Decidability of Being a Union-splitting

Tenyo Takahashi

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arXiv · 2510.14520

Decidability of Being a Union-splitting

Tenyo Takahashi

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Abstract

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 $\mathsf{NExt}\mathsf{K}$.

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BibTeX RIS

Tenyo Takahashi. 2025-10-16. Decidability of Being a Union-splitting. https://arxiv.org/abs/2510.14520

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