In this paper, we introduce a general family of sequent-style calculi over the modal language and its fragments to capture the essence of all constructively acceptable systems. Calling these calculi \emph{constructive}, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules, i.e., there is a polynomial time algorithm that reads a proof of the premise of a Visser's rule and provides a proof for its conclusion. As a positive application, we show the feasible admissibility of Visser's rules in several sequent calculi for intuitionistic modal logics, including $\mathsf{CK}$, $\mathsf{IK}$ and their extensions by the modal axioms $T$, $B$, $4$, $5$, the modal axioms of bounded width and depth and the propositional lax logic. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, then it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than $\mathsf{IPC}$ has a constructive sequent calculus.
翻译:在本文中,我们引入了一个对模式语言及其片段进行序列式计算的整体,以捕捉所有建设性可接受体系的精髓。调用这些 calculli \ emph{ compective},我们展示了任何足够强大的建设性序列计算,满足了温和的技术条件,可以令人信服地承认Visser的所有规则,即:存在着一种多元时间算法,它可以证明Visser规则的前提,并为它的结论提供了证据。作为一个积极的应用,我们展示了维瑟规则在几部直观模式逻辑的序列计算中的可接受性,包括$\ mathsf{C}、$\ mathsf{I} 。 任何强大的建设性序列计算,满足了温和的技术条件,即$, $B$, $, $, $, $, $, $, $, $, $, $, $, $, $, 5, mod, 美元, 维瑟克规则的缩逻辑。 在消极的方面,我们显示,如果一个最强的直率的逻辑, 一个不具有温和的中间的逻辑。