Fuzzy Logic: Brief . . . Mappings Which . . . From Single-Valued . . . Reasonable Properties . . . Set-Valued Extensions It Is Sufficient to . . . Main Result of Fuzzy Logic: Discussion and . . . Auxiliary Classification . . . Classification Theorems Discussion Gilbert Ornelas and Vladik Kreinovich Proof: Main Idea Proof: Main Idea (cont-d) Department of Computer Science Acknowledgments University of Texas at El Paso Title Page El Paso, Texas 79968, USA emails gtornelas@gmail.com, vladik@utep.edu ◭◭ ◮◮ http://www.cs.utep.edu/vladik ◭ ◮ http://www.cs.utep.edu/interval-comp Page 1 of 14 Go Back Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 1. Outline From Single-Valued . . . Reasonable Properties . . . • Fact: experts are often not 100% confident. It Is Sufficient to . . . • Traditional fuzzy logic: use numbers from [0 , 1]. Main Result • Problem: an expert often cannot describe degree by a Discussion and . . . single number. Auxiliary Classification . . . Discussion • Solution: use a set of numbers. Proof: Main Idea • Our result: the class of such sets coincides: Proof: Main Idea (cont-d) Acknowledgments – with all 1-point sets (i.e., with the traditional fuzzy Title Page logic), or ◭◭ ◮◮ – with all subintervals of [0 , 1], or – with all (closed) subsets of [0 , 1]. ◭ ◮ Page 2 of 14 • Conclusion: if we want to go beyond standard fuzzy logic and still avoid sets of arbitrary complexity, we Go Back have to use intervals . Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 2. Fuzzy Logic: Brief Reminder From Single-Valued . . . Reasonable Properties . . . • Classical (2-valued) logic: every statement is either It Is Sufficient to . . . true or false. Main Result • Problem: not adequate for expert knowledge, because Discussion and . . . experts are not fully confident about their statements. Auxiliary Classification . . . • Traditional fuzzy logic: a person’s degree of confidence Discussion is described by a number from the interval [0 , 1]: Proof: Main Idea – absolute confidence in a statement corresponds to 1, Proof: Main Idea (cont-d) Acknowledgments – absolute confidence in its negation corresponds to 0. Title Page • Operations: ◭◭ ◮◮ – we know: the degree of confidence a in a statement ◭ ◮ A and the degree of confidence b in a statement B , – we estimate the degree of confidence in statements Page 3 of 14 A ∧ B and A ∨ B as Go Back def def a ∧ b = min( a, b ) and a ∨ b = max( a, b ) . Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 3. Mappings Which Preserve Standard Fuzzy Logic From Single-Valued . . . Operations Reasonable Properties . . . It Is Sufficient to . . . • Important: there is no absolute scale of degrees. Main Result • Question: what possible rescalings ϕ : [0 , 1] → [0 , 1] Discussion and . . . preserve operations ∧ and ∨ , in the sense that Auxiliary Classification . . . ϕ ( a ) ∧ ϕ ( b ) = ϕ ( a ∧ b ) and ϕ ( a ) ∨ ϕ ( b ) = ϕ ( a ∨ b ) . Discussion Proof: Main Idea • Known result: if a bijection (1-1 onto mapping) is Proof: Main Idea (cont-d) monotonic, then it preserves both ∧ and ∨ . Acknowledgments • Known result: vice versa, if a bijection ϕ preserves the Title Page operations ∧ and ∨ , then it is monotonic. ◭◭ ◮◮ • Terminology: a strictly monotonic continuous function ◭ ◮ from [0 , 1] to [0 , 1] for which ϕ (0) = 0 and ϕ (1) = 1 is thus an automorphism of the structure ([0 , 1] , ∧ , ∨ ). Page 4 of 14 • The set of all automorphisms is called the automor- Go Back phism group of the structure ([0 , 1] , ∧ , ∨ ). Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 4. From Single-Valued Fuzzy Logic to Interval-Valued From Single-Valued . . . and Set-Valued Ones Reasonable Properties . . . It Is Sufficient to . . . • Need for sets: reminder. Main Result – An expert often cannot describe his or her degree Discussion and . . . by a single number. Auxiliary Classification . . . – It is therefore reasonable to describe this degree by, Discussion e.g., a set of possible values (e.g., an interval). Proof: Main Idea • Operations on sets: motivation: Proof: Main Idea (cont-d) Acknowledgments – a set A means that all values a ∈ A are possible, Title Page – B means that all the values b ∈ B are possible; ◭◭ ◮◮ – so, the set A ∧ B of possible values of a ∧ b is formed ◭ ◮ by all the values a ∧ b where a ∈ A and b ∈ B : Page 5 of 14 def A ∧ B = { a ∧ b : a ∈ A, b ∈ B } . Go Back def • Similarly, A ∨ B = { a ∨ b : a ∈ A, b ∈ B } . Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 5. Reasonable Properties of Set Extensions From Single-Valued . . . Reasonable Properties . . . • Problem: we want to allow sets from a given class S . It Is Sufficient to . . . • We want an extension of the traditional fuzzy logic: S Main Result must contain all one-element sets. Discussion and . . . • We want invariance: if S ∈ S , and ϕ ( x ) is an automor- Auxiliary Classification . . . phism, then the image ϕ ( S ) = { ϕ ( s ) : s ∈ S } should Discussion also be possible, i.e., ϕ ( S ) ∈ S . Proof: Main Idea Proof: Main Idea (cont-d) • We want closure under naturally defined ∧ and ∨ . Acknowledgments • Situation: S ∈ S , values s 1 , s 2 , . . . , s k , . . . are all possi- Title Page ble (i.e., s k ∈ S ), and s k → s . ◭◭ ◮◮ • Analysis: no matter how accurately we compute s , we ◭ ◮ will always find s k that is indistinguishable from s . Page 6 of 14 • Conclusion: it is natural to assume that this limit value Go Back s is also possible, i.e., that every set S ∈ S be closed. Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 6. It Is Sufficient to Consider Closed Classes of Sets From Single-Valued . . . Reasonable Properties . . . • Known: on the class of all bounded closed sets, there It Is Sufficient to . . . is a natural metric – Hausdorff distance d H ( S, S ′ ). Main Result • Definition: the smallest ε > 0 for which S is contained Discussion and . . . in the ε -neighborhood of S ′ and S ′ is contained in the Auxiliary Classification . . . ε -neighborhood of S . Discussion • Interpretation: if d H ( S, S ′ ) ≤ ε , and we only know the Proof: Main Idea values s ∈ S and s ′ ∈ S ′ with accuracy ε , then we Proof: Main Idea (cont-d) cannot distinguish between the sets S and S ′ . Acknowledgments • Situation: S 1 , S 2 , . . . , S k , . . . are all possible ( S i ∈ S ), Title Page and d H ( S k , S ) → 0. ◭◭ ◮◮ • Analysis: no matter how accurately we compute the ◭ ◮ values, we will always find a set S k that is indistin- Page 7 of 14 guishable from the set S (and possible). Go Back • Conclusion: the limit set S is also possible, i.e., S Full Screen is closed under the Hausdorff metric. Close
Fuzzy Logic: Brief . . . Mappings Which . . . 7. Main Result From Single-Valued . . . Reasonable Properties . . . Definition 1. A class S of closed non-empty subsets of It Is Sufficient to . . . the interval [0 , 1] is called a set-valued extension of fuzzy Main Result logic if it satisfies the following conditions: Discussion and . . . (i) the class S contains all 1-element sets { s } , s ∈ [0 , 1] ; Auxiliary Classification . . . (ii) the class S is closed under “and” and “or” operations; Discussion Proof: Main Idea (iii) the class S is closed under arbitrary automorphisms; Proof: Main Idea (cont-d) (iv) the class S is closed under Hausdorff metric. Acknowledgments Title Page Theorem 1. Every set-valued extension of fuzzy logic co- incides with one of the following three classes: ◭◭ ◮◮ • the class P of all one-point sets { s } ; ◭ ◮ • the class I of all subintervals [ s, s ] ⊆ [0 , 1] of the inter- Page 8 of 14 val [0 , 1] ; Go Back • the class C of all closed subsets S of the interval [0 , 1] . Full Screen Close
Fuzzy Logic: Brief . . . Mappings Which . . . 8. Discussion and Auxiliary Results From Single-Valued . . . Reasonable Properties . . . • Main result in plain English: if we do not want ar- It Is Sufficient to . . . bitrarily complex sets, we must restrict ourselves to Main Result intervals. Discussion and . . . • We required: that all single-valued fuzzy sets are pos- Auxiliary Classification . . . sible. Discussion • Problem: as we mentioned, single values are not real- Proof: Main Idea istic. Proof: Main Idea (cont-d) Acknowledgments • Question: what if we do not make this requirement? Title Page • First case: the class S contains a set S which contains ◭◭ ◮◮ neither 0 not 1. ◭ ◮ • Result: same as before. Page 9 of 14 • Remaining case: every S ∈ S contains 0 or 1. Go Back • Result: new classification theorem. Full Screen Close
Recommend
More recommend
