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Translations from natural deduction to sequent calculus

机译:从自然演绎到后续演算的转换

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摘要

Gentzen's "Untersuchungen" [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut-free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut-free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination.
机译:Gentzen的“ Untersuchungen” [1]将自然演绎转换为后续演算,其特性是正常导数可以转换为带削减的导数。 Prawitz在[8]中给出了翻译,而是产生了无割的派生词。结果表明,通过以相切消除的方式编写所有消除规则,具有任意结果,可以实现正态导数和无割导数之间的同构转换。标准的消除规则不允许使用完全正常的形式,这解释了根岑翻译的缩影。同样地,它显示了Prawitz的翻译中包含一个切分消除的隐式过程。

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