This article examines the computational content of the classical Gentzen sequent calculus. There are a number of well-known methods that extract computational content from first-order logic but applying these to the sequent calculus involves first translating proofs into other formalisms, Hilbert calculi or Natural Deduction for example. A direct approach which mirrors the symmetry inherent in sequent calculus has potential merits in relation to proof-theoretic considerations such as the (non-)confluence of cut elimination, the problem of cut introduction, proof compression and proof equivalence. Motivated by such applications, we provide a representation of sequent calculus proofs as higher order recursion schemes. Our approach associates to an LK proof π of $\Rightarrow \exists vF$, where F is quantifier free, an acyclic higher order recursion scheme $\mathcal{H}$ with a finite language yielding a Herbrand disjunction for $\exists vF$. More generally, we show that the language of $\mathcal{H}$ contains all Herbrand disjunctions computable from π via a broad range of cut elimination strategies.

本文研究了经典Gentzen后续演算的计算内容。有许多众所周知的方法可以从一阶逻辑中提取计算内容，但将其应用于后续演算时，首先需要将证明转换为其他形式主义，例如希尔伯特计算或自然演绎。反映后续演算中固有的对称性的直接方法相对于证明理论上的考虑具有潜在的优点，例如，消除消除（非）融合，引入切口，证明压缩和证明等效等问题。受此类应用程序的激励，我们将后续演算证明表示为高阶递归方案。我们的方法与LK证明π有关$\Rightarrow \exists vF$，其中F是无量词的非循环高阶递归方案$\mathcal{H}$ 用有限的语言产生Herbrand析取 $\exists vF$。更笼统地说，我们证明$\mathcal{H}$包含可通过广泛的切消策略从π计算得到的所有Herbrand析取。