And- and Or-Operations A Natural Idea for Double, Triple, etc. We - PowerPoint PPT Presentation

Outline Traditional Fuzzy . . . Need for “And’- and . . . Need to Go Beyond . . . “And”- and “Or”-Operations A Natural Idea for “Double”, “Triple”, etc. We Need to Extend . . . “And”-Operations in . . . Fuzzy Sets “And”-Operations for . . . MaxEnt Approach . . . Hung T. Nguyen 1 , 2 , Olga Kosheleva 3 , and Vladik Kreinovich 3 Home Page Title Page 1 Department of Mathematical Sciences, New Mexico State University Las Cruces, NM, 88003, USA, hunguyen@nmsu.edu ◭◭ ◮◮ 2 Department of Economics, Chiang Mai University, Thailand ◭ ◮ 3 University of Texas at El Paso, El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Page 1 of 14 Go Back Full Screen Close Quit

Outline Traditional Fuzzy . . . 1. Outline Need for “And’- and . . . • In the traditional fuzzy logic: Need to Go Beyond . . . A Natural Idea – the expert’s degree of confidence d ( A & B ) in a com- We Need to Extend . . . plex statement A & B “And”-Operations in . . . – is uniquely determined by his/her degrees of confi- “And”-Operations for . . . dence d ( A ) and d ( B ) in the statements A and B . MaxEnt Approach . . . • In practice, for the same degrees d ( A ) and d ( B ), we Home Page may have different degrees d ( A & B ). Title Page • The best way to take this relation into account is to ◭◭ ◮◮ explicitly elicit the corresponding degrees d ( A & B ). ◭ ◮ • If we only elicit information about pairs of statements, Page 2 of 14 then we still need to estimate, e.g., the degree d ( A & B & C ). Go Back • In this talk, we explain how to produce such “and”- Full Screen operations for “double” fuzzy sets. Close Quit

Outline Traditional Fuzzy . . . 2. Traditional Fuzzy Techniques: A Brief Reminder Need for “And’- and . . . • Experts often describe their knowledge by using impre- Need to Go Beyond . . . cise (“fuzzy”) words like “small” or “fast”. A Natural Idea We Need to Extend . . . • We need to describe this knowledge in computer un- “And”-Operations in . . . derstandable terms. “And”-Operations for . . . • A natural idea is to assign degrees of certainty MaxEnt Approach . . . d ( S ) ∈ [0 , 1] to expert statements S . Home Page • We can ask an expert to mark his/her degree of cer- Title Page tainty by a mark m on a scale from 0 to n , and take ◭◭ ◮◮ d ( S ) = m/n. ◭ ◮ Page 3 of 14 • We can also poll n experts; if m of them think that S is true, we take d ( S ) = m/n . Go Back Full Screen Close Quit

Outline Traditional Fuzzy . . . 3. Need for “And’- and “Or”-Operations Need for “And’- and . . . • We use expert knowledge to answer queries. Need to Go Beyond . . . A Natural Idea • The answer to a query Q usually depends on several We Need to Extend . . . statements. “And”-Operations in . . . • What is d ( Q )? “And”-Operations for . . . • For example, Q holds if either S 1 and S 2 hold, or if S 3 , MaxEnt Approach . . . S 3 , and S 5 hold. Home Page Title Page • Thus, to estimate d ( Q ), we must estimate the degree of certainty in propositional combinations like ◭◭ ◮◮ ( S 1 & S 2 ) ∨ ( S 3 & S 4 & S 5 ) . ◭ ◮ Page 4 of 14 • Ideally, we should ask the expert’s opinion about all such combinations. Go Back • However, for n statements, we have 2 n such combina- Full Screen tions, so we cannot ask about all of them. Close Quit

Outline Traditional Fuzzy . . . 4. Need for “And’- and “Or”-Operations (cont-d) Need for “And’- and . . . • We cannot ask the expert about degree of certainty in Need to Go Beyond . . . all possible propositional combinations. A Natural Idea We Need to Extend . . . • It is therefore necessary to estimate d ( A & B ) based on “And”-Operations in . . . d ( A ) and d ( B ). “And”-Operations for . . . • The estimate f & ( a, b ) for d ( A & B ) based on a = d ( A ) MaxEnt Approach . . . and b = d ( B ) is known as an “and”-operation ( t-norm ). Home Page • Similarly, we need an “or”-operation f ∨ ( a, b ) and a Title Page negation operation f ¬ ( a ). ◭◭ ◮◮ • The most widely used operations are: ◭ ◮ f & ( a, b ) = min( a, b ) , f & ( a, b ) = a · b, Page 5 of 14 f ∨ ( a, b ) = max( a, b ) , f ∨ ( a, b ) = a + b − a · b, Go Back f ¬ ( a ) = 1 − a. Full Screen Close Quit

Outline Traditional Fuzzy . . . 5. Need to Go Beyond Traditional Fuzzy Need for “And’- and . . . • In the traditional fuzzy techniques, we base our esti- Need to Go Beyond . . . mate of d ( A & B ) only on d ( A ) and d ( B ). A Natural Idea We Need to Extend . . . • In reality, for the same degrees of belief in A and B , “And”-Operations in . . . we may have different degrees of belief in A & B . “And”-Operations for . . . • Example 1: if d ( A ) = 0 . 5, then d ( ¬ A ) = 1 − 0 . 5 = 0 . 5. MaxEnt Approach . . . • For B = A , d ( A ) = d ( B ) = 0 . 5 and d ( A & B ) = Home Page d ( A ) = 0 . 5. Title Page • For B = ¬ A , d ( A ) = d ( B ) = 0 . 5 and d ( A & B ) = 0. ◭◭ ◮◮ • Example 2 : d (50-year-old is old) = 0 . 1, ◭ ◮ d (60-year-old is old) = 0 . 8 , so Page 6 of 14 def d 0 = d (50-year-old is old & 60-year-old is not old) = Go Back f & (0 . 1 , 1 − 0 . 2) > 0 for min( a, b ) and a · b. Full Screen • However, intuitively, d 0 = 0. Close Quit

Outline Traditional Fuzzy . . . 6. A Natural Idea Need for “And’- and . . . • A natural solution to the above problem is to explicitly Need to Go Beyond . . . elicit and store: A Natural Idea We Need to Extend . . . – not only the expert’s degree of confidence µ P ( x ) “And”-Operations in . . . that a given value x satisfies the property x “And”-Operations for . . . – but also the degree of confidence µ PP ( x, x ′ ) that both x and x ′ satisfy the property P . MaxEnt Approach . . . Home Page • In this approach, to describe a property, we need two Title Page functions: ◭◭ ◮◮ – a function µ P : X → [0 , 1], and ◭ ◮ – a function µ PP : X × X → [0 , 1] for which µ PP ( x, x ′ ) = µ PP ( x ′ , x ) and µ PP ( x, x ′ ) ≤ µ P ( x ) . Page 7 of 14 Go Back • Since we need two functions, it is natural to call such pairs ( µ P , µ PP ) double fuzzy sets . Full Screen • We can also ask about the triples ( x, x ′ , x ′′ ) etc. Close Quit

Outline Traditional Fuzzy . . . 7. We Need to Extend “And”- and “Or”-Operations Need for “And’- and . . . to “Double”, “Triple” etc. Fuzzy Sets Need to Go Beyond . . . • If we explicitly elicit d ( A & B ), we do not need the A Natural Idea usual “and”-operation. We Need to Extend . . . “And”-Operations in . . . • However, we still need to estimate d ( A & B & C ) based “And”-Operations for . . . on the available values: MaxEnt Approach . . . d ( A ) , d ( B ) , d ( C ) , d ( A & B ) , d ( A & C ) , d ( B & C ) . Home Page Title Page • We will show that: ◭◭ ◮◮ – the ideas behind the most popular t-norms and t- conorms ◭ ◮ – can be used describe the desired “and”- and “or”- Page 8 of 14 operations for the “double” fuzzy sets. Go Back Full Screen Close Quit

Outline Traditional Fuzzy . . . 8. “And”-Operations in Traditional Fuzzy Logic: Need for “And’- and . . . Reminder Need to Go Beyond . . . • Traditionally, expert’s degrees of certainty are also called A Natural Idea subjective probabilities . We Need to Extend . . . “And”-Operations in . . . • In probabilistic terms: “And”-Operations for . . . – we know the probabilities p ( s 1 ) and p ( s 2 ) of two MaxEnt Approach . . . statements s 1 and s 2 ; Home Page – we want to estimate the probability p ( s 1 & s 2 ). Title Page • Depending on the dependence between s 1 and s 2 , we ◭◭ ◮◮ may have different values of p ( s 1 & s 2 ). ◭ ◮ • There are two main approaches to deal with this non- Page 9 of 14 uniqueness: Go Back – we can find the range of all possible values p ( s 1 & s 2 ); – or we can select a single “most probable” value Full Screen p ( s 1 & s 2 ). Close Quit

Outline Traditional Fuzzy . . . 9. Inequalities (Linear Programming) Approach Need for “And’- and . . . • We need to know the probabilities of all basic combi- Need to Go Beyond . . . nations s 1 & s 2 , s 1 & ¬ s 2 , ¬ s 1 & s 2 , and ¬ s 1 & ¬ s 2 . A Natural Idea We Need to Extend . . . • We know d 1 = p ( s 1 ) and d 2 = p ( s 2 ); based on def “And”-Operations in . . . x = p ( s 1 & s 2 ), we get: “And”-Operations for . . . p ( s 1 & ¬ s 2 ) = p ( s 1 ) − p ( s 1 & s 2 ) = d 1 − x, MaxEnt Approach . . . p ( ¬ s 1 & s 2 ) = p ( s 2 ) − p ( s 1 & s 2 ) = d 2 − x, and Home Page p ( ¬ s 1 & ¬ s 2 ) = 1 − p ( s 1 ) − p ( s 2 )+ p ( s 1 & s 2 ) = 1 − d 1 − d 2 + x. Title Page • All the basic probabilities must be non-negative: ◭◭ ◮◮ x ≥ 0; d 1 − x ≥ 0; d 2 − x ≥ 0; 1 − d 1 − d 2 + x ≥ 0 , i.e., ◭ ◮ x ≥ 0; x ≤ d 1 ; x ≤ d 2 ; x ≥ d 1 + d 2 − 1 . Page 10 of 14 • So, the range of possible values is Go Back max( d 1 + d 2 − 1 , 0) ≤ x ≤ min( d 1 , d 2 ) . Full Screen • Both endpoints serve as possible t-norms. Close Quit