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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Course-of-values recursion ： ウィキペディア英語版 Course-of-values recursion In computability theory, course-of-values recursion is a technique for defining number-theoretic functions by recursion. In a definition of a function ''f'' by course-of-values recursion, the value of ''f''(''n''+1) is computed from the sequence $\backslash langle\; f(1),f(2),\backslash ldots,f(n)\backslash rangle$. The fact that such definitions can be converted into definitions using a simpler form of recursion is often used to prove that functions defined by course-of-values recursion are primitive recursive. This article uses the convention that the natural numbers are the set . ==Definition and examples == The factorial function ''n''! is recursively defined by the rules :0! = 1, :(''n''+1)! = (''n''+1) *(''n''!). This recursion is a primitive recursion because it computes the next value (''n''+1)! of the function based on the value of ''n'' and the previous value ''n''! of the function. On the other hand, the function Fib(''n''), which returns the ''n''th Fibonacci number, is defined with the recursion equations :Fib(0) = 0, :Fib(1) = 1, :Fib(''n''+2) = Fib(''n''+1) + Fib(''n''). In order to compute Fib(''n''+2), the last ''two'' values of the Fib function are required. Finally, consider the function ''g'' defined with the recursion equations :g(0) = 0, :$g(n+1)\; =\; \backslash sum\_^\; g(i)^$. To compute ''g''(''n''+1) using these equations, all the previous values of ''g'' must be computed; no fixed finite number of previous values is sufficient in general for the computation of ''g''. The functions Fib and ''g'' are examples of functions defined by course-of-values recursion. In general, a function ''f'' is defined by course-of-values recursion if there is a fixed primitive recursive function ''h'' such that for all ''n'', :$f(n)\; =\; h(n,\backslash langle\; f(0),\; f(1),\; \backslash ldots,\; f(n-1)\backslash rangle)$ where $\backslash langle\; f(0),\; f(1),\; \backslash ldots,\; f(n-1)\backslash rangle$ is a Gödel number encoding the indicated sequence. In particular :$f(0)\; =\; h(0,\backslash langle\backslash rangle),$ provides the initial value of the recursion. The function ''h'' might test its first argument to provide explicit initial values, for instance for Fib one could use the function defined by : where ''s''() denotes extraction of the element ''i'' from an encoded sequence ''s''; this is easily seen to be a primitive recursive function (assuming an appropriate Gödel numbering is used). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Course-of-values recursion」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.