Wieners Conjecture About Transformations . . . Transformation - PowerPoint PPT Presentation

Formulation of the . . . Wiener’s Conjecture: . . . How Wiener’s . . . Different Assignment . . . Wiener’s Conjecture About Transformations . . . Transformation Groups Examples of . . . Definition and Main . . . Helps Predict Which Fuzzy Selecting Membership . . . “And”-Operations Techniques Work Better Home Page Title Page Francisco Zapata 1 , Olga Kosheleva 2 , and Vladik Kreinovich 3 ◭◭ ◮◮ ◭ ◮ 1 Research Institute for Manufacturing & Engineering Systems Departments of 2 Teacher Education and 3 Computer Science Page 1 of 17 University of Texas at El Paso, El Paso, Texas 79968, USA fazg74@gmail.com, olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 1. Formulation of the Problem How Wiener’s . . . • Often, application succeeds only when we select spe- Different Assignment . . . cific fuzzy techniques (t-norm, membership f-n, etc.). Transformations . . . Examples of . . . • In different applications, different techniques are the Definition and Main . . . best. Selecting Membership . . . • How to find the best technique? “And”-Operations Home Page • Exhaustive search of all techniques is not an option: there are too many of them. Title Page • We need to come up with a narrow class of promising ◭◭ ◮◮ techniques, so that trying them all is realistic. ◭ ◮ • We show that transformation groups – motivated by Page 2 of 17 N. Wiener’s conjecture – lead to such a narrowing. Go Back • This conjecture was, in its turn, motivated by obser- Full Screen vations about human vision. Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 2. Wiener’s Conjecture: Reminder How Wiener’s . . . • The closer we are to an object, the better we can de- Different Assignment . . . termine its shape. Transformations . . . Examples of . . . • Experiments show that there are distinct phases in this Definition and Main . . . determination. Selecting Membership . . . • When the object is very far, all we see is a formless “And”-Operations blurb. Home Page • In other words, objects obtained from other by arbi- Title Page trary smooth transformations cannot be distinguished. ◭◭ ◮◮ • When the object gets closer, we can detect whether it ◭ ◮ is smooth or has sharp angles. Page 3 of 17 • We may see a circle as an ellipse, a square as a rhombus (diamond). Go Back Full Screen • At this stage, images obtained by a projective trans- formation are indistinguishable. Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 3. Wiener’s Conjecture (cont-d) How Wiener’s . . . • When the object gets closer, we can detect which lines Different Assignment . . . are parallel but we may not yet detect the angles. Transformations . . . Examples of . . . • For example, we are not sure whether what we see is a Definition and Main . . . rectangle or a parallelogram. Selecting Membership . . . • This stage corresponds to affine transformation. “And”-Operations Home Page • Then, we have a stage of similarity transformations – when we detect the shape but cannot yet detect its Title Page size. ◭◭ ◮◮ • Finally, when the object is close enough, we can detect ◭ ◮ both its shape and its size. Page 4 of 17 • Each stage can be this described by an appropriate Go Back transformation group (see a formal description below). Full Screen Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 4. Wiener’s Conjecture: Result How Wiener’s . . . • Humans result from billions of years of evolution. So, Different Assignment . . . Wiener conjectured that: Transformations . . . Examples of . . . – if there was a group intermediate between, e.g., all Definition and Main . . . projective and all continuous transformations, Selecting Membership . . . – our vision mechanism would have used it. “And”-Operations • Thus, according to the 1940s Wiener’s conjecture, such Home Page intermediate groups are not possible. Title Page • In the 1960s, Wiener’s conjecture was proven. ◭◭ ◮◮ • In the 1-D case, projective transformations are simply ◭ ◮ fractionally linear, and affine are simply linear. Page 5 of 17 • Thus, any group containing all 1-D linear transforma- tion is: Go Back – either the group of all fractionally-linear transf. Full Screen – or the group of all transformations. Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 5. How Wiener’s Conjecture Helps: General Idea How Wiener’s . . . • Fuzzy degrees are not uniquely determined. Different Assignment . . . Transformations . . . • Different elicitation techniques lead, in general, to dif- Examples of . . . ferent values. Definition and Main . . . • Sometimes, different scales are related by a linear trans- Selecting Membership . . . formation, sometimes by a non-linear one. “And”-Operations Home Page • In practice, we want a description with finitely many parameters. Title Page • Thus, we want a finite-dimensional transformation group. ◭◭ ◮◮ • Due to the above result, all such transformations are ◭ ◮ fractionally linear. Page 6 of 17 • We show that this can explain why some t-norms, mem- Go Back bership functions, etc., are empirically more successful. Full Screen Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 6. Different Assignment Procedures Are In Use How Wiener’s . . . • Intelligent systems use several different procedures for Different Assignment . . . assigning numeric values that describe uncertainty. Transformations . . . Examples of . . . • The same expert’s degree of uncertainty that he ex- Definition and Main . . . presses, e.g., by the expression “for sure”, can lead: Selecting Membership . . . – to 0.9 if we apply one procedure, and “And”-Operations – to 0.8 if another procedure is used. Home Page • 1 foot and 12 inches describe the same length, but in Title Page different scales. ◭◭ ◮◮ • We can say that 0.9 and 0.8 represent the same degree ◭ ◮ of certainty in two different scales . Page 7 of 17 • Some scales are different even in the fact that they use Go Back an interval different from [0 , 1] to represent uncertainty. Full Screen • For example, the famous MYCIN system uses the in- terval [ − 1 , 1]. Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 7. Transformations Between Reasonable Scales How Wiener’s . . . • Let F denote the class of reasonable transformations Different Assignment . . . of degrees of uncertainty. If: Transformations . . . Examples of . . . – a function x → f ( x ) is a reasonable transformation Definition and Main . . . from a scale A to some scale B , and Selecting Membership . . . – a function y → g ( y ) is a reasonable transformation “And”-Operations from B into some other scale C , Home Page – then the transformation x → g ( f ( x )) from A to C is also reasonable. Title Page ◭◭ ◮◮ • In other words, the class F of all reasonable transfor- mations must be closed under composition. Also: ◭ ◮ – if x → f ( x ) is a reasonable transformation from a Page 8 of 17 scale A to scale B , Go Back – then the inverse function is a reasonable transfor- Full Screen mation from B to A . Close • Thus, F must be a transformation group . Quit

Formulation of the . . . Wiener’s Conjecture: . . . 8. Examples of Reasonable Transformations How Wiener’s . . . • A natural method to assign a truth value t ( S ) to a Different Assignment . . . statement S is to ask several experts and take Transformations . . . Examples of . . . t ( S ) = N ( S ) . Definition and Main . . . N Selecting Membership . . . • The more expert we ask, the more reliable is this esti- “And”-Operations mate. Home Page • However, in the presence of Nobelists, experts may say Title Page nothing or follow the majority. ◭◭ ◮◮ • After we add M experts who do not answer anything ◭ ◮ and M ′ who follow the majority, we get Page 9 of 17 t ′ = N ( S ) + M ′ N · t + M ′ N + M + M ′ = N + M + M ′ = a · t + b. Go Back Full Screen • The transformation from an old scale t ( S ) to a new scale t ′ is a linear function. Close Quit

Formulation of the . . . Wiener’s Conjecture: . . . 9. Definition and Main Result How Wiener’s . . . • By a rescaling we mean a strictly increasing continuous Different Assignment . . . function f that is defined on an interval [ a, b ] ⊆ I R. Transformations . . . Examples of . . . • Suppose a set F of rescalings is a connected Lie group which contains, for all N, M, M ′ ≥ 0, a transformation Definition and Main . . . Selecting Membership . . . N · t + M ′ t → N + M + M ′ . “And”-Operations Home Page • Elements of this set F will be called reasonable trans- Title Page formations . ◭◭ ◮◮ • Result: Every reasonable transformation f ( x ) is frac- ◭ ◮ tionally linear: f ( x ) = a · x + b c · x + d . Page 10 of 17 Go Back Full Screen Close Quit