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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 Complexity index ： ウィキペディア英語版 Complexity index Besides complexity intended as a difficulty to compute a function (see computational complexity), in modern computer science and in statistics another complexity index of a function stands for denoting its information content, in turn affecting the difficulty of learning the function from examples. ''Complexity indices'' in this sense characterize the entire class of functions to which the one we are interested in belongs. Focusing on Boolean functions, the detail of a class $\backslash mathsf\; C$ of Boolean functions ''c'' essentially denotes how deeply the class is articulated. To identify this index we must first define a ''sentry function'' of $\backslash mathsf\; C$. Let us focus for a moment on a single function ''c'', call it a ''concept'' defined on a set $\backslash mathcal\; X$ of elements that we may figure as points in a Euclidean space. In this framework, the above function associates to ''c'' a set of points that, since are defined to be external to the concept, prevent it from expanding into another function of $\backslash mathsf\; C$. We may dually define these points in terms of sentinelling a given concept ''c'' from being fully enclosed (invaded) by another concept within the class. Therefore we call these points either ''sentinels'' or ''sentry points''; they are assigned by the sentry function $\backslash boldsymbol\; S$ to each concept of $\backslash mathsf\; C$ in such a way that: # the sentry points are external to the concept ''c'' to be sentineled and internal to at least one other including it, # each concept $c\text{'}$ including ''c'' has at least one of the sentry points of ''c'' either in the gap between ''c'' and $c\text{'}$, or outside $c\text{'}$ and distinct from the sentry points of $c\text{'}$, and # they constitute a minimal set with these properties. The technical definition coming from is rooted in the inclusion of an augmented concept $c^+$ made up of ''c'' plus its sentry points by another $\backslash left(c\text{'}\backslash right)^+$ in the same class. == Definition of sentry function == For a concept class $\backslash mathsf\; C$ on a space $\backslash mathfrak\; X$, a ''sentry function'' is a total function $\backslash boldsymbol\; S:\; \backslash mathsf\; C\backslash cup\backslash \backslash mapsto\; 2^$ satisfying the following conditions: # Sentinels are outside the sentineled concept ($c\backslash cap(c)=\backslash emptyset$ for all $c\backslash in\; \backslash mathsf\; C$). # Sentinels are inside the invading concept (Having introduced the sets $c^+=c\backslash cup\backslash boldsymbol\; S(c)$, an invading concept $c\text{'}\backslash in\; \backslash mathsf\; C$ is such that $c\text{'}\backslash not\backslash subseteq\; c$ and $c^+\backslash subseteq\; \backslash left(c\text{'}\backslash right)^+$. Denoting $\backslash mathrm(c)$ the set of concepts invading ''c'', we must have that if $c\_2\backslash in\backslash mathrm(c\_1)$, then $c\_2\backslash cap(c\_1)\backslash neq\backslash emptyset$). # $(c)$ is a minimal set with the above properties (No $\text{'}\backslash neq$ exists satisfying (1) and (2) and having the property that $\backslash boldsymbol\; S\text{'}(c)\backslash subseteq\; \backslash boldsymbol\; S(c)$ for every $c\backslash in\; \backslash mathsf\; C$). # Sentinels are honest guardians. It may be that $c\backslash subseteq\; \backslash left(c\text{'}\backslash right)^+$ but $(c)\backslash cap\; c\text{'}=\backslash emptyset$ so that $c\text{'}\backslash not\backslash in\backslash mathrm(c)$. This however must be a consequence of the fact that all points of $(c)$ are involved in really sentineling ''c'' against other concepts in $\backslash mathrm(c)$ and not just in avoiding inclusion of $c^+$ by $(c\text{'})^+$. Thus if we remove $c\text{'},\; (c)$ remains unchanged (Whenever $c\_1$ and $c\_2$ are such that $c\_1\backslash subset\; c\_2\backslash cup(c\_2)$ and $c\_2\backslash cap(c\_1)=\backslash emptyset$, then the restriction of $$ to $\backslash \backslash cup\backslash mathrm(c\_1)-\backslash$ is a sentry function on this set). $(c)$ is the ''frontier'' of ''c'' upon $\backslash boldsymbol\; S$. With reference to the picture on the right, $\backslash$ is a candidate frontier of $c\_0$ against $c\_1,c\_2,c\_3,c\_4$. All points are in the gap between a $c\_i$ and $c\_0$. They avoid inclusion of $c\_0\backslash cup\backslash$ in $c\_3$, provided that these points are not used by the latter for sentineling itself against other concepts. ''Vice versa'' we expect that $c\_1$ uses $x\_1$ and $x\_3$ as its own sentinels, $c\_2$ uses $x\_2$ and $x\_3$ and $c\_4$ uses $x\_1$ and $x\_2$ analogously. Point $x\_4$ is not allowed as a $c\_0$ sentry point since, like any diplomatic seat, it should be located outside all other concepts just to ensure that it is not occupied in case of invasion by $c\_0$. 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「Complexity index」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.