confidence in the above “and”-statement Relation to Possibility . . . can be obtained by applying an “and”-operation: Why Not Use . . . A Natural Question µ ( S ) = f & ( µ ( x 1 ) , µ ( x 2 ) , . . . , 1 − µ ( y 1 ) , 1 − µ ( y 2 ) , . . . ) . This Question Is Non- . . . Main Result • In fuzzy logic, there are many possible “and”- operations (t-norms). Auxiliary Result: . . . Home Page • However, for most of them (e.g., for a · b ) the result of Title Page applying this operation to infinitely many values is 0. ◭◭ ◮◮ • Among the most widely used t-norms, the only “and”- operation for which the result is non-0 is min, so ◭ ◮ Page 6 of 29 � � µ ( S ) = min x ∈ S µ ( x ) , inf inf y �∈ S (1 − µ ( y )) . Go Back Full Screen Close Quit
Outline Representing a . . . 6. Relation to Possibility and Belief Analysis of the Situation • Similar expressions describe possibility degree and de- Relation to Possibility . . . gree of belief : Why Not Use . . . A Natural Question Poss( S ) = sup µ ( x ) This Question Is Non- . . . x ∈ S Main Result Bel( S ) = 1 − Poss( − S ) = inf x ∈ S µ ( x ) . Auxiliary Result: . . . Home Page • One can see that our degree ρ ( S ) cane described in terms of plausibility and belief, as Title Page µ ( S ) = min(Bel( S ) , Bel( − S )) = ◭◭ ◮◮ ◭ ◮ min(Bel( S ) , 1 − Poss( S )) . Page 7 of 29 Go Back Full Screen Close Quit
Outline Representing a . . . 7. Why Not Use Probabilities: Advantage of a Analysis of the Situation Fuzzy Approach Relation to Possibility . . . • At first glance, it may seem that in this situation, we Why Not Use . . . could also use a probabilistic approach. A Natural Question This Question Is Non- . . . • In this case, if we denote the probability that x ∈ S by p ( x ), then the probability that y �∈ S is equal to Main Result Auxiliary Result: . . . 1 − p ( y ) . Home Page • If we make a usual probabilistic assumption that events Title Page x ∈ S corresponding to different x are independent: ◭◭ ◮◮ �� � �� � Prob( S = U ) = p ( x i ) · (1 − p ( y j )) . ◭ ◮ i j Page 8 of 29 • However, when we have infinitely many values x i and Go Back y j , this product becomes a meaningless 0. Full Screen • Thus, in general, it is not possible to use the proba- bilistic approach in this situation. Close Quit
Outline Representing a . . . 8. A Natural Question Analysis of the Situation • We have shown how: Relation to Possibility . . . Why Not Use . . . – if we know the original membership function µ ( x ), A Natural Question – then we can determine the degree µ ( S ) for each This Question Is Non- . . . crisp set S . Main Result • Natural question : how uniquely can we reconstruct Auxiliary Result: . . . µ ( x ) from µ ( S )? Home Page • In other words: Title Page ◭◭ ◮◮ – if we know the value µ ( S ) for every crisp set S , – can we uniquely reconstruct the original member- ◭ ◮ ship function µ ( x )? Page 9 of 29 Go Back Full Screen Close Quit
Outline Representing a . . . 9. This Question Is Non-Trivial Analysis of the Situation • At first glance, it may seem that this reconstruction is Relation to Possibility . . . easy: e.g., to find µ ( a ), why not take S = { a } ? Why Not Use . . . A Natural Question • However, one can easily see that this simple approach This Question Is Non- . . . does not work. Main Result • For example, if µ ( x 0 ) = 1, and we want to find µ ( a ) Auxiliary Result: . . . for some a � = x 0 , then for x 0 �∈ { a } , we have Home Page 1 − µ ( x 0 ) = 1 − 1 = 0 . Title Page • Thus, inf y �∈{ a } (1 − µ ( y )) = 0, and so, irrespective of what ◭◭ ◮◮ ◭ ◮ is the actual value of µ ( a ): Page 10 of 29 � � µ ( { a } ) = min x ∈{ a } µ ( x ) , inf inf y �∈{ a } (1 − µ ( y )) = 0 . Go Back • We therefore need more sophisticated techniques for Full Screen reconstructing µ ( x ) from µ ( S ). Close Quit
Outline Representing a . . . 10. Main Result Analysis of the Situation • Proposition 1. Relation to Possibility . . . Why Not Use . . . – Let µ ( x ) and µ ′ ( x ) be membership f-ns, and let A Natural Question � � This Question Is Non- . . . µ ( S ) = min x ∈ S µ ( x ) , inf inf y �∈ S (1 − µ ( y )) and Main Result � � Auxiliary Result: . . . µ ′ ( S ) = min x ∈ S µ ′ ( x ) , inf y �∈ S (1 − µ ′ ( y )) inf . Home Page Title Page – If µ ( S ) = µ ′ ( S ) for all crisp sets S ⊆ X , then ◭◭ ◮◮ µ ( x ) = µ ′ ( x ) for all x. ◭ ◮ • So, the membership f-n µ ( x ) can indeed be uniquely Page 11 of 29 reconstructed if we know µ ( S ) for all crisp sets S . Go Back • The proof of the main result consists of several lemmas. Full Screen Close Quit
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