Towards a Physically Analysis of the Problem Meaningful Definition - PowerPoint PPT Presentation

Need to Define . . . Computable Numbers . . . Computable Functions Computable Compact Set Towards a Physically Analysis of the Problem Meaningful Definition of Additional Problem: . . . Resulting Definition Computable Discontinuous Main Result Inverse Relations: . . . and Multi-Valued Functions Home Page (Constraints) Title Page ◭◭ ◮◮ Martine Ceberio, Olga Kosheleva, and ◭ ◮ Vladik Kreinovich Page 1 of 15 University of Texas at El Paso El Paso, Texas 79968, USA Go Back mceberio@utep.edu, olgak@utep.edu Full Screen vladik@utep.edu Close Quit

Need to Define . . . Computable Numbers . . . 1. Need to Define Computable Discontinuous Func- Computable Functions tions Computable Compact Set • Many physical phenomena include discontinuous de- Analysis of the Problem pendencies y = f ( x ) (“jumps”). Additional Problem: . . . Resulting Definition • Examples: phase transitions, quantum transitions. Main Result • In other physical situations, for some values x , we may Inverse Relations: . . . have several possible values y . Home Page • From the mathematical viewpoint, this means that the Title Page relation between x and y is no longer a function. ◭◭ ◮◮ • It is a relation , aka constraint R ⊆ X × Y , or a multi- ◭ ◮ valued function . Page 2 of 15 • We thus need to know when a discontinuous and/or multi-valued function to be computable. Go Back Full Screen • Alas, the current definitions of computable functions are mostly limited to continuous case. Close Quit

Need to Define . . . Computable Numbers . . . 2. Computable Numbers and Metric Spaces: Re- Computable Functions minder Computable Compact Set • Intuitively, a real number x is computable if we can Analysis of the Problem compute it with any desired accuracy. Additional Problem: . . . Resulting Definition • Formally, x is computable if ∃ an algorithm that, given n ∈ N, returns a rational number r n s.t. | x − r n | ≤ 2 − n . Main Result Inverse Relations: . . . • A similar notion of computable elements can be defined Home Page for general metric spaces. Title Page • At each moment of time, we only have a finite amount ◭◭ ◮◮ of information about x . ◭ ◮ • Based on this information, we produce an approxima- Page 3 of 15 tion corresponding to this information. Go Back • Any information can be represented, in the computer, as a sequence of 0s and 1s. Full Screen Close Quit

Need to Define . . . Computable Numbers . . . 3. Computable Metric Spaces (cont-d) Computable Functions • Any 0-1 sequence can be, in turn, interpreted as a bi- Computable Compact Set nary integer n . Analysis of the Problem Additional Problem: . . . • Let � x n denote an approximation corresponding to an Resulting Definition integer n . Main Result • So, we require that in a computable metric space, there Inverse Relations: . . . is a sequence of such approximating elements { � x n } . Home Page • Computable means, in particular, that the distance Title Page d X ( � x m ) between such elements should be computable. x n , � ◭◭ ◮◮ • A metric space X with a sequence { � x n } is called com- ◭ ◮ putable if ∃ an algorithm m, n → d X ( � x n ). x m , � Page 4 of 15 • An element x ∈ X is called computable if there exists Go Back x k n , x ) ≤ 2 − n . an algorithm n → k n s.t. d X ( � Full Screen Close Quit

Need to Define . . . Computable Numbers . . . 4. Computable Functions Computable Functions • A f-n f : X → Y from comp. metric space X to Computable Compact Set comp. metric space Y is computable if ∃ algorithm s.t.: Analysis of the Problem Additional Problem: . . . – it uses x as an input, and Resulting Definition – it computes, for each integer n , a 2 − n -approximation Main Result y k to f ( x ). Inverse Relations: . . . • By “uses x as an input”, we mean that this algorithm Home Page can request, for each m , a 2 − m -approximation x ℓ to x . Title Page • Alas, all the functions computable according to this ◭◭ ◮◮ definition are continuous. ◭ ◮ • Thus, we cannot use this definition to check how well Page 5 of 15 we can compute a discontinuous function. Go Back Full Screen Close Quit

Need to Define . . . Computable Numbers . . . 5. Continuity Explained Computable Functions • Continuity of continuous functions is easy to under- Computable Compact Set stand. Analysis of the Problem Additional Problem: . . . • Lets us consider a simple discontinuous function f ( x ) = Resulting Definition sign( x ): Main Result • sign( x ) = 1 for x > 0; Inverse Relations: . . . • sign( x ) = 0 for x = 0; Home Page • sign( x ) = − 1 for x < 0. Title Page • Let us assume that we can compute sign( x ) with accu- ◭◭ ◮◮ racy 2 − 2 . ◭ ◮ • Then we would be able, given a comp. real number x , Page 6 of 15 to tell whether x = 0. Go Back • This is known to be algorithmically impossible. Full Screen Close Quit

Need to Define . . . Computable Numbers . . . 6. Computable Compact Set Computable Functions • In analyzing computability, it is often useful to start Computable Compact Set with pre-compact metric spaces X , where: Analysis of the Problem Additional Problem: . . . – for every positive real number ε > 0, Resulting Definition – there exists a finite ε -net L , i.e., Main Result ∀ x ∈ X ∃ y ∈ L ( d X ( x, y ) ≤ ε ) . Inverse Relations: . . . Home Page • A pre-compact set is compact if every converging se- Title Page quence has a limit. ◭◭ ◮◮ • A compact metric space X computable compact if: ◭ ◮ – X is a computable metric space, and Page 7 of 15 – there exists an algorithm that, given an integer n , returns a 2 − n -net L n for X . Go Back Full Screen Close Quit

Need to Define . . . Computable Numbers . . . 7. Simplifying Comment Computable Functions • Functions can also be undefined for some inputs x . Computable Compact Set Analysis of the Problem • This is easy to repair: if a relation is not everywhere Additional Problem: . . . defined: Resulting Definition – we can make it everywhere defined Main Result – if we consider, instead of the original set X , a pro- Inverse Relations: . . . jection of R on this set. Home Page • For example, a function √ x : Title Page – is not everywhere defined on the real line, but ◭◭ ◮◮ – it is everywhere defined on the set of all non-negative ◭ ◮ real numbers. Page 8 of 15 • Thus, without losing generality, we can assume that Go Back our relation R is everywhere defined: Full Screen ∀ x ∈ X ∃ y ∈ Y (( x, y ) ∈ R ) . Close Quit

Need to Define . . . Computable Numbers . . . 8. Analysis of the Problem Computable Functions • From the physical viewpoint, what does it mean that Computable Compact Set the dependence between x and y is computable? Analysis of the Problem Additional Problem: . . . • For a multi-valued function, for the same input x , we Resulting Definition may get several different values y . Main Result • In this case, it is desirable to compute the set of all Inverse Relations: . . . possible value y corresponding to a given x . Home Page • For compact Y , the set of x -possible values of y is pre- Title Page compact. ◭◭ ◮◮ • Thus, with any given accuracy, this set can be de- ◭ ◮ scribed by a finite list L of possible values: Page 9 of 15 – if y is a possible value of f ( x ), then y should be Go Back close to one of the values from L ; – vice versa, each value from L should be close to Full Screen some f ( x ). Close Quit

Need to Define . . . Computable Numbers . . . 9. Additional Problem: Discontinuity Computable Functions • Let us consider f ( x ) = sign( x ). Computable Compact Set Analysis of the Problem • At each stage of the computation, we only know an Additional Problem: . . . approximate value of x . Resulting Definition • So, even when actually x = 0, we cannot exclude that Main Result x > 0 or x < 0, so all 3 values (0, ± 1) are possible. Inverse Relations: . . . Home Page • In general, we need to take into account not only f ( x ) but also f ( x ′ ) for close x ′ . Title Page • In view of this, the above properties of the list L must ◭◭ ◮◮ be appropriately modified: ◭ ◮ – if y is a possible value of f ( x ′ ) for some x ′ ≈ x , Page 10 of 15 then y should be close to one of the values from L ; Go Back – for every value from L , there must exist a close y which is a possible value of f ( x ′ ) for some x ′ ≈ x . Full Screen Close Quit

Need to Define . . . Computable Numbers . . . 10. Resulting Definition Computable Functions • Let X and Y be computable compact sets with metrics Computable Compact Set d X and d Y . Analysis of the Problem Additional Problem: . . . • An everywhere defined relation R ⊆ X × Y is called Resulting Definition computable if there exists an algorithm that: Main Result – given a computable element x ∈ X and computable Inverse Relations: . . . positive numbers 0 < ε < ε ′ and 0 < δ , Home Page – produces a finite list { y 1 , . . . , y m } ⊆ Y Title Page such that: ◭◭ ◮◮ (1) if ( x ′ , y ) ∈ R for some x ′ for which d X ( x ′ , x ) ≤ ε , ◭ ◮ then there exists an i for which d Y ( y, y i ) ≤ δ ; Page 11 of 15 (2) ∀ element y i from this list, ∃ values x ′ and y for Go Back which d X ( x, x ′ ) ≤ ε ′ , d Y ( y i , y ) ≤ δ , and ( x ′ , y ) ∈ R . Full Screen Close Quit