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Computation in the limit
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Computation in the limit : ウィキペディア英語版
Computation in the limit
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in the limit and limit recursive are also used. One can think of limit computable functions as those admitting an eventually correct computable guessing procedure at their true value. A set is limit computable just when its characteristic function is limit computable.
If the sequence is uniformly computable relative to ''D'', then the function is limit computable in ''D''.
== Formal definition ==

A total function r(x) is limit computable if there is a total computable function \hat(x,s) such that
: \displaystyle \hat = \lim_ \hat(x,s)
The total function r(x) is limit computable in ''D'' if there is a total function \hat(x,s) computable in ''D'' also satisfying
: \displaystyle \hat = \lim_ \hat(x,s)
A set of natural numbers is defined to be computable in the limit if and only if its characteristic function is computable in the limit. In contrast, the set is computable if and only if it is computable in the limit by a function \phi(t,i) and there is a second computable function that takes input ''i'' and returns a value of ''t'' large enough the \phi(t,i) has stabilized.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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