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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 Quantifier rank ： ウィキペディア英語版 Quantifier rank In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. Notice that the quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two logically equivalent formulae can have different quantifier ranks, when they express the same thing in different ways. ==Definition== Quantifier Rank of a Formula in First-order language (FO) Let φ be a FO formula. The quantifier rank of φ, written qr(φ), is defined as * $qr(\backslash varphi)\; =\; 0$, if φ is atomic. * $qr(\backslash varphi\_1\; \backslash land\; \backslash varphi\_2)\; =\; qr(\backslash varphi\_1\; \backslash lor\; \backslash varphi\_2)\; =\; max(qr(\backslash varphi\_1),\; qr(\backslash varphi\_2))$. * $qr(\backslash lnot\; \backslash varphi)\; =\; qr(\backslash varphi)$. * $qr(\backslash exists\_x\; \backslash varphi)\; =\; qr(\backslash varphi)\; +\; 1$. Remarks * We write FO() for the set of all first-order formulas φ with $qr(\backslash varphi)\; \backslash le\; n$. * Relational FO() (without function symbols) is always of finite size, i.e. contains a finite number of formulas * Notice that in Prenex normal form the Quantifier Rank of φ is exactly the number of quantifiers appearing in φ. Quantifier Rank of a higher order Formula * For Fixpoint logic, with a least fix point operator LFP: : qr(()y) = 1 + qr( φ) ... 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「Quantifier rank」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.