Decidability of Being a Union-splitting
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912652123439104 |
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| author | Takahashi, Tenyo |
| author_facet | Takahashi, Tenyo |
| contents | 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 $\mathsf{NExt}\mathsf{K}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decidability of Being a Union-splitting Takahashi, Tenyo Logic 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 $\mathsf{NExt}\mathsf{K}$. |
| title | Decidability of Being a Union-splitting |
| topic | Logic |
| url | https://arxiv.org/abs/2510.14520 |