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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 Zhegalkin polynomial ： ウィキペディア英語版 Zhegalkin polynomial Zhegalkin (also Zegalkin or Gegalkine) polynomials form one of many possible representations of the operations of boolean algebra. Introduced by the Russian mathematician I. I. Zhegalkin in 1927, they are the polynomials of ordinary high school algebra interpreted over the integers mod 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, ''x''2 = ''x''. Hence a polynomial such as 3''x''2''y''5''z'' is congruent to, and can therefore be rewritten as, ''xyz''. ==Boolean equivalent== Prior to 1927 boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, etc. Zhegalkin showed that all boolean operations could be written as ordinary numeric polynomials, thinking of the logical constants 0 and 1 as integers mod 2. The logical operation of conjunction is realized as the arithmetic operation of multiplication ''xy'', and logical exclusive-or as arithmetic addition mod 2, (written here as ''x''⊕''y'' to avoid confusion with the common use of + as a synonym for inclusive-or ∨). Logical complement ¬''x'' is then derived from 1 and ⊕ as ''x''⊕1. Since ∧ and ¬ form a sufficient basis for the whole of boolean algebra, meaning that all other logical operations are obtainable as composites of these basic operations, it follows that the polynomials of ordinary algebra can represent all boolean operations, allowing boolean reasoning to be performed reliably by appealing to the familiar laws of high school algebra without the distraction of the differences from high school algebra that arise with disjunction in place of addition mod 2. An example application is the representation of the boolean 2-out-of-3 threshold or median operation as the Zhegalkin polynomial ''xy''⊕''yz''⊕''zx'', which is 1 when at least two of the variables are 1 and 0 otherwise. 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「Zhegalkin polynomial」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.