标题： 可判定性作为并集分割 显示英文标题

标题： Decidability of Being a Union-splitting

摘要： 许多逻辑性质对于正常的模态逻辑来说是不可判定的，仅有少数例外，例如一致性以及与$\mathsf{K}$相同。 本文表明，在$\mathsf{NExt}\mathsf{K}$，即正常模态逻辑的格中，成为并集分裂的性质是可判定的，从而回答了开放问题 [WZ07, 问题 2]。 这是通过用有限模态代数来提供并集分裂的语义特征来实现的。 此外，通过澄清与并集分裂的联系，我们证明在$\mathsf{NExt}\mathsf{K}$中，具有可判定的公理化问题以及成为（不可）判定公式也是可判定的。 后者回答了 [CZ97, 问题 17.3] 对于$\mathsf{NExt}\mathsf{K}$的情况。 显示英文摘要

摘要： Many logical properties are known to be undecidable for normal modal logics, with few exceptions such as consistency and coincidence with $\mathsf{K}$. This paper shows that the property of being a union-splitting in $\mathsf{NExt}\mathsf{K}$, the lattice of normal modal logics, is decidable, thus answering the open problem [WZ07, Problem 2]. This is done by providing a semantic characterization of union-splittings in terms of finite modal algebras. Moreover, by clarifying the connection to union-splittings, we show that in $\mathsf{NExt}\mathsf{K}$, having a decidable axiomatization problem and being a (un)decidable formula are also decidable. The latter answers [CZ97, Problem 17.3] for $\mathsf{NExt}\mathsf{K}$.