V2EX cut elimination

Cut Elimination

释义 Definition

(逻辑/证明论)割消去:指在序列演算等形式系统中,通过一系列变换把证明里使用的 cut(割)规则去掉,同时仍能证明同一个结论。直观上,它把“先引入一个中间命题再消去”的推理步骤改写为更“直接”的推导,常用于说明系统的一致性、可判定性、子公式性质等。该术语在更广义上也可指“消除中间引理式步骤”的证明正规化思想。

发音 Pronunciation (IPA)

/kt lmen/

例句 Examples

Cut elimination can simplify a proof by removing unnecessary intermediate steps.
割消去可以通过移除不必要的中间步骤来简化一个证明。

In sequent calculus, the cut elimination theorem shows that any proof using cut can be transformed into a cut-free proof, often yielding the subformula property.
在序列演算中,割消去定理表明任何使用割规则的证明都能转换为不含割的证明,并且往往由此得到子公式性质。

词源 Etymology

cut 在证明论里是“割规则”的名称(把一个中间公式当作“切口/接口”,把两段推导接起来);elimination 来自拉丁语 eliminare,有“移出、去除”的意思。术语整体主要与德国逻辑学家 Gentzen(根岑)在 1930 年代提出的证明论工作相关,他用割消去来刻画证明的“正规化/纯化”。

相关词 Related Words

文学与名著中的用例 Literary Works

  • Gerhard Gentzen, Investigations into Logical Deduction(19341935):提出并系统讨论割消去思想(常与“割消去定理”相关联)。
  • Gaisi Takeuti, Proof Theory:以“cut-elimination”作为证明论核心技术之一反复使用与推广。
  • S. Troelstra & H. Schwichtenberg, Basic Proof Theory:在序列演算与自然演绎框架中讲解割消去与正规化的关系。
  • Jean-Yves Girard, Proofs and Types:在类型论与证明论对应(CurryHoward)背景下涉及相关正规化/消去结果。
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