How Do You Know If A Function Is Computable?

If S=∅ or S=RE then the property is said to be trivial, and the induced LS is computable. An example for a simple S is one the contains only a single language, say Scomplete={Σ∗}.

What is Turing computable function TOC?

A function is Turing computable if the function’s value can be computed with a Turing machine . More specifically, let D be a set of words in a given alphabet and let f be a function which maps elements of D to words on the same alphabet.

What makes a problem 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?

: capable of being computed.

What isn’t computable?

A non-computable is a problem for which there is no algorithm that can be used to solve it. Most famous example of a non-computablity (or undecidability) is the Halting Problem.

What are computable predicates?

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?

A partial function f : N k → N. 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.

Are irrational numbers 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!

What is effectively computable function?

computable function

(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 are computable numbers?

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.

What did Alan Turing’s 1936 paper on computable numbers prove?

Turing showed, by means of his universal machine, that mathematics was also undecidable. To demonstrate this, Turing came up with the concept of “computable numbers,” which are numbers defined by some definite rule, and thus calculable on the universal machine.

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.

Are all primitive recursive functions computable?

Since the primitive recursive functions are a subset of µ-recursive functions they are clearly computable in the sense of µ recursive functions. Since all µ recursive functions are Turing computable, clearly all primitive recursive functions are Turing computable as well.

What is total computable function?

we define a total computable function F(x, y) of two variables whose range is A: F(x, y) = { x if R(x, y) holds. a otherwise. Then A = ran(F).

Why we define the Turing reducibility by computability relative to set?

Every set is Turing equivalent to its complement. Every computable set is Turing reducible to every other set. Because any computable set can be computed with no oracle, it can be computed by an oracle machine that ignores the given oracle.

What is a Turing machine in theory of computation?

A Turing machine is a mathematical model of computation that defines an abstract machine that manipulates symbols on a strip of tape according to a table of rules. … The Turing machine was invented in 1936 by Alan Turing, who called it an “a-machine” (automatic machine).

What are the types of computability?

The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power.

Is the empty set computable?

The empty set is 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.

Is Uncomputable a word?

Not computable; that cannot be computed.

What is computability theory in computer science?

Computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. A central question of computer science is to address the limits of computing devices by understanding the problems we can use computers to solve.

Are all integers 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.

Why are computable numbers countable?

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. … This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals.

Is Pi computable number?

1 Answer. Yes, π is computable. There are a few equivalent definitions of computable, but the most useful one here is the one you have given above: a real number r is computable if there exists an algorithm to find its n th digit.