Are all functions computable? I’d like to share a simple proof I’ve discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function!
What functions are not computable?
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.
How do you know if a function is computable?
As φp(x)↓ for all x≥1, g(p)=1 if and only if φp(p)↓ by the definition of φp, which is actually the function g. Hence, if g would be computable, the halting problem would be computable as well.
Are all continuous functions computable?
A famous result in intuitionistic mathematics is that all real-valued total functions are continuous. Since the requirements for a function to be admitted intuitionistically is that it must define a procedure or algorithm,
all functions are computable
. This seems to suggest that all computable functions are continuous.
Are all real numbers computable?
Real numbers used in any explicit way in traditional mathematics are always computable in this sense. But as Turing pointed out,
the overwhelming majority of all possible real numbers are not computable
. For certainly there can be no more computable real numbers than there are possible Turing machines.
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).
Are all solvable problems computable?
A mathematical problem is computable if it can be solved in principle by a computing device. Some common synonyms for “computable” are “solvable”, “decidable”, and “recursive”.
Hilbert believed that all mathematical problems were solvable, but in the 1930’s Gödel, Turing, and Church showed that this is not the case
.
What is the meaning of computable?
Definition of computable
:
capable of being computed
.
Why is busy beaver not computable?
The busy beaver function is non-computable,
because it grows faster than any computable function
! Proof: Let f be any computable function. Consider M: X then M
F
then M
F
where X: blank → x 1’s. Note X has x states.
What is a total 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.
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.
What is a partially computable function?
Definition: A partial function is
a function f : (N ∪ {∞})n → N ∪ {∞},n ≥ 0 such that f(c1, …, cn) = ∞ if some ci = ∞
. In the context of computability theory, whenever we refer to a function on N, we mean a partial function in the above sense. Definitions: Domain(f) = { x ∈ Nn | f( x) = ∞}
Are rational numbers computable?
It turns out that almost every number is uncomputable. To understand this we first introduce the concept of a set being countable. A set is called countable if it can be put in one-to-one coorespondence with the integers. For instance,
rational numbers are countable
.
Are all rational numbers constructible?
All rational numbers are constructible
, and all constructible numbers are algebraic numbers (Courant and Robbins 1996, p. 133). If a cubic equation with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins 1996, p. 136).
Is an irrational number computable?
However, the set of all irrational numbers is uncountable, so there must be some irrational number whose decimal expansion is not computable! In fact, since only countably many irrational numbers can be computed,
“most” irrational numbers are not computable
!
Why is Ackermann function not primitive recursive?
Not primitive recursiveEdit
The Ackermann function
grows faster than any primitive recursive function
and therefore is not itself primitive recursive.
Does Ackermann function terminate?
The Ackermann function does indeed terminate for all natural number inputs
, but there’s no way to give a natural number mea- sure which proves it. Instead we need to use something more general. In fact, as a measure we can use any well-founded relation.
How do we define the Ackermann’s function?
(algorithm) Definition:
A function of two parameters whose value grows very fast
. Formal Definition: A(0, j)=j+1 for j ≥ 0.
What problems are not computable?
Non-Computable Problems – A non-computable is a problem for which there is no algorithm that can be used to solve it. The most famous example of a non-computability (or undecidability) is the
Halting Problem
.
Are all NP problems solvable?
(i)
All NP-complete problems are solvable in polynomial time
: Yes. Every problem in NP is polynomially reducible to SAT, and SAT is reducible to every NP-hard problem. Therefore, a polynomial time solution to any NP-hard problem (such as 3Col) implies that every problem in NP can be solved in polynomial time.
What are tractable and intractable problems?
Tractable Problem:
a problem that is solvable by a polynomial-time algorithm. The upper bound is polynomial. Intractable Problem: a problem that cannot be solved by a polynomial-time al- gorithm
.
What is the difference between computability and complexity?
Put succinctly,
computability theory is concerned with what can be computed versus what cannot; complexity is concerned with the resources required to compute the things that are computable
.
What do you mean by Optimisation?
Definition of optimization
:
an act, process, or methodology of making something (such as a design, system, or decision) as fully perfect, functional, or effective as possible
specifically : the mathematical procedures (such as finding the maximum of a function) involved in this.
What do you mean by compatibility?
Compatibility is
the capacity for two systems to work together without having to be altered to do so
. Compatible software applications use the same data formats. For example, if word processor applications are compatible, the user should be able to open their document files in either product.
Is Busy Beaver Undecidable?
The Busy Beaver numbers are an undecidable set
; if they were decidable, we could figure out BB(n) for each n, enabling us to decide the halting problem. They are also not recursively enumerable, but for a trickier reason.
What does Busy Beaver mean?
Hardworking, very industrious
, as in With all her activities, Sue is always busy as a bee, or Bob’s busy as a beaver trying to finish painting before it rains. The comparison to beavers dates from the late 1700s, the variant from the late 1300s. Also see eager beaver; work like a beaver.
Does Busy Beaver grow faster than tree?
Since TREE is a computable function
the BB function grows faster than it
, but TREE seems to grow much more quickly early on, so when does Busy Beaver surpass it?
What does it mean for a function to be total?
(definition) Definition:
A function which is defined for all inputs of the right type, that is, for all of a domain
. See also partial function. Note: Square (x2) is a total function.
Is Uncomputable a word?
Uncomputable definition
Not computable; that cannot be computed
.
What does it mean to compute a function?
1. The term compute can have multiple meanings, but the basic meaning is
the act of calculating a value using mathematical formulas or other methods
. A computer processes many functions by computing information, which in its simplest form, is a series of ones and zeroes (called binary code).
What is a total 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.
What does it mean for a problem to be computable?
A mathematical problem is computable
if it can be solved in principle by a computing device
. Some common synonyms for “computable” are “solvable”, “decidable”, and “recursive”. Hilbert believed that all mathematical problems were solvable, but in the 1930’s Gödel, Turing, and Church showed that this is not the case.
What is the meaning of computable?
Definition of computable
:
capable of being computed
.
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.
