What is a non-computable function?
Yet there are also problems and functions that that are non-computable (or undecidable or uncomputable), meaning that there exists no algorithm that can compute an answer or output for all inputs in a finite number of simple steps.
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 every function computable?
There is a Turing machine program with the property that for any function f : N → N on the natural numbers, including non-computable functions, there is a model of arithmetic or set theory inside of which the function computed by agrees exactly with on all standard finite input. …
Is Ackermann function computable?
The Ackermann function is the simplest example of a well-defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Dötzel 1991).
How do you know if a function is computable?
To summarise, based on this view a function is computable if: (a) given an input from its domain, possibly relying on unbounded storage space, it can give the corresponding output by following a procedure (program, algorithm) that is formed by a finite number of exact unambiguous instructions; (b) it returns such …
Is a problem computable?
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 the attributes are used in computable functions?
The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute the function. The models of computation listed above give different interpretations of what a procedure is and how it is used, but these interpretations share many properties.
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.
Can Ackermann’s function be coded using for or do loops?
The Ackermann function was discovered and studied by Wilhelm Ackermann (1896–1962) in 1928. The Ackermann function can only be calculated using a while loop, which keeps repeating an action until an associated test returns false.
Are recursive functions computable?
A general recursive function is called total recursive function if it is defined for every input, or, equivalently, if it can be computed by a total Turing machine. There is no way to computably tell if a given general recursive function is total – see Halting problem.
The function f such that f ( n) = 1 if there is a sequence of at least n consecutive fives in the decimal expansion of π, and f ( n) = 0 otherwise, is computable. (The function f is either the constant 1 function, which is computable, or else there is a k such that f ( n) = 1 if n < k and f ( n) = 0 if n ≥ k.
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 .
What are the two types of computable functions?
Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions . Before the precise definition of computable function, mathematicians often used the informal term effectively calculable.
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).