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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 Shannon expansion ： ウィキペディア英語版 Boole's expansion theorem Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: $F\; =\; x\; \backslash cdot\; F\_x\; +\; x\text{'}\; \backslash cdot\; F\_x\text{'}$, where $F$ is any Boolean function and $F\_x$and $F\_x\text{'}$ are $F$ with the argument $x$ equal to $1$ and to $0$ respectively. The terms $F\_x$ and $F\_x\text{'}$ are sometimes called the positive and negative Shannon cofactors, respectively, of $F$ with respect to $x$. These are functions, computed by restrict operator, $restrict(F,\; x,\; 0)$ and $restrict(F,\; x,\; 1)$ (see valuation (logic) and partial application). It has been called the "fundamental theorem of Boolean algebra".〔Paul C. Rosenbloom, ''The Elements of Mathematical Logic'', 1950, p. 5〕 Besides its theoretical importance, it paved the way for binary decision diagrams, satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits. ==Statement of the theorem== A more explicit way of stating the theorem is- : $f(X\_1,\; X\_2,\; \backslash dots\; ,\; X\_n)\; =\; X\_1\; \backslash cdot\; f(1,\; X\_2,\; \backslash dots\; ,\; X\_n)\; +\; X\_1\text{'}\; \backslash cdot\; f(0,\; X\_2,\; \backslash dots\; ,\; X\_n)$ Proof for the statement follows from direct use of mathematical induction, from the observation that $f(X\_1)\; =\; X\_1.f(1)\; +\; X\_1\text{'}.f(0)$ and expanding a 2-ary and n-ary Boolean functions identically. 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「Boole's expansion theorem」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.