Computable

For example, the least upper bound of a bounded increasing computable sequence of computable real numbers need not be a computable real number (Bridges and Richman, 1987:58).

A complex number is called computable if its real and imaginary parts are computable.

Alan Turing writes, "all digital computers are in a sense equivalent."sfn The widely accepted Church-Turing thesis holds that any function computable by an effective procedure is computable by a Turing machine.

All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive.

A real number is computable if and only if the set of natural numbers it represents (when written in binary and viewed as a characteristic function) is computable.

Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods.

Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them.

Digit strings and the Cantor and Baire spaces Turing's original paper defined computable numbers as follows: :A real number is computable if its digit sequence can be produced by some algorithm or Turing machine.

Every computable number is arithmetical, but not every arithmetical number is computable.

For example, Rice's theorem shows that each of the following sets of computable functions is undecidable: * The class of computable functions that return 0 for every input, and its complement.

Formal statement Let be an admissible numbering of the computable functions ; a map from the natural numbers to the class of unary (partial) computable functions.

His argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable.

Jürgen Schmidhuber (2000) constructed a limit-computable "Super Omega" which in a sense is much more random than the original limit-computable Omega, as one cannot significantly compress the Super Omega by any enumerating non-halting algorithm.

The entire set of computable numbers is countable, so most reals are not computable.

These definitions are equivalent from the point of view of Turing computability: a function is oracle-computable from a given oracle under all of these definitions if it is oracle-computable under any of them.

Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers.

While the full order relation is not computable, the restriction of it to pairs of unequal numbers is computable.

The note is prepared using Computable General Equilibrium Models developed by UNDP for Afghanistan.

A year later, he published the groundbreaking paper “On Computable Numbers, With an Application to the Entscheidungsproblem” (or “decidability problem”), a reference in German to a celebrated riddle that the American logician had also explained.

After finite time the first n bits of the output will never change any more (it does not matter that this time itself is not computable by a halting program).