Deciding the Existence of Interpolants and Definitions in First-Order Modal Logic
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915554084782080 |
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| author | Kurucz, Agi Wolter, Frank Zakharyaschev, Michael |
| author_facet | Kurucz, Agi Wolter, Frank Zakharyaschev, Michael |
| contents | None of the first-order modal logics between $\mathsf{K}$ and $\mathsf{S5}$ under the constant domain semantics enjoys Craig interpolation or projective Beth definability, even in the language restricted to a single individual variable. It follows that the existence of a Craig interpolant for a given implication or of an explicit definition for a given predicate cannot be directly reduced to validity as in classical first-order and many other logics. Our concern here is the decidability and computational complexity of the interpolant and definition existence problems. We first consider two decidable fragments of first-order modal logic $\mathsf{S5}$: the one-variable fragment $\mathsf{Q^1S5}$ and its extension $\mathsf{S5}_{\mathcal{ALC}^u}$ that combines $\mathsf{S5}$ and the description logic$\mathcal{ALC}$ with the universal role. We prove that interpolant and definition existence in $\mathsf{Q^1S5}$ and $\mathsf{S5}_{\mathcal{ALC}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $\mathsf{FO^2}$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment $\mathsf{Q^1K}$ of first-order modal logic $\mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_04598 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Deciding the Existence of Interpolants and Definitions in First-Order Modal Logic Kurucz, Agi Wolter, Frank Zakharyaschev, Michael Logic in Computer Science None of the first-order modal logics between $\mathsf{K}$ and $\mathsf{S5}$ under the constant domain semantics enjoys Craig interpolation or projective Beth definability, even in the language restricted to a single individual variable. It follows that the existence of a Craig interpolant for a given implication or of an explicit definition for a given predicate cannot be directly reduced to validity as in classical first-order and many other logics. Our concern here is the decidability and computational complexity of the interpolant and definition existence problems. We first consider two decidable fragments of first-order modal logic $\mathsf{S5}$: the one-variable fragment $\mathsf{Q^1S5}$ and its extension $\mathsf{S5}_{\mathcal{ALC}^u}$ that combines $\mathsf{S5}$ and the description logic$\mathcal{ALC}$ with the universal role. We prove that interpolant and definition existence in $\mathsf{Q^1S5}$ and $\mathsf{S5}_{\mathcal{ALC}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $\mathsf{FO^2}$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment $\mathsf{Q^1K}$ of first-order modal logic $\mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable. |
| title | Deciding the Existence of Interpolants and Definitions in First-Order Modal Logic |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2303.04598 |