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.
Read more about this topic: Computable Number
Famous quotes containing the words informal, definition and/or machine:
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The white man regards the universe as a gigantic machine hurtling through time and space to its final destruction: individuals in it are but tiny organisms with private lives that lead to private deaths: personal power, success and fame are the absolute measures of values, the things to live for. This outlook on life divides the universe into a host of individual little entities which cannot help being in constant conflict thereby hastening the approach of the hour of their final destruction.”
—Policy statement, 1944, of the Youth League of the African National Congress. pt. 2, ch. 4, Fatima Meer, Higher than Hope (1988)