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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 Alpha recursion theory ： ウィキペディア英語版 Alpha recursion theory In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals $\backslash alpha$. An admissible ordinal is closed under $\backslash Sigma\_1(L\_\backslash alpha)$ functions. Admissible ordinals are models of Kripke–Platek set theory. In what follows $\backslash alpha$ is considered to be fixed. The objects of study in $\backslash alpha$ recursion are subsets of $\backslash alpha$. A is said to be $\backslash alpha$ recursively enumerable if it is $\backslash Sigma\_1$ definable over $L\_\backslash alpha$. A is recursive if both A and $\backslash alpha\; /\; A$ (its complement in $\backslash alpha$) are $\backslash alpha$ recursively enumerable. Members of $L\_\backslash alpha$ are called $\backslash alpha$ finite and play a similar role to the finite numbers in classical recursion theory. We say R is a reduction procedure if it is $\backslash alpha$ recursively enumerable and every member of R is of the form $\backslash langle\; H,J,K\; \backslash rangle$ where ''H'', ''J'', ''K'' are all α-finite. ''A'' is said to be α-recursive in ''B'' if there exist $R\_0,R\_1$ reduction procedures such that: : $K\; \backslash subseteq\; A\; \backslash leftrightarrow\; \backslash exists\; H:\; \backslash exists\; J:(H,J,K\; \backslash rangle\; \backslash in\; R\_0\; \backslash wedge\; H\; \backslash subseteq\; B\; \backslash wedge\; J\; \backslash subseteq\; \backslash alpha\; /\; B),$ : $K\; \backslash subseteq\; \backslash alpha\; /\; A\; \backslash leftrightarrow\; \backslash exists\; H:\; \backslash exists\; J:(H,J,K\; \backslash rangle\; \backslash in\; R\_1\; \backslash wedge\; H\; \backslash subseteq\; B\; \backslash wedge\; J\; \backslash subseteq\; \backslash alpha\; /\; B).$ If ''A'' is recursive in ''B'' this is written $\backslash scriptstyle\; A\; \backslash le\_\backslash alpha\; B$. By this definition ''A'' is recursive in $\backslash scriptstyle\backslash varnothing$ (the empty set) if and only if ''A'' is recursive. However A being recursive in B is not equivalent to A being $\backslash Sigma\_1(L\_\backslash alpha())$. We say ''A'' is regular if $\backslash forall\; \backslash beta\; \backslash in\; \backslash alpha:\; A\; \backslash cap\; \backslash beta\; \backslash in\; L\_\backslash alpha$ or in other words if every initial portion of ''A'' is α-finite. ==Results in $\backslash alpha$ recursion== Shore's splitting theorem: Let A be $\backslash alpha$ recursively enumerable and regular. There exist $\backslash alpha$ recursively enumerable $B\_0,B\_1$ such that Shore's density theorem: Let ''A'', ''C'' be α-regular recursively enumerable sets such that then there exists a regular α-recursively enumerable set ''B'' such that . 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「Alpha recursion theory」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.