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 …

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.

Why is busy beaver not computable?

The busy beaver function BB(n) describes the maximum number of steps that an n-state Turing machine can execute before it halts (assuming it halts at all). It is not a computable function because computing it allows you to solve the halting problem.

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.

What does it mean to be effectively computable?

is effectively computable if there is an effective procedure or algorithm that correctly calculates f. An effective procedure is one that meets the following specifications.

How might we recognize that a process in nature computes a function not computable by a Turing machine?

If we can describe it (in principle) using quantum mechanics, then a Turing machine can simulate the physical process and compute the same function. This by itself cuts out a lot of possible physical systems. We know that there are functions not computable by a Turing machine.

How did the term busy beaver come about?

Taken from the phrase “busy as a beaver,” referring to beavers’ reputation for being extremely industrious. Between working two part-time jobs, volunteering on the weekends, and looking after his little brother, Sam’s been a busy beaver this summer.

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. …

What is the meaning of computable?

Definition of computable : capable of being computed.

Can Church Turing thesis be proved?

There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle.

What is Church’s Theorem?

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. If first-order logic were decidable, would also be decidable.

Is there a total unary function that is not computable?

Source: Computability, An introduction to recursive function theory by Nigel Cutland Cambridge UP 1980 Chapter 4 Numbering computable functions Theorem 2.6 There is a total unary function that is not computable. I am pleased to inform you that you can stop looking; it is well known that such a function cannot exist.

What is an example of a non-computable function?

The function constructed in the theorem you cite (Theorem 2.6 of Chapter 4 of Cutland’s book) is a good (and I would even say, ‘concrete’) example of an non-computable function. Note that most functions from N to N are incomputable. However, (roughly speaking) most such functions that arise in mathematics are computable.

Is Cutland’s total function non-computable?

However, (roughly speaking) most such functions that arise in mathematics are computable. You have a misunderstanding of what a “primitive recursive function” is. Cutland indeed proves nicely the existence of a total function which is non-computable; however, it is not primitive recursive!