Computable Numbers, What is a Computable . . . Computable Sets, - PowerPoint PPT Presentation

What Is a Computable . . . What Is a Computable Set What Is a Computable . . . What Is a Computable . . . Computable Numbers, What is a Computable . . . Computable Sets, What is a Computable . . . What Is a Computable . . . Computable Functions, Examples of Positive . . . Examples of Negative . . . And How It Is All Related Home Page to Interval Computations Title Page ◭◭ ◮◮ Vladik Kreinovich ◭ ◮ Department of Computer Science Page 1 of 14 University of Texas at El Paso El Paso, TX 79968, USA Go Back vladik@utep.edu Full Screen http://www.cs.utep.edu/vladik Close Quit

What Is a Computable . . . What Is a Computable Set 1. What Is a Computable Number What Is a Computable . . . • From the physical viewpoint, real numbers x describe What Is a Computable . . . values of different quantities. What is a Computable . . . What is a Computable . . . • We get values of real numbers by measurements. What Is a Computable . . . • Measurements are never 100% accurate, so after a mea- Examples of Positive . . . surement, we get an approximate value r k of x . Examples of Negative . . . • In principle, we can measure x with higher and higher Home Page accuracy. Title Page • So, from the computational viewpoint, a real number ◭◭ ◮◮ is a sequence of rational numbers r k for which, e.g., ◭ ◮ | x − r k | ≤ 2 − k . Page 2 of 14 • By an algorithm processing real numbers, we mean an Go Back algorithm using r k as an “oracle” (subroutine). Full Screen • This is how computations with real numbers are de- fined in computable analysis . Close Quit

What Is a Computable . . . What Is a Computable Set 2. Relation to Interval Analysis What Is a Computable . . . • Once we know: What Is a Computable . . . What is a Computable . . . – the measurement result � x and What is a Computable . . . – the upper bound ∆ on the measurement error What Is a Computable . . . def ∆ x = � x − x , Examples of Positive . . . we can conclude that the actual value x belongs to the Examples of Negative . . . interval [ � x − ∆ , � x + ∆]. Home Page Title Page • In interval analysis, this is all we know: ◭◭ ◮◮ – we performed measurements (or estimates), ◭ ◮ – we get intervals, and – we want to extract as much information as possible Page 3 of 14 from these results. Go Back • In particular, we want to know what can we conclude Full Screen about y = f ( x 1 , . . . , x n ), where f is a known algorithm. Close Quit

What Is a Computable . . . What Is a Computable Set 3. Computable vs. Interval Analysis (cont-d) What Is a Computable . . . • In computable (constructive) analysis: What Is a Computable . . . What is a Computable . . . – we take into account that eventually, What is a Computable . . . – we will be able to measure each x i with higher and What Is a Computable . . . higher accuracy. Examples of Positive . . . • In other words, for each quantity, Examples of Negative . . . Home Page – instead of a single interval, Title Page – we have a sequence of narrower and narrower inter- vals, ◭◭ ◮◮ – a sequence that eventually converging to the actual ◭ ◮ value. Page 4 of 14 • “ Interval analysis is applied constructive analysis ” Go Back (Yuri Matiyasevich, of 10th Hilbert problem fame). Full Screen Close Quit

What Is a Computable . . . What Is a Computable Set 4. Constructive vs. Computable Analysis What Is a Computable . . . • There is a subtle difference between constructive anal- What Is a Computable . . . ysis and computable analysis . What is a Computable . . . What is a Computable . . . • Crudely speaking, constructive analysis only considers What Is a Computable . . . objects that can be algorithmically constructed. Examples of Positive . . . • E.g., we only allow computable real numbers. Examples of Negative . . . Home Page • In contrast, computable analysis takes into account that inputs can be non-computable. Title Page • E.g., measurement results are often random. ◭◭ ◮◮ • Computable analysis checks what we can compute ◭ ◮ based on such – possibly non-computable – inputs. Page 5 of 14 Go Back Full Screen Close Quit

What Is a Computable . . . What Is a Computable Set 5. Computable Analysis: Typical Questions What Is a Computable . . . • In general: What Is a Computable . . . What is a Computable . . . – once we know x i with more and more accuracy, What is a Computable . . . – we can usually find y = f ( x 1 , . . . , x n ) with more What Is a Computable . . . and more accuracy. Examples of Positive . . . • This means that the corresponding function Examples of Negative . . . f ( x 1 , . . . , x n ) is computable. Home Page • One of the possible questions is: which questions about Title Page y = f ( x 1 , . . . , x n ) we will be able to eventually answer? ◭◭ ◮◮ • Example: if y > 0, then we will eventually be able to ◭ ◮ confirm this. Page 6 of 14 • On the other hand, no matter how accurately we mea- Go Back sure, we will never be able to check whether y = 0. Full Screen Close Quit

What Is a Computable . . . What Is a Computable Set 6. What Is a Computable Set What Is a Computable . . . • In a computer, we can only store finitely many objects What Is a Computable . . . – i.e., a finite set, with computable distances. What is a Computable . . . What is a Computable . . . • It is therefore reasonable to define a computable set as What Is a Computable . . . a set S that: Examples of Positive . . . – can be algorithmically approximated, with any Examples of Negative . . . given accuracy, Home Page – by finite sets. Title Page • Approximated means that every x ∈ S is 2 − n -close to ◭◭ ◮◮ one of the elements from the approximating finite set. ◭ ◮ • Elements of these finite sets approximate our set with Page 7 of 14 higher and higher accuracy. Go Back Full Screen Close Quit

What Is a Computable . . . What Is a Computable Set 7. What Is a Computable Set (cont-d) What Is a Computable . . . • A computer has a linear memory, so it is convenient to What Is a Computable . . . place these elements into an infinite sequence What is a Computable . . . What is a Computable . . . x 1 , x 2 , . . . What Is a Computable . . . • Elements from this sequence approximate any element Examples of Positive . . . from the given set. Examples of Negative . . . • Thus, this sequence must be everywhere dense in this Home Page set. Title Page • In practice, we do not know the exact values of the ◭◭ ◮◮ elements. ◭ ◮ • We only have approximations to elements of the set. Page 8 of 14 • Based on these approximations, we can never know Go Back whether the resulting set is closed or not. Full Screen • For example, whether a set of real numbers is the in- terval [ − 1 , 1] or the same interval minus 0 point. Close Quit

What Is a Computable . . . What Is a Computable Set 8. What Is a Computable Set (final) What Is a Computable . . . • To ignore such un-detectable differences, it is reason- What Is a Computable . . . able to assume: What is a Computable . . . What is a Computable . . . – that our set is complete , What Is a Computable . . . – i.e., that it includes the limit of each converging Examples of Positive . . . sequence. Examples of Negative . . . • Thus, we arrive at the following definition. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 14 Go Back Full Screen Close Quit

What Is a Computable . . . What Is a Computable Set 9. What is a Computable Set: Definition What Is a Computable . . . • By a computable set , we mean a complete metric space What Is a Computable . . . with an everywhere dense sequence { x i } for which: What is a Computable . . . What is a Computable . . . • ∃ an algorithm that, given i and j , computes the What Is a Computable . . . distance d ( x i , x j ) (with any given accuracy), and Examples of Positive . . . • There exists an algorithm that: Examples of Negative . . . • given a natural number n , Home Page • returns a natural number N ( n ) for which every Title Page point x 1 , x 2 , . . . is 2 − n -close to one of the points ◭◭ ◮◮ x 1 , . . . , x N ( n ) . ◭ ◮ • By a computable element x of a computable set, we Page 10 of 14 mean an algorithm that: Go Back • given a natural number n , Full Screen • returns an integer i ( n ) for which d ( x, x i ( n ) ) ≤ 2 − n . Close Quit

What Is a Computable . . . What Is a Computable Set 10. What is a Computable Function: What Is a Computable . . . Intuitive Idea What Is a Computable . . . • A computable function f should be able: What is a Computable . . . What is a Computable . . . • given a computable real number (or, more gener- What Is a Computable . . . ally, a computable element of a computable set), Examples of Positive . . . • to compute the value f ( x ) with any given accuracy. Examples of Negative . . . • Computable elements x are given by their approxima- Home Page tions. Title Page • Thus, to compute f ( x ) with a given accuracy 2 − n , we ◭◭ ◮◮ need to: ◭ ◮ • determine how accurately we need to compute x to achieve the desired accuracy 2 − n in f ( x ), Page 11 of 14 • then use the corresponding approximation to x to Go Back compute the desired approximation to f ( x ). Full Screen • So, we arrive at the following definition. Close Quit