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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 Herbrand's theorem (proof theory) ： ウィキペディア英語版 Herbrand's theorem Herbrand's theorem is a fundamental result of mathematical logic obtained by Jacques Herbrand (1930).〔J. Herbrand: Recherches sur la theorie de la demonstration. Travaux de la Societe des Sciences et des Lettres de Varsovie, Class III, Sciences Mathematiques et Physiques, 33, 1930.〕 It essentially allows a certain kind of reduction of first-order logic to propositional logic. Although Herbrand originally proved his theorem for arbitrary formulas of first-order logic,〔Samuel R. Buss: "Handbook of Proof Theory". Chapter 1, "An Introduction to Proof Theory". Elsevier, 1998.〕 the simpler version shown here, restricted to formulas in prenex form containing only existential quantifiers became more popular. Let :$(\backslash exists\; y\_1,\backslash ldots,y\_n)F(y\_1,\backslash ldots,y\_n)$ be a formula of first-order logic with :$F(y\_1,\backslash ldots,y\_n)$ quantifier-free. Then :$(\backslash exists\; y\_1,\backslash ldots,y\_n)F(y\_1,\backslash ldots,y\_n)$ is valid if and only if there exists a finite sequence of terms $t\_$, possibly in an expansion of the language, with :$1\; \backslash le\; i\; \backslash le\; k$ and $1\; \backslash le\; j\; \backslash le\; n$, such that :$F(t\_,\backslash ldots,t\_)\; \backslash vee\; \backslash ldots\; \backslash vee\; F(t\_,\backslash ldots,t\_)$ is valid. If it is valid, :$F(t\_,\backslash ldots,t\_)\; \backslash vee\; \backslash ldots\; \backslash vee\; F(t\_,\backslash ldots,t\_)$ is called a ''Herbrand disjunction'' for :$(\backslash exists\; y\_1,\backslash ldots,y\_n)F(y\_1,\backslash ldots,y\_n).$ Informally: a formula $A$ in prenex form containing existential quantifiers only is provable (valid) in first-order logic if and only if a disjunction composed of substitution instances of the quantifier-free subformula of $A$ is a tautology (propositionally derivable). The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by Herbrandization. Conversion to prenex form can be avoided, if ''structural'' Herbrandization is performed. Herbrandization can be avoided by imposing additional restrictions on the variable dependencies allowed in the Herbrand disjunction. ==Proof Sketch== A proof of the non-trivial direction of the theorem can be constructed according to the following steps: # If the formula $(\backslash exists\; y\_1,\backslash ldots,y\_n)F(y\_1,\backslash ldots,y\_n)$ is valid, then by completeness of cut-free sequent calculus, which follows from Gentzen's cut-elimination theorem, there is a cut-free proof of $\backslash vdash\; (\backslash exists\; y\_1,\backslash ldots,y\_n)F(y\_1,\backslash ldots,y\_n)$. # Starting from above downwards, remove the inferences that introduce existential quantifiers. # Remove contraction-inferences on previously existentially quantified formulas, since the formulas (now with terms substituted for previously quantified variables) might not be identical anymore after the removal of the quantifier inferences. # The removal of contractions accumulates all the relevant substitution instances of $F(y\_1,\backslash ldots,y\_n)$ in the right side of the sequent, thus resulting in a proof of $\backslash vdash\; F(t\_,\backslash ldots,t\_),\; \backslash ldots,\; F(t\_,\backslash ldots,t\_)$, from which the Herbrand disjunction can be obtained. However, sequent calculus and cut-elimination were not known at the time of Herbrand's theorem, and Herbrand had to prove his theorem in a more complicated way. 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「Herbrand's 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.