What makes a number computable?

What makes a number computable?

In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals.

Are the computable numbers complete?

While the set of real numbers is uncountable, the set of computable numbers is only countable and thus almost all real numbers are not computable. That the computable numbers are at most countable intuitively comes from the fact that they are produced by Turing machines, of which there are only countably many.

What is effectively computable function?

computable function [kəm¦pyüd·ə·bəl ′fəŋk·shən] (mathematics) A function whose value can be calculated by some Turing machine in a finite number of steps. Also known as effectively computable function.

Is the set of natural numbers computable?

The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbers less than a given natural number is computable.

What is the largest computable number?

c, a C program by Ralph Loader that came in first place for the Bignum Bakeoff contest, whose objective was to write a C program (in 512 characters or less) that generates the largest possible output on a theoretical machine with infinite memory. It is among the largest computable numbers ever devised.

Are the computable reals countable?

In the usual set theoretic tradition computable real numbers are countable since there are only a countable quantity of Turing Machines.

How do you know if a function is computable?

According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that a function is computable if and only if it has an algorithm.

How do you prove a set is computable?

The set A is computable via the following algorithm: on input x run both Ma(x) and Mb(x) simul- taneously; if Ma(x) halts then output YES, if Mb(x) halts then output NO. Since either x ∈ A or x ∈ A, one of these two events must happen.

Are all finite sets computable?

According to wikipedia, every finite set is computable. Definition: set S⊂N is computable if there exists an algorithm which defines in finite time if a given number n is in Set.

What is computability in theory of computation?

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.

What are computable functions?

Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output.

Why do we use a number system?

A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects. Common examples of numberings include Gödel numberings in first-order logic and admissible numberings of the set of partial computable functions.

How do you formalize computable functions?

For example, one can formalize computable functions as μ-recursive functions, which are partial functions that take finite tuples of natural numbers and return a single natural number (just as above).

What is an example of an uncountable function?

Uncomputable functions and unsolvable problems. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin’s constant .