Computable Number - Informal Definition Using A Turing Machine As Example

Informal Definition Using A Turing Machine As Example

In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936, i.e. as "sequences of digits interpreted as decimal fractions" between 0 and 1:

"A computable number one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number ." (Minsky 1967:159)

The key notions in the definition are (1) that some n is specified at the start, (2) for any n the computation only takes a finite number of steps, after which the machine produces the desired output and terminates.

An alternate form of (2) – the machine successively prints all n of the digits on its tape, halting after printing the nth – emphasizes Minsky's observation: (3) That by use of a Turing machine, a finite definition – in the form of the machine's table – is being used to define what is a potentially-infinite string of decimal digits.

This is however not the modern definition which only requires the result be accurate to within any given accuracy. The informal definition above is subject to a rounding problem called the table-maker's dilemma whereas the modern definition is not.