Polynomial (Berwald-Moor) Computational . . . Finsler Metrics - PowerPoint PPT Presentation

Introduction: Main . . . Symmetry of . . . Towards a General . . . A New Model: . . . Polynomial (Berwald-Moor) Computational . . . Finsler Metrics Euclidean Space: Proof From Euclidean to . . . and Related Partial Orders From Minkowski to . . . Discussion Beyond Space-Time: Towards Analyzing . . . Towards Applications to Title Page ◭◭ ◮◮ Logic and Decision Making ◭ ◮ Olga Kosheleva and Vladik Kreinovich Page 1 of 25 University of Texas at El Paso Go Back El Paso, Texas 79968, USA emails olgak@utep.edu, vladik@utep.edu Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 1. Objective of Science and Engineering Towards a General . . . • One of the main objectives: help people select decisions A New Model: . . . which are the most beneficial to them. Computational . . . Euclidean Space: Proof • To make these decisions, From Euclidean to . . . – we must know people’s preferences , From Minkowski to . . . – we must have the information about different events Discussion – possible consequences of different decisions, and Towards Analyzing . . . – we must also have information about the degree of Title Page certainty ◭◭ ◮◮ ∗ (since information is never absolutely accurate ◭ ◮ and precise). Page 2 of 25 Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 2. Partial Orders Naturally Appear in Many Applica- tion Areas Towards a General . . . A New Model: . . . • Reminder: we need info re preferences , events , and de- Computational . . . grees of certainty . Euclidean Space: Proof From Euclidean to . . . • All these types of information naturally lead to partial orders: From Minkowski to . . . Discussion – For preferences , a < b means that b is preferable Towards Analyzing . . . to a . Title Page ∗ This relation is used in decision theory . ◭◭ ◮◮ – For events , a < b means that a can influence b . ◭ ◮ ∗ This causality relation is used in space-time physics . Page 3 of 25 – For degrees of certainty , a < b means that a is less certain than b . Go Back ∗ This relation is used in logics describing uncer- Full Screen tainty – such as fuzzy logic . Close Quit

Introduction: Main . . . Symmetry of . . . 3. Numerical Characteristics Related to Partial Or- ders Towards a General . . . A New Model: . . . + An order is a natural way of describing a relation. Computational . . . Euclidean Space: Proof − Orders are difficult to process, since most data process- From Euclidean to . . . ing algorithms process numbers . From Minkowski to . . . • Natural idea: use numerical characteristics to describe Discussion the orders. Towards Analyzing . . . • Fact: this idea is used in all three application areas: Title Page – in decision making, utility describes preferences: ◭◭ ◮◮ a < b if and only if u ( a ) < u ( b ); ◭ ◮ – in space-time physics, metric (and time coordinates) Page 4 of 25 describes causality relation; Go Back – in logic and soft constraints, numbers from the in- Full Screen terval [0 , 1] are used to describe degrees of certainty. Close Quit

Introduction: Main . . . Symmetry of . . . 4. Need to Combine Numerical Characteristics: Emergence of Polynomial Aggregation Formulas Towards a General . . . A New Model: . . . • In decision making, we need to combine utilities u 1 , Computational . . . . . . , u n of different participants. Euclidean Space: Proof – Nobelist Josh Nash showed that reasonable condi- From Euclidean to . . . tions lead to u = u 1 · . . . · u n . From Minkowski to . . . Discussion • In space-time geometry, we need to combine coordi- Towards Analyzing . . . nates x i into a metric. Title Page – Reasonable conditions lead to polynomial metrics ◭◭ ◮◮ s 2 = c 2 · ( x 0 − x ′ 0 ) 2 − ( x 1 − x ′ 1 ) 2 − ( x 2 − x ′ 2 ) 2 − ( x 3 − x ′ 3 ) 2 ; ◭ ◮ s 4 = ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 ) · ( x 3 − x ′ 3 ) · ( x 4 − x ′ 4 ) . Page 5 of 25 • In fuzzy logic, we must combine degrees of certainty d i Go Back in A i into a degree d for A 1 & A 2 . Full Screen – Reasonable conditions lead to polynomial functions like d = d 1 · d 2 . Close Quit

Introduction: Main . . . Symmetry of . . . 5. Mathematical Observation: Polynomial Formulas Are Tensor-Related Towards a General . . . A New Model: . . . • Fact: in many areas, we have a general polynomial Computational . . . dependence Euclidean Space: Proof f ( x 1 , . . . , x n ) = f 0 + From Euclidean to . . . n From Minkowski to . . . � f i · x i + Discussion i =1 Towards Analyzing . . . n n � � Title Page f ij · x i · x j + ◭◭ ◮◮ i =1 j =1 n n n ◭ ◮ � � � f ijk · x i · x j · x k + Page 6 of 25 i =1 j =1 k =1 . . . Go Back • In mathematical terms: to describe this dependence, Full Screen we need a finite set of tensors f 0 , f i , f ij , f ijk , . . . Close Quit

Introduction: Main . . . Symmetry of . . . 6. Towards a General Justification of Polynomial For- mulas Towards a General . . . A New Model: . . . • Fact: similar polynomials appear in different applica- Computational . . . tion areas. Euclidean Space: Proof From Euclidean to . . . • Reasonable conclusion: there must be a common rea- son behind them. From Minkowski to . . . Discussion • What we do: we provide such a general reason. Towards Analyzing . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 25 Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 7. Class of Functions Towards a General . . . • Objective: find a finite-parametric class F of analytical A New Model: . . . functions f ( x 1 , . . . , x n ). Computational . . . Euclidean Space: Proof • Meaning: f ( x 1 , . . . , x n ) approximate the actual com- From Euclidean to . . . plex aggregation function. From Minkowski to . . . • Reasonable requirement: this class F is invariant with Discussion respect to addition and multiplication by a constant. Towards Analyzing . . . • Conclusion: the class F is a (finite-dimensional) linear Title Page space of functions. ◭◭ ◮◮ • Meaning: invariance w.r.t. multiplication by a constant ◭ ◮ corresponds to the choice of a measuring unit. Page 8 of 25 • If we replace the original measuring unit by a one which Go Back is λ times smaller, then all the numerical values · λ : Full Screen f ( x 1 , . . . , x n ) is replaced with λ · f ( x 1 , . . . , x n ). Close Quit

Introduction: Main . . . Symmetry of . . . 8. Similar Scale-Invariance for the Inputs x i Towards a General . . . • Similarly: in all three areas, the numerical values x i A New Model: . . . are defined modulo the choice of a measuring unit. Computational . . . – If we replace the original measuring unit by a one Euclidean Space: Proof which is λ times smaller, From Euclidean to . . . From Minkowski to . . . – then all the numerical values get multiplied by this factor λ : Discussion Towards Analyzing . . . x i is replaced with λ · x i . Title Page • Conclusion: it is reasonable to require that the finite- ◭◭ ◮◮ dimensional linear space F be invariant with respect to such re-scalings: ◭ ◮ – if f ( x 1 , . . . , x n ) ∈ F , Page 9 of 25 – then for every λ > 0, the function Go Back def f λ ( x 1 , . . . , x n ) = f ( λ · x 1 , . . . , λ · x n ) Full Screen also belongs to the family F . Close Quit

Introduction: Main . . . Symmetry of . . . 9. Definition and the Main Result Towards a General . . . Definition. Let n be an arbitrary integer. We say that A New Model: . . . a finite-dimensional linear space F of analytical functions Computational . . . of n variables is scale-invariant if for every f ∈ F and for Euclidean Space: Proof every λ > 0 , the function From Euclidean to . . . def From Minkowski to . . . f λ ( x 1 , . . . , x n ) = f ( λ · x 1 , . . . , λ · x n ) Discussion also belongs to the family F . Towards Analyzing . . . Title Page Main result. For every scale-invariant finite-dimensional linear space F of analytical functions, every element f ∈ F ◭◭ ◮◮ is a polynomial. ◭ ◮ Page 10 of 25 Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 10. Proof (Part 1) Towards a General . . . • Let F be a scale-invariant finite-dimensional linear space A New Model: . . . F of analytical functions. Computational . . . Euclidean Space: Proof • Let f ( x 1 , . . . , x n ) be a function from this family F . From Euclidean to . . . • By definition, an analytical function f ( x 1 , . . . , x n ) is an From Minkowski to . . . infinite series consisting of monomials m ( x 1 , . . . , x n ): Discussion m ( x 1 , . . . , x n ) = a i 1 ...i n · x i 1 1 · . . . · x i n n . Towards Analyzing . . . Title Page • For each such term, by its total order , we will under- ◭◭ ◮◮ stand the sum i 1 + . . . + i n . ◭ ◮ – if we multiply each input of this monomial by λ , Page 11 of 25 – then the value of the monomial is multiplied by λ k : Go Back m ( λ · x 1 , . . . λ · x n ) = a i 1 ...i n · ( λ · x 1 ) i 1 · . . . · ( λ · x n ) i n = Full Screen n = λ k · m ( x 1 , . . . , x n ) . λ i 1 + ... + i n · a i 1 ...i n · x i 1 1 · . . . · x i n Close Quit