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 JJ II Vladik Kreinovich J I

Department of Computer Science Page1 of 14 University of Texas at El Paso El Paso, TX 79968, USA Go Back [email protected] Full Screen

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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 . . . • We get values of real numbers by measurements. What is a Computable . . . What Is a Computable . . . • Measurements are never 100% accurate, so after a mea- Examples of Positive . . . surement, we get an approximate value rk 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 JJ II is a sequence of rational numbers rk for which, e.g., −k J I |x − rk| ≤ 2 . Page2 of 14 • By an algorithm processing real numbers, we mean an Go Back algorithm using rk as an “oracle” (subroutine). • This is how computations with real numbers are de- Full Screen fined in computable analysis. Close

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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 e What is a Computable . . . – the upper bound ∆ on the measurement error What Is a Computable . . . ∆x def= x − x, e Examples of Positive . . . we can conclude that the actual value x belongs to the Examples of Negative . . . Home Page interval [xe − ∆, xe + ∆]. • In interval analysis, this is all we know: Title Page – we performed measurements (or estimates), JJ II – we get intervals, and J I – we want to extract as much information as possible Page3 of 14

from these results. Go Back

• In particular, we want to know what can we conclude Full Screen about y = f(x , . . . , x ), where f is a known algorithm. 1 n Close

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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 xi 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, – we have a sequence of narrower and narrower inter- Title Page vals, JJ II

– a sequence that eventually converging to the actual J I

value. Page4 of 14

• “Interval analysis is applied constructive analysis” Go Back (Yuri Matiyasevich, of 10th Hilbert problem fame). Full Screen

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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 objects that can be algorithmically constructed. What Is a Computable . . . Examples of Positive . . .

• E.g., we only allow computable real numbers. Examples of Negative . . . • In contrast, computable analysis takes into account Home Page that inputs can be non-computable. Title Page

• E.g., measurement results are often random. JJ II

• Computable analysis checks what we can compute J I

based on such – possibly non-computable – inputs. Page5 of 14

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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 xi with more and more accuracy, What is a Computable . . . – we can usually find y = f(x , . . . , x ) with more 1 n What Is a Computable . . . and more accuracy. Examples of Positive . . .

• This means that the corresponding function Examples of Negative . . . f(x1, . . . , xn) is computable. Home Page • One of the possible questions is: which questions about Title Page

y = f(x1, . . . , xn) we will be able to eventually answer? JJ II

• Example: if y > 0, then we will eventually be able to J I confirm this. Page6 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

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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 a set S that: What Is a Computable . . . Examples of Positive . . .

– can be algorithmically approximated, with any Examples of Negative . . .

given accuracy, Home Page

– by finite sets. Title Page −n • Approximated means that every x ∈ S is 2 -close to JJ II one of the elements from the approximating finite set. J I • Elements of these finite sets approximate our set with Page7 of 14 higher and higher accuracy. Go Back

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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 . . . x1, x2,... 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 JJ II elements. J I • We only have approximations to elements of the set. Page8 of 14 • Based on these approximations, we can never know whether the resulting set is closed or not. Go Back • For example, whether a set of real numbers is the in- Full Screen terval [−1, 1] or the same interval minus 0 point. Close

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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

JJ II

J I

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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 {xi} 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(xi, xj) (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 −n point x1, x2,... is 2 -close to one of the points JJ II x1, . . . , xN(n). J I

• 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 −n • returns an integer i(n) for which d(x, xi(n)) ≤ 2 . Close

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What Is a Computable Set 10. What is a Computable Function: Intuitive Idea What Is a Computable . . . What Is a Computable . . .

• A computable function f should be able: What is a Computable . . . • given a computable real number (or, more gener- What is a Computable . . . ally, a computable element of a computable set), What Is a Computable . . . 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 JJ II need to: J I • 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

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What Is a Computable Set 11. What Is a Computable Function: Definition What Is a Computable . . . • We say that a function f(x) from a computable set to What Is a Computable . . . real numbers is computable if: What is a Computable . . .

What is a Computable . . . • first, we have an algorithm that, given n, returns m for which d(x, x0) ≤ 2−m implies that What Is a Computable . . . Examples of Positive . . . 0 −n |f(x) − f(x )| ≤ 2 , and Examples of Negative . . .

Home Page • second, we have an algorithm that, given i, com- Title Page putes f(xi). • The existence of m for every n is nothing else but uni- JJ II form continuity. J I • So, in effect, we want f(x) to be effectively uniformly Page 12 of 14

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What Is a Computable Set 12. Examples of Positive Results What Is a Computable . . . • We can algorithmically compute the maximum M of a What Is a Computable . . . computable function f(x) on a computable set X: What is a Computable . . .

What is a Computable . . . – for given ε > 0, we know what accuracy δ > 0 we need for x to get f(x) with accuracy ε; What Is a Computable . . . Examples of Positive . . . – so, we find a δ-net {x1, . . . , xn} ⊆ X, Examples of Negative . . .

– then max(f(x1), . . . , f(xn)) is the desired ε- Home Page approximation to M. Title Page • A more complex example: for every computable func- JJ II tion f(x) on a computable set: J I • for every four rational numbers r1 < r1 < r2 < r2, Page 13 of 14 • we can algorithmically find values b1 ∈ (r1, r1) and Go Back b2 ∈ (b2, b2) for which Full Screen {x : b1 ≤ f(x) ≤ b2} is a computable set. Close

Quit 13. Examples of Negative Results What Is a Computable . . . What Is a Computable Set What Is a Computable . . . • No algorithm is possible that, given two numbers x and What Is a Computable . . . y, would check whether x = y. What is a Computable . . . What is a Computable . . . • This follows from the halting problem: it is not possible What Is a Computable . . . to check whether a given algorithm halts on given data. Examples of Positive . . . Examples of Negative . . . • More complex examples: Home Page

– No algorithm is possible that, given f, returns x Title Page such that f(x) = 0. JJ II – No algorithm is possible that, given f, returns x s.t. f(x) = max f(y) (but max f(y) is computable.) J I y∈K y∈K Page 14 of 14 – No algorithm is possible that, given f, returns x such that f(x) = x. Go Back

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